Research ArticleComputational Modelling of Couette Flow ofNanofluids with Viscous Heating and Convective Cooling

Oluwole Daniel Makinde1 Ahmada Omar2 and M Samuel Tshehla1

1Faculty of Military Science Stellenbosch University Private Bag X2 Saldanha 7395 South Africa2Mathematics and Computational Science and Engineering Nelson Mandela African Institution ofScience and Technology (NM-AIST) Arusha Tanzania

Correspondence should be addressed to Ahmada Omar alianm-aistactz

Received 28 April 2014 Accepted 18 November 2014 Published 14 December 2014

Academic Editor Zhijie Xu

Copyright copy 2014 Oluwole Daniel Makinde et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The combined effect of viscous heating and convective cooling on Couette flow and heat transfer characteristics of water basenanofluids containing Copper Oxide (CuO) and Alumina (Al

2O3) as nanoparticles is investigated It is assumed that the nanofluid

flows in a channel between two parallel plates with the channelrsquos upper plate accelerating and exchange heat with the ambientsurrounding following the Newtonrsquos law of cooling while the lower plate is stationary and maintained at a constant temperatureUsing appropriate similarity transformation the governingNavier-Stokes and the energy equations are reduced to a set of nonlinearordinary differential equations These equations are solved analytically by regular perturbation method with series improvementtechnique andnumerically by an efficient Runge-Kutta-Fehlberg integration technique coupledwith shootingmethodThe effects ofthe governing parameters on the dimensionless velocity temperature skin friction pressure drop andNusselt number are presentedgraphically and discussed quantitatively

1 Introduction

Studies related to laminar flow and heat transfer of a viscousfluid in the space between two parallel plates one of whichis moving relative to the other have received the attentionof several researchers due to their numerous industrial andengineering applications This type of flow is named inhonour of Maurice Marie Alfred Couette a professor ofphysics at the French University of Angers in the late 19thcentury [1] Couette flow has been used to estimate thedrag force in many wall driven applications such as lubri-cation engineering power generators and pumps polymertechnology petroleum industry and purification of crudeoil Literature survey indicates that interest in the Couetteflows has grown during the past decades Jana and Datta [2]examined the effects of Coriolis force on the Couette flowand heat transfer between two parallel plates in a rotatingsystem Singh [3] studied unsteady free convection flow ofan incompressible viscous fluid between two vertical parallelplates in which one is fixed and the other is impulsively

started in its own plane Kearsley [4] investigated the problemof steady state Couette flow with viscous heating Jha [5]numerically examined the effects ofmagnetic field onCouetteflow between two vertical parallel plates The combinedeffects of variable viscosity and thermal conductivity ongeneralized Couette flow and heat transfer in the presenceof transversely imposed magnetic field have been studiednumerically by Makinde and Onyejekwe [6] Seth et al [7]presented a closed form solution for hydromagnetic unsteadyCouette flow of a viscous incompressible electrically con-ducting fluid between two parallel porous plates Deka andBhattacharya [8] obtained an exact solution of unsteady freeconvective Couette flow of a viscous incompressible heat gen-eratingabsorbing fluid confined between two vertical platesin a porous medium Meanwhile the enhancement of heattransfer in a Couette flow of fluid subjected to a temperaturegradient is an important issue that is expected to improvethe efficient operation of several engineering and tribologicaldevices Other relevant applications can be found in enginecooling solar water heating cooling of electronics cooling

Hindawi Publishing CorporationInternational Journal of Computational MathematicsVolume 2014 Article ID 631749 12 pageshttpdxdoiorg1011552014631749

2 International Journal of Computational Mathematics

of transformer oil improving diesel generator efficiencycooling of heat exchanging devices improving heat trans-fer efficiency of chillers domestic refrigerator-freezers andcooling in machining and in nuclear reactor The commonheat transfer fluids such as water ethylene glycol and engineoil have limited heat transfer capabilities owing to theirlow thermal conductivity whereas metals have much higherthermal conductivities than these fluids With the recentimprovements in nanotechnology the production of particleswith sizes on the order of nanometers can be achievedConsequently the idea of dispersing these nanoparticles ina base liquid for improving thermal conductivity has beenproposed [9] Such suspension of nanoparticles in a base fluidis called a nanofluid Due to their small size nanoparticlesfluidize easily inside the base fluid and as a consequenceclogging of channels and erosion in channel walls are nolonger a problem It is even possible to use nanofluids inmicrochannels [10] Niu et al [11] theoretically studied theslip-flow and heat transfer of a non-Newtonian nanofluidin a microtube by means of theoretical method In theirresearch the power-law rheology was adopted to describethe non-Newtonian characteristics of the flow in which thefluid consistency coefficient and the flow behaviour indexdepend on the nanoparticle volume fraction Motsumi andMakinde [12] reported a numerical solution for the effects ofthermal radiation and viscous dissipation on boundary layerflow of nanofluids over a permeable moving flat plate Choiet al [13] studied the Couette flow of nanofluids composed ofnegatively charged nanoparticles dispersed in aqueous NaClsolutions They found that the velocity profile of nanofluidscontaining charged nanoparticles deviates significantly fromthe classical linear velocity profile of Couette flow

In the studies mentioned above the combined effectsof viscous heating and convective cooling on Couette flowof nanofluids have not been discussed while such flows arevery important in lubrication technology and tribologicalproblems Therefore the objective of the present paper is toanalyze the effects of viscous heating and convective coolingon the Couette flow of water base nanofluids containingCopper Oxide (CuO) and Alumina (Al

2O3) as nanoparticles

In Sections 2ndash4 the model nonlinear governing equationstogether with the analytical and numerical solution areobtained Pertinent results are presented graphically anddiscussed quantitatively in Section 5while the conclusions aredrawn in Section 6

2 Problem Formulation

We consider a two-dimensional steady Couette flow of vis-cous incompressible water base nanofluids containing Cop-per Oxide (CuO) and Alumina (Al

2O3) as nanoparticles

in which an accelerating upper plate drags adjacent fluidalong with it and thereby imparts a motion to the rest ofthe fluid The lower plate is fixed and kept at a constanttemperature119879

0while the upper accelerating plate is subjected

to a convective heat exchange with the ambient surroundingfollowing Newtonrsquos law of cooling We choose a Cartesiancoordinates system in such away that the119909-axis is taken alongthe channel and the119910-axis is normal to it as shown in Figure 1

u(x y) T(x y)

Nanofluid

v(x a) = 0 minuskf120597T

120597y(x a) = h(T minus Ta) u(x a) = xE

y = 0

y = a

v(x 0) = 0 T(x 0) = T0 u(x 0) = 0

Figure 1 Schematic diagram of the physical system

The governing equations which are those of conservationof mass momentum and energy are [1 4 13]

120597119906

120597119909+120597V120597119910= 0 (1)

120588nf (119906120597119906

120597119909+ V120597119906

120597119910) = minus

120597119875

120597119909+ 120583nf (

1205972119906

1205971199102+1205972119906

1205971199092) (2)

120588nf (119906120597V120597119909+ V120597V120597119910) = minus

120597119875

120597119910+ 120583nf (

1205972V1205971199102

+1205972V1205971199092

) (3)

119906120597119879

120597119909+ V120597119879

120597119910=

119896nf

(120588119888119901)nf

(1205972119879

1205971199092+1205972119879

1205971199102)

+2120583nf

(120588119888119901)nf

[(120597119906

120597119909)

2

+ (120597V120597119910)

2

]

+120583nf

(120588119888119901)nf

(120597V120597119909+120597119906

120597119910)

2

(4)

where (119906 V) are the velocity components of the nanofluidin the (119909 119910) directions respectively 119864 is the upper plateacceleration parameter 119886 is the channel width 119875 is thepressure 119879 is the nanofluid temperature 120583nf is the Maxwell-Garnetts [14] approximation for effective dynamic viscosity ofthe nanofluid 119896nf is the effective thermal conductivity of thenanofluid given byBrinkman [15]120588nf is the nanofluid densityand (120588119888

119901)nf is the heat capacitance of the nanofluid which are

given by [16 17]

120583nf =120583119891

(1 minus 120593)25 120588nf = (1 minus 120593) 120588119891 + 120593120588119904

120572nf =119896nf

(120588119888119901)nf

119896nf119896119891

=

(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(120588119888119901)nf= (1 minus 120593) (120588119888

119901)119891+ 120593 (120588119888

119901)119904

(5)

In (5) 120593 is the nanoparticles solid volume fraction 120588119891is

the reference density of the fluid fraction 120588119904is the reference

International Journal of Computational Mathematics 3

density of the solid fraction 120583119891is the viscosity of the fluid

fraction 119896119891is the thermal conductivity of the fluid fraction 119888

119901

is the specific heat at constant pressure and 119896119904is the thermal

conductivity of the solid volume fraction The boundaryconditions at the channel walls may be written as

119906 (119909 0) = 0 119879 = 1198790 V (119909 0) = 0

119906 (119909 119886) = 119864119909 V (119909 119886) = 0

minus119896nf120597119879

120597119910119879 (119909 119886) = ℎ [119879 (119909 119886) minus 119879

119886]

(6)

where119879119886is the ambient surrounding temperature and ℎ is the

coefficient of heat transfer Introducing the stream function120595and vorticityΩ into the governing equations (1)ndash(4) we havethe following

119906 =120597120595

120597119910 V = minus

120597120595

120597119909

Ω =120597119906

120597119910minus120597V120597119909=1205972120595

1205971199102+1205972120595

1205971199092

(7)

After eliminating the pressure 119875 from (2) and (3) we obtain

120588nf (120597120595

120597119910

120597Ω

120597119909minus120597120595

120597119909

120597Ω

120597119910) = 120583nf (

1205972Ω

1205971199102+1205972Ω

1205971199092) (8)

(120597120595

120597119910

120597119879

120597119909minus120597120595

120597119909

120597119879

120597119910)

=119896nf

(120588119888119901)nf

(1205972119879

1205971199102+1205972119879

1205971199092)

+4120583nf

(120588119888119901)nf

(1205972120595

120597119909120597119910)

2

+120583nf

(120588119888119901)nf

(minus1205972120595

1205971199092+1205972120595

1205971199102)

2

(9)

The following dimensionless variables and parameters areintroduced into (8) and (9) together with their correspondingboundary conditions

120578 =119910

119886 120595 =

120595

1198641198862 Ω =

Ω

119864

Φ =119879 minus 1198790

119879119886minus 1198790

119877 =1198641198862

120592119891

119883 =119909

119886

120592119891=

120583119891

120588119891

Pr =120583119891119888119901

119896119891

1198981= (1 minus 120593 +

120593120588119904

120588119891

) (1 minus 120593)25

120582 =

120583119891

120573119886

119875 =119875

119864120583119891

1198982= 1198984(1 minus 120593 +

120593 (120588119888119901)119904

(120588119888119901)119891

)

Bi = ℎ119886119896119891

Ec = 11986421198862

119888119901(119879119886minus 1198790)

1198983=

1198984

(1 minus 120593)25 119898

4=

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(10)

and we obtain

1198981119877(

120597120595

120597120578

120597Ω

120597119883minus120597120595

120597119883

120597Ω

120597120578) = (

1205972Ω

1205971205782+1205972Ω

1205971198832)

1198982119877Pr(

120597120595

120597120578

120597Φ

120597119883minus120597120595

120597119883

120597Φ

120597120578)

= (1205972Φ

1205971205782+1205972Φ

1205971198832) + 4119898

3Ec Pr(

1205972120595

120597119883120597120578)

2

+ 1198983Ec Pr(minus

1205972120595

1205971198832+1205972120595

1205971205782)

2

(11)

with

120597120595

120597120578(119883 0) = 0 Φ (119883 0) = 0

120597120595

120597119883(119883 0) = 0

120597120595

120597120578(119883 1) = 119883

120597120595

120597119883(119883 1) = 0

120597Φ

120597120578(119883 1) = minus119898

4Bi [Φ (119883 1) minus 1]

(12)

where 119877 is the Reynolds number Ec is the Eckert numberBi is the Biot number Pr is the base fluid Prandtl numberand 119898

1 1198982 1198983 and 119898

4can be easily determined from

the thermophysical properties of the base fluid and thenanoparticles We seek a similarity form of solution of theform

120595 (119883 120578) = 119883119865 (120578) 119882 (119883 120578) = 119883119889119865

119889120578

119881 (120578) = minus119865 (120578) Φ (119883 120578) = 119867 (120578) + 1198832120579 (120578)

(13)

Substituting (13) into (11)-(12) we obtain

1198894119865

1198891205784= 1198981119877(

119889119865

119889120578

1198892119865

1198891205782minus 119865

1198893119865

1198891205783) (14)

1198892120579

1198891205782= minus1198983Ec Pr(119889

2119865

1198891205782)

2

+ 1198982119877Pr(2120579119889119865

119889120578minus 119865

119889120579

119889120578)

(15)

1198892119867

1198891205782= minus2120579 minus 4119898

3Ec Pr(119889119865

119889120578)

2

minus 1198982119877Pr119865119889119867

119889120578 (16)

119889119865

119889120578(0) = 0 120579 (0) = 0 119867 (0) = 0

119865 (0) = 0

(17)

4 International Journal of Computational Mathematics

119889119865

119889120578(1) = 1 119865 (1) = 0

119889120579

119889120578(1) = minus119898

4Bi 120579 (1)

119889119867

119889120578(1) = minus119898

4Bi [119867 (1) minus 1]

(18)

The dimensionless fluid axial pressure gradient is given as

minus120597119875

120597119883= 119883119860 (19)

where

(1 minus 120593)25

119860 =1198893119865

1198891205783minus 1198981119877[(

119889119865

119889120578)

2

minus 1198651198892119865

1198891205782] (20)

Other quantities of practical interest in this study are the localskin friction coefficient 119862

119891and the local Nusselt number Nu

which are defined as

119862119891=120591119908

120583119891119864 Nu =

119886119902119908

119896119891119879119908

(21)

where 120591119908is the skin friction and 119902

119908is the heat flux at the

channel upper accelerating wall which are given by

120591119908= 120583nf

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

119902119908= minus119896nf

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

(22)

Using (10) and (13) we substitute (22) into (21) and obtain

119862119891=

119883

(1 minus 120593)25

1198892119865

1198891205782(1)

Nu = minus119896nf119896119891

[119889119867

119889120578(1) + 119883

2 119889120579

119889120578(1)]

(23)

In the following section the boundary value problems in(14)ndash(18) were solved analytically using regular perturba-tion method and numerically by the Runge-Kutta-Fehlbergmethod with shooting technique [18] The results are utilisedto compute the fluid pressure gradient local skin friction andlocal Nusselt number as highlighted in (20) and (23)

3 Perturbation Method

The governing model equations (14)ndash(18) are nonlinear andthis makes its exact solution very intractable Howeverapproximate solution can be easily obtained by forminga power series expansion in the parameter 119877 It is worthnoting that an exact solution may exist for this problemunder certain assumptions The existence of exact solutionprovides remarkable and more accurate results Assume aseries solution of the form

119865 (120578) =

infin

sum

119894=0

119865119894119877119894 120579 (120578) =

infin

sum

119894=0

120579119894119877119894 (24)

Substituting the solution series in (24) into (14)ndash(18) andcollecting the coefficients of like powers of 119877 we obtain thefollowing

Zeroth Order Consider

11988941198650

1198891205784= 0

11988921205790

1198891205782= minus1198983Ec Pr(

11988921198650

1198891205782)

2

11988921198670

1198891205782= minus2120579

0minus 41198983Ec Pr(

1198891198650

119889120578)

2

(25)

with

1198891198650

119889120578(0) = 0 120579

0(0) = 0 119867

0(0) = 0

1198650(0) = 0

1198891198650

119889120578(1) = 1 119865

0(1) = 0

1198891205790

119889120578(1) = minus119898

4Bi 1205790(1)

1198891198670

119889120578(1) = minus119898

4Bi [1198670(1) minus 1]

(26)

Higher Order (119899 ge 1) Consider

1198894119865119899

1198891205784= 1198981119877

119899minus1

sum

119894=0

(119889119865119894

119889120578

1198892119865119899minus119894minus1

1198891205782minus 119865119894

1198893119865119899minus119894minus1

1198891205783)

1198892120579119899

1198891205782= minus 119898

3Ec Pr

119899

sum

119894=0

(1198892119865119894

1198891205782

1198892119865119899minus119894

1198891205782)

+ 1198982119877Pr119899minus1

sum

119894=0

(2120579119894

119889119865119899minus119894minus1

119889120578minus 119865119894

119889120579119899minus119894minus1

119889120578)

1198892119867119899

1198891205782= minus 2120579

119899minus 41198983Ec Pr

119899

sum

119894=0

(119889119865119894

119889120578

119889119865119899minus119894

119889120578)

minus 1198982119877Pr119899minus1

sum

119894=0

(119865119894

119889119867119899minus119894minus1

119889120578)

(27)

with

119889119865119899

119889120578(0) = 0 120579

119899(0) = 0 119867

119899(0) = 0

119865119899(0) = 0

119889119865119899

119889120578(1) = 0 119865

119899(1) = 0

119889120579119899

119889120578(1) = minus119898

4Bi 120579119899(1)

119889119867119899

119889120578(1) = minus119898

4Bi119867119899(1)

(28)

International Journal of Computational Mathematics 5

The equations are solved iteratively and the series solutionsfor the velocity and temperature fields are given as

119865 (120578) = 1205782(120578 minus 1) +

11989811198771205782

210(31205783minus 1205782+ 2120578 + 5) (120578 minus 1)

2

minus1198982

111987721205782

5821200(120578 minus 1)

2

times (5041205787minus 840120578

6minus 644120578

5minus 1603120578

4+ 3774120578

3

minus 28611205782minus 256120578 + 2349) + 119874 (119877

3)

120579 (120578) = minus1198983Ec Pr 120578119866

1

1 + 1198984Bi

minus1198983Ec Pr 120578119877119866

2

420 (1 + 1198984Bi)2

+ 119874 (1198772)

119867 (120578) = minus1198663120578

3 (1 + 1198984Bi)2

+ 119874 (119877)

(29)

where the expressions for 119866119894 119894 = 1 2 3 are given in the

appendix Using a computer symbolic algebra package(MAPLE) several terms of the above solution series in (29)are obtained From (29) together with (20) and (23) weobtained the series solutions for the skin friction Nusseltnumber and axial pressure gradient as

119862119891=

119883

(1 minus 120601)25

times (4 +31198981119877

35minus471198982

11198772

323400minus2511198983

11198773

39239200+ 119874 (119877

4))

119860 =1

(1 minus 120601)25

times (6 minus81198981119877

35+2477119898

2

11198772

485100minus43793119898

3

11198773

529729200

+ 119874 (1198774))

Nu = 1

31198984(1 + 119898

4Bi)2

times (minus31198982

4Bi2 + 2119898

3Ec Pr1198982

4Bi2 minus 3119898

4Bi

+ 81198983119864Pr119898

4Bi)

minusPr119877119873

1260 (1 + 1198984Bi)3119898

4

+ 1198832[31198983Ec Pr119898

4Bi

(1 + 1198984Bi)1198984

+1198983Pr Ec119877119872

420 (1 + 1198984Bi)2119898

4

]

+ 119874 (1198772)

(30)

The expressions for119873 and119872 are given in the appendixThesepower series solutions are valid for very small parameter

values of 119877 however using Pade approximation technique[19] that is based on the series summation and improvementmethod the usability of the solution series is extendedbeyond small parameter values of 119877

4 Numerical Procedure

Here we employed Runge-Kutta-Fehlberg method withshooting technique [18] to numerically solve the couplednonlinear ordinary differential equations (14)ndash(16) subjectto the boundary conditions (17)-(18) for different set ofparameter values The procedure involves transforming thenonlinear boundary value problem (BVP) into an initial valueproblem (IVP) as follows let

1199111= 119865 119911

2=119889119865

119889120578 119911

3=1198892119865

1198891205782 119911

4=1198893119865

1198891205783

1199115= 120579 119911

6=119889120579

119889120578 119911

7= 119867 119911

8=119889119867

119889120578

(31)

Equation (31) is substituted into (14)ndash(18) and a system of firstorder differential equations is obtained

1198891199111

119889120578= 1199112

1198891199112

119889120578= 1199113

1198891199113

119889120578= 1199114

1198891199114

119889120578= 1198981119877 (11991121199113minus 11991111199114)

1198891199115

119889120578= 1199116

1198891199116

119889120578= minus1198983Ec Pr 1199112

3+ 1198982119877Pr (2119911

51199112minus 11991111199116)

1198891199117

119889120578= 1199118

1198891199118

119889120578= minus2119911

5minus 41198983Ec Pr 1199112

2minus 1198982119877Pr (119911

11199118)

(32)

subject to the initial conditions

1199111(0) = 0 119911

2(0) = 0 119911

3(0) = 119904

1

1199114(0) = 119904

2 119911

5(0) = 0 119911

6(0) = 119904

3

1199117(0) = 0 119911

8(0) = 119904

4

(33)

To determine the unspecified initial conditions 1199041 1199042 1199043 and

1199044in (33) the initial guesses are supplied and resulting IVP

is solved using Runge-Kutta scheme The process is repeateduntil the solution at 120578 = 1 shoots to the boundary conditionsat that point The best way to achieve this is to supply theinitial condition in to the Runge-Kutta scheme as unknownand solve the resulting nonlinear algebraic system of equa-tions iteratively using Newton-Raphson It is important tonote that the resulting nonlinear system of algebraic equationmay not have unique solution therefore a carefully choseninitial guess which is close to the desired solution must besupplied The scheme has quadratic convergence with stepsize taken as Δ120578 = 0001 and tolerance set to 10minus7

6 International Journal of Computational Mathematics

Table 1 Thermophysical properties of the fluid phase (water) andnanoparticles [16 17]

Physicalproperties

Fluid phase(water)

CuO Al2O3

119888119901(JkgK) 4179 6500 765

120588 (kgm3) 9971 5356 3970

119896 (WmK) 0613 20 40

Table 2 Computation showing the skin friction for 119877 = 1119883 = 1

120593

119862119891

CuO-water(series)

119862119891

CuO-water(numerical)

119862119891

Al2O3-water(series)

119862119891

Al2O3-water(numerical)

000 4085563 40855627 4085563 40855627001 4186948 41869479 4189890 41898897005 4630877 46308766 4645589 46455887010 5287037 52870375 5316469 53164695015 6084673 60846727 6128833 61288327020 7065422 70654222 7124319 71243186025 8286944 82869436 8360585 83605848030 9830736 98307360 9919130 99191299

5 Results and Discussion

The Couette flow and heat transfer characteristics of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

with viscous heating and convective cooling have beeninvestigated The nonlinear similarity ordinary differentialequations governing the boundary value problemwere solvedboth analytically using the perturbation method coupledwith series improvement technique and numerically usingRunge-Kutta-Fehlberg integration technique coupled withshooting scheme Thermophysical properties of base fluidand nanoparticles are presented in Table 1 For pure water thePrandtl number is taken as Pr = 62 [12 16 17] and momen-tum diffusivity is dominant within the fluid in comparisonto pure conduction The nanoparticle volume fraction in thebase fluid is taken as 120593 = 0 to 03 (ie ranging from 0 to 30percent) while the case of120593 = 0 corresponds to the absence ofnanoparticle in the based fluid (water) Numerical solutionsare displayed in Tables 2 and 3 together with Figures 2ndash17In Tables 2 and 3 the numerical values of the skin frictionand the Nusselt number are displayed when 119877 = 1 for boththe series solution and the numerical solutionwith increasingnanoparticle volume fraction It is noteworthy that perfectagreement is achieved between the improved series solutionand the numerical solution Moreover it is observed thatboth the skin friction and the Nusselt number increase withincreasing nanoparticles volume fraction Interestingly theskin friction produced by Al

2O3-water nanofluid is higher

than that of CuO-water nanofluid while Nusselt numberproduced by CuO-water nanofluid is higher than that ofAl2O3-water nanofluid

Table 3 Computation showing the Nusselt number for Bi = 1 119877 =1 Ec = 01 and119883 = 1

120593

NuCuO-water(series)

NuCuO-water(numerical)

NuAl2O3-water

(series)

NuAl2O3-water(numerical)

000 0315380 03153802 0315380 03153802001 0326419 03264199 0326493 03264928005 0376286 03762861 0376626 03766263010 0452718 04527175 0453179 04531794015 0547443 05474432 0547555 05475550020 0664319 06643189 0663318 06633184025 0808605 08086051 0805338 08053384030 0987501 09875014 0980249 09802487

08

08

06

06

04

04

02

02

0

0

1

1

CuO-waterAl2O3-water

minus02

minus04

WV

W

V

120578

120593 = 02 R = 30 X = 1

Figure 2 Nanofluids velocity profiles

51 Velocity Profiles with Parameter Variation Figures 2ndash4illustrate the effects of parameter variation on both the axialand normal velocity profiles Generally flow reversal isobserved in the region near the lower fixed plate as repre-sented by negative value of the axial velocity (119882) while thefluid near the upper accelerating plate moves faster forwardin the axial direction Also the normal velocity profiles (119881)are skewed towards the upper plate due to the drag forceexerted on the fluid by the upper plate acceleration Figure 2shows that the intensity of flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate produced by CuO-water are higher thanthat of Al

2O3-water under the same parametric conditions

With CuO-water as the working nanofluid it is observedthat increasing nanoparticles volume fraction concentrationfrom 0 to 30 increases the flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate as shown in Figure 3 With increasingacceleration of the upper plate (ie 119877 increasing) a decrease

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Computational Mathematics

of transformer oil improving diesel generator efficiencycooling of heat exchanging devices improving heat trans-fer efficiency of chillers domestic refrigerator-freezers andcooling in machining and in nuclear reactor The commonheat transfer fluids such as water ethylene glycol and engineoil have limited heat transfer capabilities owing to theirlow thermal conductivity whereas metals have much higherthermal conductivities than these fluids With the recentimprovements in nanotechnology the production of particleswith sizes on the order of nanometers can be achievedConsequently the idea of dispersing these nanoparticles ina base liquid for improving thermal conductivity has beenproposed [9] Such suspension of nanoparticles in a base fluidis called a nanofluid Due to their small size nanoparticlesfluidize easily inside the base fluid and as a consequenceclogging of channels and erosion in channel walls are nolonger a problem It is even possible to use nanofluids inmicrochannels [10] Niu et al [11] theoretically studied theslip-flow and heat transfer of a non-Newtonian nanofluidin a microtube by means of theoretical method In theirresearch the power-law rheology was adopted to describethe non-Newtonian characteristics of the flow in which thefluid consistency coefficient and the flow behaviour indexdepend on the nanoparticle volume fraction Motsumi andMakinde [12] reported a numerical solution for the effects ofthermal radiation and viscous dissipation on boundary layerflow of nanofluids over a permeable moving flat plate Choiet al [13] studied the Couette flow of nanofluids composed ofnegatively charged nanoparticles dispersed in aqueous NaClsolutions They found that the velocity profile of nanofluidscontaining charged nanoparticles deviates significantly fromthe classical linear velocity profile of Couette flow

In the studies mentioned above the combined effectsof viscous heating and convective cooling on Couette flowof nanofluids have not been discussed while such flows arevery important in lubrication technology and tribologicalproblems Therefore the objective of the present paper is toanalyze the effects of viscous heating and convective coolingon the Couette flow of water base nanofluids containingCopper Oxide (CuO) and Alumina (Al

2O3) as nanoparticles

In Sections 2ndash4 the model nonlinear governing equationstogether with the analytical and numerical solution areobtained Pertinent results are presented graphically anddiscussed quantitatively in Section 5while the conclusions aredrawn in Section 6

2 Problem Formulation

We consider a two-dimensional steady Couette flow of vis-cous incompressible water base nanofluids containing Cop-per Oxide (CuO) and Alumina (Al

2O3) as nanoparticles

in which an accelerating upper plate drags adjacent fluidalong with it and thereby imparts a motion to the rest ofthe fluid The lower plate is fixed and kept at a constanttemperature119879

0while the upper accelerating plate is subjected

to a convective heat exchange with the ambient surroundingfollowing Newtonrsquos law of cooling We choose a Cartesiancoordinates system in such away that the119909-axis is taken alongthe channel and the119910-axis is normal to it as shown in Figure 1

u(x y) T(x y)

Nanofluid

v(x a) = 0 minuskf120597T

120597y(x a) = h(T minus Ta) u(x a) = xE

y = 0

y = a

v(x 0) = 0 T(x 0) = T0 u(x 0) = 0

Figure 1 Schematic diagram of the physical system

The governing equations which are those of conservationof mass momentum and energy are [1 4 13]

120597119906

120597119909+120597V120597119910= 0 (1)

120588nf (119906120597119906

120597119909+ V120597119906

120597119910) = minus

120597119875

120597119909+ 120583nf (

1205972119906

1205971199102+1205972119906

1205971199092) (2)

120588nf (119906120597V120597119909+ V120597V120597119910) = minus

120597119875

120597119910+ 120583nf (

1205972V1205971199102

+1205972V1205971199092

) (3)

119906120597119879

120597119909+ V120597119879

120597119910=

119896nf

(120588119888119901)nf

(1205972119879

1205971199092+1205972119879

1205971199102)

+2120583nf

(120588119888119901)nf

[(120597119906

120597119909)

2

+ (120597V120597119910)

2

]

+120583nf

(120588119888119901)nf

(120597V120597119909+120597119906

120597119910)

2

(4)

where (119906 V) are the velocity components of the nanofluidin the (119909 119910) directions respectively 119864 is the upper plateacceleration parameter 119886 is the channel width 119875 is thepressure 119879 is the nanofluid temperature 120583nf is the Maxwell-Garnetts [14] approximation for effective dynamic viscosity ofthe nanofluid 119896nf is the effective thermal conductivity of thenanofluid given byBrinkman [15]120588nf is the nanofluid densityand (120588119888

119901)nf is the heat capacitance of the nanofluid which are

given by [16 17]

120583nf =120583119891

(1 minus 120593)25 120588nf = (1 minus 120593) 120588119891 + 120593120588119904

120572nf =119896nf

(120588119888119901)nf

119896nf119896119891

=

(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(120588119888119901)nf= (1 minus 120593) (120588119888

119901)119891+ 120593 (120588119888

119901)119904

(5)

In (5) 120593 is the nanoparticles solid volume fraction 120588119891is

the reference density of the fluid fraction 120588119904is the reference

International Journal of Computational Mathematics 3

density of the solid fraction 120583119891is the viscosity of the fluid

fraction 119896119891is the thermal conductivity of the fluid fraction 119888

119901

is the specific heat at constant pressure and 119896119904is the thermal

conductivity of the solid volume fraction The boundaryconditions at the channel walls may be written as

119906 (119909 0) = 0 119879 = 1198790 V (119909 0) = 0

119906 (119909 119886) = 119864119909 V (119909 119886) = 0

minus119896nf120597119879

120597119910119879 (119909 119886) = ℎ [119879 (119909 119886) minus 119879

119886]

(6)

where119879119886is the ambient surrounding temperature and ℎ is the

coefficient of heat transfer Introducing the stream function120595and vorticityΩ into the governing equations (1)ndash(4) we havethe following

119906 =120597120595

120597119910 V = minus

120597120595

120597119909

Ω =120597119906

120597119910minus120597V120597119909=1205972120595

1205971199102+1205972120595

1205971199092

(7)

After eliminating the pressure 119875 from (2) and (3) we obtain

120588nf (120597120595

120597119910

120597Ω

120597119909minus120597120595

120597119909

120597Ω

120597119910) = 120583nf (

1205972Ω

1205971199102+1205972Ω

1205971199092) (8)

(120597120595

120597119910

120597119879

120597119909minus120597120595

120597119909

120597119879

120597119910)

=119896nf

(120588119888119901)nf

(1205972119879

1205971199102+1205972119879

1205971199092)

+4120583nf

(120588119888119901)nf

(1205972120595

120597119909120597119910)

2

+120583nf

(120588119888119901)nf

(minus1205972120595

1205971199092+1205972120595

1205971199102)

2

(9)

The following dimensionless variables and parameters areintroduced into (8) and (9) together with their correspondingboundary conditions

120578 =119910

119886 120595 =

120595

1198641198862 Ω =

Ω

119864

Φ =119879 minus 1198790

119879119886minus 1198790

119877 =1198641198862

120592119891

119883 =119909

119886

120592119891=

120583119891

120588119891

Pr =120583119891119888119901

119896119891

1198981= (1 minus 120593 +

120593120588119904

120588119891

) (1 minus 120593)25

120582 =

120583119891

120573119886

119875 =119875

119864120583119891

1198982= 1198984(1 minus 120593 +

120593 (120588119888119901)119904

(120588119888119901)119891

)

Bi = ℎ119886119896119891

Ec = 11986421198862

119888119901(119879119886minus 1198790)

1198983=

1198984

(1 minus 120593)25 119898

4=

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(10)

and we obtain

1198981119877(

120597120595

120597120578

120597Ω

120597119883minus120597120595

120597119883

120597Ω

120597120578) = (

1205972Ω

1205971205782+1205972Ω

1205971198832)

1198982119877Pr(

120597120595

120597120578

120597Φ

120597119883minus120597120595

120597119883

120597Φ

120597120578)

= (1205972Φ

1205971205782+1205972Φ

1205971198832) + 4119898

3Ec Pr(

1205972120595

120597119883120597120578)

2

+ 1198983Ec Pr(minus

1205972120595

1205971198832+1205972120595

1205971205782)

2

(11)

with

120597120595

120597120578(119883 0) = 0 Φ (119883 0) = 0

120597120595

120597119883(119883 0) = 0

120597120595

120597120578(119883 1) = 119883

120597120595

120597119883(119883 1) = 0

120597Φ

120597120578(119883 1) = minus119898

4Bi [Φ (119883 1) minus 1]

(12)

where 119877 is the Reynolds number Ec is the Eckert numberBi is the Biot number Pr is the base fluid Prandtl numberand 119898

1 1198982 1198983 and 119898

4can be easily determined from

the thermophysical properties of the base fluid and thenanoparticles We seek a similarity form of solution of theform

120595 (119883 120578) = 119883119865 (120578) 119882 (119883 120578) = 119883119889119865

119889120578

119881 (120578) = minus119865 (120578) Φ (119883 120578) = 119867 (120578) + 1198832120579 (120578)

(13)

Substituting (13) into (11)-(12) we obtain

1198894119865

1198891205784= 1198981119877(

119889119865

119889120578

1198892119865

1198891205782minus 119865

1198893119865

1198891205783) (14)

1198892120579

1198891205782= minus1198983Ec Pr(119889

2119865

1198891205782)

2

+ 1198982119877Pr(2120579119889119865

119889120578minus 119865

119889120579

119889120578)

(15)

1198892119867

1198891205782= minus2120579 minus 4119898

3Ec Pr(119889119865

119889120578)

2

minus 1198982119877Pr119865119889119867

119889120578 (16)

119889119865

119889120578(0) = 0 120579 (0) = 0 119867 (0) = 0

119865 (0) = 0

(17)

4 International Journal of Computational Mathematics

119889119865

119889120578(1) = 1 119865 (1) = 0

119889120579

119889120578(1) = minus119898

4Bi 120579 (1)

119889119867

119889120578(1) = minus119898

4Bi [119867 (1) minus 1]

(18)

The dimensionless fluid axial pressure gradient is given as

minus120597119875

120597119883= 119883119860 (19)

where

(1 minus 120593)25

119860 =1198893119865

1198891205783minus 1198981119877[(

119889119865

119889120578)

2

minus 1198651198892119865

1198891205782] (20)

Other quantities of practical interest in this study are the localskin friction coefficient 119862

119891and the local Nusselt number Nu

which are defined as

119862119891=120591119908

120583119891119864 Nu =

119886119902119908

119896119891119879119908

(21)

where 120591119908is the skin friction and 119902

119908is the heat flux at the

channel upper accelerating wall which are given by

120591119908= 120583nf

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

119902119908= minus119896nf

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

(22)

Using (10) and (13) we substitute (22) into (21) and obtain

119862119891=

119883

(1 minus 120593)25

1198892119865

1198891205782(1)

Nu = minus119896nf119896119891

[119889119867

119889120578(1) + 119883

2 119889120579

119889120578(1)]

(23)

In the following section the boundary value problems in(14)ndash(18) were solved analytically using regular perturba-tion method and numerically by the Runge-Kutta-Fehlbergmethod with shooting technique [18] The results are utilisedto compute the fluid pressure gradient local skin friction andlocal Nusselt number as highlighted in (20) and (23)

3 Perturbation Method

The governing model equations (14)ndash(18) are nonlinear andthis makes its exact solution very intractable Howeverapproximate solution can be easily obtained by forminga power series expansion in the parameter 119877 It is worthnoting that an exact solution may exist for this problemunder certain assumptions The existence of exact solutionprovides remarkable and more accurate results Assume aseries solution of the form

119865 (120578) =

infin

sum

119894=0

119865119894119877119894 120579 (120578) =

infin

sum

119894=0

120579119894119877119894 (24)

Substituting the solution series in (24) into (14)ndash(18) andcollecting the coefficients of like powers of 119877 we obtain thefollowing

Zeroth Order Consider

11988941198650

1198891205784= 0

11988921205790

1198891205782= minus1198983Ec Pr(

11988921198650

1198891205782)

2

11988921198670

1198891205782= minus2120579

0minus 41198983Ec Pr(

1198891198650

119889120578)

2

(25)

with

1198891198650

119889120578(0) = 0 120579

0(0) = 0 119867

0(0) = 0

1198650(0) = 0

1198891198650

119889120578(1) = 1 119865

0(1) = 0

1198891205790

119889120578(1) = minus119898

4Bi 1205790(1)

1198891198670

119889120578(1) = minus119898

4Bi [1198670(1) minus 1]

(26)

Higher Order (119899 ge 1) Consider

1198894119865119899

1198891205784= 1198981119877

119899minus1

sum

119894=0

(119889119865119894

119889120578

1198892119865119899minus119894minus1

1198891205782minus 119865119894

1198893119865119899minus119894minus1

1198891205783)

1198892120579119899

1198891205782= minus 119898

3Ec Pr

119899

sum

119894=0

(1198892119865119894

1198891205782

1198892119865119899minus119894

1198891205782)

+ 1198982119877Pr119899minus1

sum

119894=0

(2120579119894

119889119865119899minus119894minus1

119889120578minus 119865119894

119889120579119899minus119894minus1

119889120578)

1198892119867119899

1198891205782= minus 2120579

119899minus 41198983Ec Pr

119899

sum

119894=0

(119889119865119894

119889120578

119889119865119899minus119894

119889120578)

minus 1198982119877Pr119899minus1

sum

119894=0

(119865119894

119889119867119899minus119894minus1

119889120578)

(27)

with

119889119865119899

119889120578(0) = 0 120579

119899(0) = 0 119867

119899(0) = 0

119865119899(0) = 0

119889119865119899

119889120578(1) = 0 119865

119899(1) = 0

119889120579119899

119889120578(1) = minus119898

4Bi 120579119899(1)

119889119867119899

119889120578(1) = minus119898

4Bi119867119899(1)

(28)

International Journal of Computational Mathematics 5

The equations are solved iteratively and the series solutionsfor the velocity and temperature fields are given as

119865 (120578) = 1205782(120578 minus 1) +

11989811198771205782

210(31205783minus 1205782+ 2120578 + 5) (120578 minus 1)

2

minus1198982

111987721205782

5821200(120578 minus 1)

2

times (5041205787minus 840120578

6minus 644120578

5minus 1603120578

4+ 3774120578

3

minus 28611205782minus 256120578 + 2349) + 119874 (119877

3)

120579 (120578) = minus1198983Ec Pr 120578119866

1

1 + 1198984Bi

minus1198983Ec Pr 120578119877119866

2

420 (1 + 1198984Bi)2

+ 119874 (1198772)

119867 (120578) = minus1198663120578

3 (1 + 1198984Bi)2

+ 119874 (119877)

(29)

where the expressions for 119866119894 119894 = 1 2 3 are given in the

appendix Using a computer symbolic algebra package(MAPLE) several terms of the above solution series in (29)are obtained From (29) together with (20) and (23) weobtained the series solutions for the skin friction Nusseltnumber and axial pressure gradient as

119862119891=

119883

(1 minus 120601)25

times (4 +31198981119877

35minus471198982

11198772

323400minus2511198983

11198773

39239200+ 119874 (119877

4))

119860 =1

(1 minus 120601)25

times (6 minus81198981119877

35+2477119898

2

11198772

485100minus43793119898

3

11198773

529729200

+ 119874 (1198774))

Nu = 1

31198984(1 + 119898

4Bi)2

times (minus31198982

4Bi2 + 2119898

3Ec Pr1198982

4Bi2 minus 3119898

4Bi

+ 81198983119864Pr119898

4Bi)

minusPr119877119873

1260 (1 + 1198984Bi)3119898

4

+ 1198832[31198983Ec Pr119898

4Bi

(1 + 1198984Bi)1198984

+1198983Pr Ec119877119872

420 (1 + 1198984Bi)2119898

4

]

+ 119874 (1198772)

(30)

The expressions for119873 and119872 are given in the appendixThesepower series solutions are valid for very small parameter

values of 119877 however using Pade approximation technique[19] that is based on the series summation and improvementmethod the usability of the solution series is extendedbeyond small parameter values of 119877

4 Numerical Procedure

Here we employed Runge-Kutta-Fehlberg method withshooting technique [18] to numerically solve the couplednonlinear ordinary differential equations (14)ndash(16) subjectto the boundary conditions (17)-(18) for different set ofparameter values The procedure involves transforming thenonlinear boundary value problem (BVP) into an initial valueproblem (IVP) as follows let

1199111= 119865 119911

2=119889119865

119889120578 119911

3=1198892119865

1198891205782 119911

4=1198893119865

1198891205783

1199115= 120579 119911

6=119889120579

119889120578 119911

7= 119867 119911

8=119889119867

119889120578

(31)

Equation (31) is substituted into (14)ndash(18) and a system of firstorder differential equations is obtained

1198891199111

119889120578= 1199112

1198891199112

119889120578= 1199113

1198891199113

119889120578= 1199114

1198891199114

119889120578= 1198981119877 (11991121199113minus 11991111199114)

1198891199115

119889120578= 1199116

1198891199116

119889120578= minus1198983Ec Pr 1199112

3+ 1198982119877Pr (2119911

51199112minus 11991111199116)

1198891199117

119889120578= 1199118

1198891199118

119889120578= minus2119911

5minus 41198983Ec Pr 1199112

2minus 1198982119877Pr (119911

11199118)

(32)

subject to the initial conditions

1199111(0) = 0 119911

2(0) = 0 119911

3(0) = 119904

1

1199114(0) = 119904

2 119911

5(0) = 0 119911

6(0) = 119904

3

1199117(0) = 0 119911

8(0) = 119904

4

(33)

To determine the unspecified initial conditions 1199041 1199042 1199043 and

1199044in (33) the initial guesses are supplied and resulting IVP

is solved using Runge-Kutta scheme The process is repeateduntil the solution at 120578 = 1 shoots to the boundary conditionsat that point The best way to achieve this is to supply theinitial condition in to the Runge-Kutta scheme as unknownand solve the resulting nonlinear algebraic system of equa-tions iteratively using Newton-Raphson It is important tonote that the resulting nonlinear system of algebraic equationmay not have unique solution therefore a carefully choseninitial guess which is close to the desired solution must besupplied The scheme has quadratic convergence with stepsize taken as Δ120578 = 0001 and tolerance set to 10minus7

6 International Journal of Computational Mathematics

Table 1 Thermophysical properties of the fluid phase (water) andnanoparticles [16 17]

Physicalproperties

Fluid phase(water)

CuO Al2O3

119888119901(JkgK) 4179 6500 765

120588 (kgm3) 9971 5356 3970

119896 (WmK) 0613 20 40

Table 2 Computation showing the skin friction for 119877 = 1119883 = 1

120593

119862119891

CuO-water(series)

119862119891

CuO-water(numerical)

119862119891

Al2O3-water(series)

119862119891

Al2O3-water(numerical)

000 4085563 40855627 4085563 40855627001 4186948 41869479 4189890 41898897005 4630877 46308766 4645589 46455887010 5287037 52870375 5316469 53164695015 6084673 60846727 6128833 61288327020 7065422 70654222 7124319 71243186025 8286944 82869436 8360585 83605848030 9830736 98307360 9919130 99191299

5 Results and Discussion

The Couette flow and heat transfer characteristics of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

with viscous heating and convective cooling have beeninvestigated The nonlinear similarity ordinary differentialequations governing the boundary value problemwere solvedboth analytically using the perturbation method coupledwith series improvement technique and numerically usingRunge-Kutta-Fehlberg integration technique coupled withshooting scheme Thermophysical properties of base fluidand nanoparticles are presented in Table 1 For pure water thePrandtl number is taken as Pr = 62 [12 16 17] and momen-tum diffusivity is dominant within the fluid in comparisonto pure conduction The nanoparticle volume fraction in thebase fluid is taken as 120593 = 0 to 03 (ie ranging from 0 to 30percent) while the case of120593 = 0 corresponds to the absence ofnanoparticle in the based fluid (water) Numerical solutionsare displayed in Tables 2 and 3 together with Figures 2ndash17In Tables 2 and 3 the numerical values of the skin frictionand the Nusselt number are displayed when 119877 = 1 for boththe series solution and the numerical solutionwith increasingnanoparticle volume fraction It is noteworthy that perfectagreement is achieved between the improved series solutionand the numerical solution Moreover it is observed thatboth the skin friction and the Nusselt number increase withincreasing nanoparticles volume fraction Interestingly theskin friction produced by Al

2O3-water nanofluid is higher

than that of CuO-water nanofluid while Nusselt numberproduced by CuO-water nanofluid is higher than that ofAl2O3-water nanofluid

Table 3 Computation showing the Nusselt number for Bi = 1 119877 =1 Ec = 01 and119883 = 1

120593

NuCuO-water(series)

NuCuO-water(numerical)

NuAl2O3-water

(series)

NuAl2O3-water(numerical)

000 0315380 03153802 0315380 03153802001 0326419 03264199 0326493 03264928005 0376286 03762861 0376626 03766263010 0452718 04527175 0453179 04531794015 0547443 05474432 0547555 05475550020 0664319 06643189 0663318 06633184025 0808605 08086051 0805338 08053384030 0987501 09875014 0980249 09802487

08

08

06

06

04

04

02

02

0

0

1

1

CuO-waterAl2O3-water

minus02

minus04

WV

W

V

120578

120593 = 02 R = 30 X = 1

Figure 2 Nanofluids velocity profiles

51 Velocity Profiles with Parameter Variation Figures 2ndash4illustrate the effects of parameter variation on both the axialand normal velocity profiles Generally flow reversal isobserved in the region near the lower fixed plate as repre-sented by negative value of the axial velocity (119882) while thefluid near the upper accelerating plate moves faster forwardin the axial direction Also the normal velocity profiles (119881)are skewed towards the upper plate due to the drag forceexerted on the fluid by the upper plate acceleration Figure 2shows that the intensity of flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate produced by CuO-water are higher thanthat of Al

2O3-water under the same parametric conditions

With CuO-water as the working nanofluid it is observedthat increasing nanoparticles volume fraction concentrationfrom 0 to 30 increases the flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate as shown in Figure 3 With increasingacceleration of the upper plate (ie 119877 increasing) a decrease

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 3

density of the solid fraction 120583119891is the viscosity of the fluid

fraction 119896119891is the thermal conductivity of the fluid fraction 119888

119901

is the specific heat at constant pressure and 119896119904is the thermal

conductivity of the solid volume fraction The boundaryconditions at the channel walls may be written as

119906 (119909 0) = 0 119879 = 1198790 V (119909 0) = 0

119906 (119909 119886) = 119864119909 V (119909 119886) = 0

minus119896nf120597119879

120597119910119879 (119909 119886) = ℎ [119879 (119909 119886) minus 119879

119886]

(6)

where119879119886is the ambient surrounding temperature and ℎ is the

coefficient of heat transfer Introducing the stream function120595and vorticityΩ into the governing equations (1)ndash(4) we havethe following

119906 =120597120595

120597119910 V = minus

120597120595

120597119909

Ω =120597119906

120597119910minus120597V120597119909=1205972120595

1205971199102+1205972120595

1205971199092

(7)

After eliminating the pressure 119875 from (2) and (3) we obtain

120588nf (120597120595

120597119910

120597Ω

120597119909minus120597120595

120597119909

120597Ω

120597119910) = 120583nf (

1205972Ω

1205971199102+1205972Ω

1205971199092) (8)

(120597120595

120597119910

120597119879

120597119909minus120597120595

120597119909

120597119879

120597119910)

=119896nf

(120588119888119901)nf

(1205972119879

1205971199102+1205972119879

1205971199092)

+4120583nf

(120588119888119901)nf

(1205972120595

120597119909120597119910)

2

+120583nf

(120588119888119901)nf

(minus1205972120595

1205971199092+1205972120595

1205971199102)

2

(9)

The following dimensionless variables and parameters areintroduced into (8) and (9) together with their correspondingboundary conditions

120578 =119910

119886 120595 =

120595

1198641198862 Ω =

Ω

119864

Φ =119879 minus 1198790

119879119886minus 1198790

119877 =1198641198862

120592119891

119883 =119909

119886

120592119891=

120583119891

120588119891

Pr =120583119891119888119901

119896119891

1198981= (1 minus 120593 +

120593120588119904

120588119891

) (1 minus 120593)25

120582 =

120583119891

120573119886

119875 =119875

119864120583119891

1198982= 1198984(1 minus 120593 +

120593 (120588119888119901)119904

(120588119888119901)119891

)

Bi = ℎ119886119896119891

Ec = 11986421198862

119888119901(119879119886minus 1198790)

1198983=

1198984

(1 minus 120593)25 119898

4=

(119896119904+ 2119896119891) + 120593 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) minus 2120593 (119896

119891minus 119896119904)

(10)

and we obtain

1198981119877(

120597120595

120597120578

120597Ω

120597119883minus120597120595

120597119883

120597Ω

120597120578) = (

1205972Ω

1205971205782+1205972Ω

1205971198832)

1198982119877Pr(

120597120595

120597120578

120597Φ

120597119883minus120597120595

120597119883

120597Φ

120597120578)

= (1205972Φ

1205971205782+1205972Φ

1205971198832) + 4119898

3Ec Pr(

1205972120595

120597119883120597120578)

2

+ 1198983Ec Pr(minus

1205972120595

1205971198832+1205972120595

1205971205782)

2

(11)

with

120597120595

120597120578(119883 0) = 0 Φ (119883 0) = 0

120597120595

120597119883(119883 0) = 0

120597120595

120597120578(119883 1) = 119883

120597120595

120597119883(119883 1) = 0

120597Φ

120597120578(119883 1) = minus119898

4Bi [Φ (119883 1) minus 1]

(12)

where 119877 is the Reynolds number Ec is the Eckert numberBi is the Biot number Pr is the base fluid Prandtl numberand 119898

1 1198982 1198983 and 119898

4can be easily determined from

the thermophysical properties of the base fluid and thenanoparticles We seek a similarity form of solution of theform

120595 (119883 120578) = 119883119865 (120578) 119882 (119883 120578) = 119883119889119865

119889120578

119881 (120578) = minus119865 (120578) Φ (119883 120578) = 119867 (120578) + 1198832120579 (120578)

(13)

Substituting (13) into (11)-(12) we obtain

1198894119865

1198891205784= 1198981119877(

119889119865

119889120578

1198892119865

1198891205782minus 119865

1198893119865

1198891205783) (14)

1198892120579

1198891205782= minus1198983Ec Pr(119889

2119865

1198891205782)

2

+ 1198982119877Pr(2120579119889119865

119889120578minus 119865

119889120579

119889120578)

(15)

1198892119867

1198891205782= minus2120579 minus 4119898

3Ec Pr(119889119865

119889120578)

2

minus 1198982119877Pr119865119889119867

119889120578 (16)

119889119865

119889120578(0) = 0 120579 (0) = 0 119867 (0) = 0

119865 (0) = 0

(17)

4 International Journal of Computational Mathematics

119889119865

119889120578(1) = 1 119865 (1) = 0

119889120579

119889120578(1) = minus119898

4Bi 120579 (1)

119889119867

119889120578(1) = minus119898

4Bi [119867 (1) minus 1]

(18)

The dimensionless fluid axial pressure gradient is given as

minus120597119875

120597119883= 119883119860 (19)

where

(1 minus 120593)25

119860 =1198893119865

1198891205783minus 1198981119877[(

119889119865

119889120578)

2

minus 1198651198892119865

1198891205782] (20)

Other quantities of practical interest in this study are the localskin friction coefficient 119862

119891and the local Nusselt number Nu

which are defined as

119862119891=120591119908

120583119891119864 Nu =

119886119902119908

119896119891119879119908

(21)

where 120591119908is the skin friction and 119902

119908is the heat flux at the

channel upper accelerating wall which are given by

120591119908= 120583nf

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

119902119908= minus119896nf

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

(22)

Using (10) and (13) we substitute (22) into (21) and obtain

119862119891=

119883

(1 minus 120593)25

1198892119865

1198891205782(1)

Nu = minus119896nf119896119891

[119889119867

119889120578(1) + 119883

2 119889120579

119889120578(1)]

(23)

In the following section the boundary value problems in(14)ndash(18) were solved analytically using regular perturba-tion method and numerically by the Runge-Kutta-Fehlbergmethod with shooting technique [18] The results are utilisedto compute the fluid pressure gradient local skin friction andlocal Nusselt number as highlighted in (20) and (23)

3 Perturbation Method

The governing model equations (14)ndash(18) are nonlinear andthis makes its exact solution very intractable Howeverapproximate solution can be easily obtained by forminga power series expansion in the parameter 119877 It is worthnoting that an exact solution may exist for this problemunder certain assumptions The existence of exact solutionprovides remarkable and more accurate results Assume aseries solution of the form

119865 (120578) =

infin

sum

119894=0

119865119894119877119894 120579 (120578) =

infin

sum

119894=0

120579119894119877119894 (24)

Substituting the solution series in (24) into (14)ndash(18) andcollecting the coefficients of like powers of 119877 we obtain thefollowing

Zeroth Order Consider

11988941198650

1198891205784= 0

11988921205790

1198891205782= minus1198983Ec Pr(

11988921198650

1198891205782)

2

11988921198670

1198891205782= minus2120579

0minus 41198983Ec Pr(

1198891198650

119889120578)

2

(25)

with

1198891198650

119889120578(0) = 0 120579

0(0) = 0 119867

0(0) = 0

1198650(0) = 0

1198891198650

119889120578(1) = 1 119865

0(1) = 0

1198891205790

119889120578(1) = minus119898

4Bi 1205790(1)

1198891198670

119889120578(1) = minus119898

4Bi [1198670(1) minus 1]

(26)

Higher Order (119899 ge 1) Consider

1198894119865119899

1198891205784= 1198981119877

119899minus1

sum

119894=0

(119889119865119894

119889120578

1198892119865119899minus119894minus1

1198891205782minus 119865119894

1198893119865119899minus119894minus1

1198891205783)

1198892120579119899

1198891205782= minus 119898

3Ec Pr

119899

sum

119894=0

(1198892119865119894

1198891205782

1198892119865119899minus119894

1198891205782)

+ 1198982119877Pr119899minus1

sum

119894=0

(2120579119894

119889119865119899minus119894minus1

119889120578minus 119865119894

119889120579119899minus119894minus1

119889120578)

1198892119867119899

1198891205782= minus 2120579

119899minus 41198983Ec Pr

119899

sum

119894=0

(119889119865119894

119889120578

119889119865119899minus119894

119889120578)

minus 1198982119877Pr119899minus1

sum

119894=0

(119865119894

119889119867119899minus119894minus1

119889120578)

(27)

with

119889119865119899

119889120578(0) = 0 120579

119899(0) = 0 119867

119899(0) = 0

119865119899(0) = 0

119889119865119899

119889120578(1) = 0 119865

119899(1) = 0

119889120579119899

119889120578(1) = minus119898

4Bi 120579119899(1)

119889119867119899

119889120578(1) = minus119898

4Bi119867119899(1)

(28)

International Journal of Computational Mathematics 5

The equations are solved iteratively and the series solutionsfor the velocity and temperature fields are given as

119865 (120578) = 1205782(120578 minus 1) +

11989811198771205782

210(31205783minus 1205782+ 2120578 + 5) (120578 minus 1)

2

minus1198982

111987721205782

5821200(120578 minus 1)

2

times (5041205787minus 840120578

6minus 644120578

5minus 1603120578

4+ 3774120578

3

minus 28611205782minus 256120578 + 2349) + 119874 (119877

3)

120579 (120578) = minus1198983Ec Pr 120578119866

1

1 + 1198984Bi

minus1198983Ec Pr 120578119877119866

2

420 (1 + 1198984Bi)2

+ 119874 (1198772)

119867 (120578) = minus1198663120578

3 (1 + 1198984Bi)2

+ 119874 (119877)

(29)

where the expressions for 119866119894 119894 = 1 2 3 are given in the

appendix Using a computer symbolic algebra package(MAPLE) several terms of the above solution series in (29)are obtained From (29) together with (20) and (23) weobtained the series solutions for the skin friction Nusseltnumber and axial pressure gradient as

119862119891=

119883

(1 minus 120601)25

times (4 +31198981119877

35minus471198982

11198772

323400minus2511198983

11198773

39239200+ 119874 (119877

4))

119860 =1

(1 minus 120601)25

times (6 minus81198981119877

35+2477119898

2

11198772

485100minus43793119898

3

11198773

529729200

+ 119874 (1198774))

Nu = 1

31198984(1 + 119898

4Bi)2

times (minus31198982

4Bi2 + 2119898

3Ec Pr1198982

4Bi2 minus 3119898

4Bi

+ 81198983119864Pr119898

4Bi)

minusPr119877119873

1260 (1 + 1198984Bi)3119898

4

+ 1198832[31198983Ec Pr119898

4Bi

(1 + 1198984Bi)1198984

+1198983Pr Ec119877119872

420 (1 + 1198984Bi)2119898

4

]

+ 119874 (1198772)

(30)

The expressions for119873 and119872 are given in the appendixThesepower series solutions are valid for very small parameter

values of 119877 however using Pade approximation technique[19] that is based on the series summation and improvementmethod the usability of the solution series is extendedbeyond small parameter values of 119877

4 Numerical Procedure

Here we employed Runge-Kutta-Fehlberg method withshooting technique [18] to numerically solve the couplednonlinear ordinary differential equations (14)ndash(16) subjectto the boundary conditions (17)-(18) for different set ofparameter values The procedure involves transforming thenonlinear boundary value problem (BVP) into an initial valueproblem (IVP) as follows let

1199111= 119865 119911

2=119889119865

119889120578 119911

3=1198892119865

1198891205782 119911

4=1198893119865

1198891205783

1199115= 120579 119911

6=119889120579

119889120578 119911

7= 119867 119911

8=119889119867

119889120578

(31)

Equation (31) is substituted into (14)ndash(18) and a system of firstorder differential equations is obtained

1198891199111

119889120578= 1199112

1198891199112

119889120578= 1199113

1198891199113

119889120578= 1199114

1198891199114

119889120578= 1198981119877 (11991121199113minus 11991111199114)

1198891199115

119889120578= 1199116

1198891199116

119889120578= minus1198983Ec Pr 1199112

3+ 1198982119877Pr (2119911

51199112minus 11991111199116)

1198891199117

119889120578= 1199118

1198891199118

119889120578= minus2119911

5minus 41198983Ec Pr 1199112

2minus 1198982119877Pr (119911

11199118)

(32)

subject to the initial conditions

1199111(0) = 0 119911

2(0) = 0 119911

3(0) = 119904

1

1199114(0) = 119904

2 119911

5(0) = 0 119911

6(0) = 119904

3

1199117(0) = 0 119911

8(0) = 119904

4

(33)

To determine the unspecified initial conditions 1199041 1199042 1199043 and

1199044in (33) the initial guesses are supplied and resulting IVP

is solved using Runge-Kutta scheme The process is repeateduntil the solution at 120578 = 1 shoots to the boundary conditionsat that point The best way to achieve this is to supply theinitial condition in to the Runge-Kutta scheme as unknownand solve the resulting nonlinear algebraic system of equa-tions iteratively using Newton-Raphson It is important tonote that the resulting nonlinear system of algebraic equationmay not have unique solution therefore a carefully choseninitial guess which is close to the desired solution must besupplied The scheme has quadratic convergence with stepsize taken as Δ120578 = 0001 and tolerance set to 10minus7

6 International Journal of Computational Mathematics

Table 1 Thermophysical properties of the fluid phase (water) andnanoparticles [16 17]

Physicalproperties

Fluid phase(water)

CuO Al2O3

119888119901(JkgK) 4179 6500 765

120588 (kgm3) 9971 5356 3970

119896 (WmK) 0613 20 40

Table 2 Computation showing the skin friction for 119877 = 1119883 = 1

120593

119862119891

CuO-water(series)

119862119891

CuO-water(numerical)

119862119891

Al2O3-water(series)

119862119891

Al2O3-water(numerical)

000 4085563 40855627 4085563 40855627001 4186948 41869479 4189890 41898897005 4630877 46308766 4645589 46455887010 5287037 52870375 5316469 53164695015 6084673 60846727 6128833 61288327020 7065422 70654222 7124319 71243186025 8286944 82869436 8360585 83605848030 9830736 98307360 9919130 99191299

5 Results and Discussion

The Couette flow and heat transfer characteristics of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

with viscous heating and convective cooling have beeninvestigated The nonlinear similarity ordinary differentialequations governing the boundary value problemwere solvedboth analytically using the perturbation method coupledwith series improvement technique and numerically usingRunge-Kutta-Fehlberg integration technique coupled withshooting scheme Thermophysical properties of base fluidand nanoparticles are presented in Table 1 For pure water thePrandtl number is taken as Pr = 62 [12 16 17] and momen-tum diffusivity is dominant within the fluid in comparisonto pure conduction The nanoparticle volume fraction in thebase fluid is taken as 120593 = 0 to 03 (ie ranging from 0 to 30percent) while the case of120593 = 0 corresponds to the absence ofnanoparticle in the based fluid (water) Numerical solutionsare displayed in Tables 2 and 3 together with Figures 2ndash17In Tables 2 and 3 the numerical values of the skin frictionand the Nusselt number are displayed when 119877 = 1 for boththe series solution and the numerical solutionwith increasingnanoparticle volume fraction It is noteworthy that perfectagreement is achieved between the improved series solutionand the numerical solution Moreover it is observed thatboth the skin friction and the Nusselt number increase withincreasing nanoparticles volume fraction Interestingly theskin friction produced by Al

2O3-water nanofluid is higher

than that of CuO-water nanofluid while Nusselt numberproduced by CuO-water nanofluid is higher than that ofAl2O3-water nanofluid

Table 3 Computation showing the Nusselt number for Bi = 1 119877 =1 Ec = 01 and119883 = 1

120593

NuCuO-water(series)

NuCuO-water(numerical)

NuAl2O3-water

(series)

NuAl2O3-water(numerical)

000 0315380 03153802 0315380 03153802001 0326419 03264199 0326493 03264928005 0376286 03762861 0376626 03766263010 0452718 04527175 0453179 04531794015 0547443 05474432 0547555 05475550020 0664319 06643189 0663318 06633184025 0808605 08086051 0805338 08053384030 0987501 09875014 0980249 09802487

08

08

06

06

04

04

02

02

0

0

1

1

CuO-waterAl2O3-water

minus02

minus04

WV

W

V

120578

120593 = 02 R = 30 X = 1

Figure 2 Nanofluids velocity profiles

51 Velocity Profiles with Parameter Variation Figures 2ndash4illustrate the effects of parameter variation on both the axialand normal velocity profiles Generally flow reversal isobserved in the region near the lower fixed plate as repre-sented by negative value of the axial velocity (119882) while thefluid near the upper accelerating plate moves faster forwardin the axial direction Also the normal velocity profiles (119881)are skewed towards the upper plate due to the drag forceexerted on the fluid by the upper plate acceleration Figure 2shows that the intensity of flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate produced by CuO-water are higher thanthat of Al

2O3-water under the same parametric conditions

With CuO-water as the working nanofluid it is observedthat increasing nanoparticles volume fraction concentrationfrom 0 to 30 increases the flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate as shown in Figure 3 With increasingacceleration of the upper plate (ie 119877 increasing) a decrease

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Computational Mathematics

119889119865

119889120578(1) = 1 119865 (1) = 0

119889120579

119889120578(1) = minus119898

4Bi 120579 (1)

119889119867

119889120578(1) = minus119898

4Bi [119867 (1) minus 1]

(18)

The dimensionless fluid axial pressure gradient is given as

minus120597119875

120597119883= 119883119860 (19)

where

(1 minus 120593)25

119860 =1198893119865

1198891205783minus 1198981119877[(

119889119865

119889120578)

2

minus 1198651198892119865

1198891205782] (20)

Other quantities of practical interest in this study are the localskin friction coefficient 119862

119891and the local Nusselt number Nu

which are defined as

119862119891=120591119908

120583119891119864 Nu =

119886119902119908

119896119891119879119908

(21)

where 120591119908is the skin friction and 119902

119908is the heat flux at the

channel upper accelerating wall which are given by

120591119908= 120583nf

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

119902119908= minus119896nf

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=119886

(22)

Using (10) and (13) we substitute (22) into (21) and obtain

119862119891=

119883

(1 minus 120593)25

1198892119865

1198891205782(1)

Nu = minus119896nf119896119891

[119889119867

119889120578(1) + 119883

2 119889120579

119889120578(1)]

(23)

In the following section the boundary value problems in(14)ndash(18) were solved analytically using regular perturba-tion method and numerically by the Runge-Kutta-Fehlbergmethod with shooting technique [18] The results are utilisedto compute the fluid pressure gradient local skin friction andlocal Nusselt number as highlighted in (20) and (23)

3 Perturbation Method

The governing model equations (14)ndash(18) are nonlinear andthis makes its exact solution very intractable Howeverapproximate solution can be easily obtained by forminga power series expansion in the parameter 119877 It is worthnoting that an exact solution may exist for this problemunder certain assumptions The existence of exact solutionprovides remarkable and more accurate results Assume aseries solution of the form

119865 (120578) =

infin

sum

119894=0

119865119894119877119894 120579 (120578) =

infin

sum

119894=0

120579119894119877119894 (24)

Substituting the solution series in (24) into (14)ndash(18) andcollecting the coefficients of like powers of 119877 we obtain thefollowing

Zeroth Order Consider

11988941198650

1198891205784= 0

11988921205790

1198891205782= minus1198983Ec Pr(

11988921198650

1198891205782)

2

11988921198670

1198891205782= minus2120579

0minus 41198983Ec Pr(

1198891198650

119889120578)

2

(25)

with

1198891198650

119889120578(0) = 0 120579

0(0) = 0 119867

0(0) = 0

1198650(0) = 0

1198891198650

119889120578(1) = 1 119865

0(1) = 0

1198891205790

119889120578(1) = minus119898

4Bi 1205790(1)

1198891198670

119889120578(1) = minus119898

4Bi [1198670(1) minus 1]

(26)

Higher Order (119899 ge 1) Consider

1198894119865119899

1198891205784= 1198981119877

119899minus1

sum

119894=0

(119889119865119894

119889120578

1198892119865119899minus119894minus1

1198891205782minus 119865119894

1198893119865119899minus119894minus1

1198891205783)

1198892120579119899

1198891205782= minus 119898

3Ec Pr

119899

sum

119894=0

(1198892119865119894

1198891205782

1198892119865119899minus119894

1198891205782)

+ 1198982119877Pr119899minus1

sum

119894=0

(2120579119894

119889119865119899minus119894minus1

119889120578minus 119865119894

119889120579119899minus119894minus1

119889120578)

1198892119867119899

1198891205782= minus 2120579

119899minus 41198983Ec Pr

119899

sum

119894=0

(119889119865119894

119889120578

119889119865119899minus119894

119889120578)

minus 1198982119877Pr119899minus1

sum

119894=0

(119865119894

119889119867119899minus119894minus1

119889120578)

(27)

with

119889119865119899

119889120578(0) = 0 120579

119899(0) = 0 119867

119899(0) = 0

119865119899(0) = 0

119889119865119899

119889120578(1) = 0 119865

119899(1) = 0

119889120579119899

119889120578(1) = minus119898

4Bi 120579119899(1)

119889119867119899

119889120578(1) = minus119898

4Bi119867119899(1)

(28)

International Journal of Computational Mathematics 5

The equations are solved iteratively and the series solutionsfor the velocity and temperature fields are given as

119865 (120578) = 1205782(120578 minus 1) +

11989811198771205782

210(31205783minus 1205782+ 2120578 + 5) (120578 minus 1)

2

minus1198982

111987721205782

5821200(120578 minus 1)

2

times (5041205787minus 840120578

6minus 644120578

5minus 1603120578

4+ 3774120578

3

minus 28611205782minus 256120578 + 2349) + 119874 (119877

3)

120579 (120578) = minus1198983Ec Pr 120578119866

1

1 + 1198984Bi

minus1198983Ec Pr 120578119877119866

2

420 (1 + 1198984Bi)2

+ 119874 (1198772)

119867 (120578) = minus1198663120578

3 (1 + 1198984Bi)2

+ 119874 (119877)

(29)

where the expressions for 119866119894 119894 = 1 2 3 are given in the

appendix Using a computer symbolic algebra package(MAPLE) several terms of the above solution series in (29)are obtained From (29) together with (20) and (23) weobtained the series solutions for the skin friction Nusseltnumber and axial pressure gradient as

119862119891=

119883

(1 minus 120601)25

times (4 +31198981119877

35minus471198982

11198772

323400minus2511198983

11198773

39239200+ 119874 (119877

4))

119860 =1

(1 minus 120601)25

times (6 minus81198981119877

35+2477119898

2

11198772

485100minus43793119898

3

11198773

529729200

+ 119874 (1198774))

Nu = 1

31198984(1 + 119898

4Bi)2

times (minus31198982

4Bi2 + 2119898

3Ec Pr1198982

4Bi2 minus 3119898

4Bi

+ 81198983119864Pr119898

4Bi)

minusPr119877119873

1260 (1 + 1198984Bi)3119898

4

+ 1198832[31198983Ec Pr119898

4Bi

(1 + 1198984Bi)1198984

+1198983Pr Ec119877119872

420 (1 + 1198984Bi)2119898

4

]

+ 119874 (1198772)

(30)

The expressions for119873 and119872 are given in the appendixThesepower series solutions are valid for very small parameter

values of 119877 however using Pade approximation technique[19] that is based on the series summation and improvementmethod the usability of the solution series is extendedbeyond small parameter values of 119877

4 Numerical Procedure

Here we employed Runge-Kutta-Fehlberg method withshooting technique [18] to numerically solve the couplednonlinear ordinary differential equations (14)ndash(16) subjectto the boundary conditions (17)-(18) for different set ofparameter values The procedure involves transforming thenonlinear boundary value problem (BVP) into an initial valueproblem (IVP) as follows let

1199111= 119865 119911

2=119889119865

119889120578 119911

3=1198892119865

1198891205782 119911

4=1198893119865

1198891205783

1199115= 120579 119911

6=119889120579

119889120578 119911

7= 119867 119911

8=119889119867

119889120578

(31)

Equation (31) is substituted into (14)ndash(18) and a system of firstorder differential equations is obtained

1198891199111

119889120578= 1199112

1198891199112

119889120578= 1199113

1198891199113

119889120578= 1199114

1198891199114

119889120578= 1198981119877 (11991121199113minus 11991111199114)

1198891199115

119889120578= 1199116

1198891199116

119889120578= minus1198983Ec Pr 1199112

3+ 1198982119877Pr (2119911

51199112minus 11991111199116)

1198891199117

119889120578= 1199118

1198891199118

119889120578= minus2119911

5minus 41198983Ec Pr 1199112

2minus 1198982119877Pr (119911

11199118)

(32)

subject to the initial conditions

1199111(0) = 0 119911

2(0) = 0 119911

3(0) = 119904

1

1199114(0) = 119904

2 119911

5(0) = 0 119911

6(0) = 119904

3

1199117(0) = 0 119911

8(0) = 119904

4

(33)

To determine the unspecified initial conditions 1199041 1199042 1199043 and

1199044in (33) the initial guesses are supplied and resulting IVP

is solved using Runge-Kutta scheme The process is repeateduntil the solution at 120578 = 1 shoots to the boundary conditionsat that point The best way to achieve this is to supply theinitial condition in to the Runge-Kutta scheme as unknownand solve the resulting nonlinear algebraic system of equa-tions iteratively using Newton-Raphson It is important tonote that the resulting nonlinear system of algebraic equationmay not have unique solution therefore a carefully choseninitial guess which is close to the desired solution must besupplied The scheme has quadratic convergence with stepsize taken as Δ120578 = 0001 and tolerance set to 10minus7

6 International Journal of Computational Mathematics

Table 1 Thermophysical properties of the fluid phase (water) andnanoparticles [16 17]

Physicalproperties

Fluid phase(water)

CuO Al2O3

119888119901(JkgK) 4179 6500 765

120588 (kgm3) 9971 5356 3970

119896 (WmK) 0613 20 40

Table 2 Computation showing the skin friction for 119877 = 1119883 = 1

120593

119862119891

CuO-water(series)

119862119891

CuO-water(numerical)

119862119891

Al2O3-water(series)

119862119891

Al2O3-water(numerical)

000 4085563 40855627 4085563 40855627001 4186948 41869479 4189890 41898897005 4630877 46308766 4645589 46455887010 5287037 52870375 5316469 53164695015 6084673 60846727 6128833 61288327020 7065422 70654222 7124319 71243186025 8286944 82869436 8360585 83605848030 9830736 98307360 9919130 99191299

5 Results and Discussion

The Couette flow and heat transfer characteristics of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

with viscous heating and convective cooling have beeninvestigated The nonlinear similarity ordinary differentialequations governing the boundary value problemwere solvedboth analytically using the perturbation method coupledwith series improvement technique and numerically usingRunge-Kutta-Fehlberg integration technique coupled withshooting scheme Thermophysical properties of base fluidand nanoparticles are presented in Table 1 For pure water thePrandtl number is taken as Pr = 62 [12 16 17] and momen-tum diffusivity is dominant within the fluid in comparisonto pure conduction The nanoparticle volume fraction in thebase fluid is taken as 120593 = 0 to 03 (ie ranging from 0 to 30percent) while the case of120593 = 0 corresponds to the absence ofnanoparticle in the based fluid (water) Numerical solutionsare displayed in Tables 2 and 3 together with Figures 2ndash17In Tables 2 and 3 the numerical values of the skin frictionand the Nusselt number are displayed when 119877 = 1 for boththe series solution and the numerical solutionwith increasingnanoparticle volume fraction It is noteworthy that perfectagreement is achieved between the improved series solutionand the numerical solution Moreover it is observed thatboth the skin friction and the Nusselt number increase withincreasing nanoparticles volume fraction Interestingly theskin friction produced by Al

2O3-water nanofluid is higher

than that of CuO-water nanofluid while Nusselt numberproduced by CuO-water nanofluid is higher than that ofAl2O3-water nanofluid

Table 3 Computation showing the Nusselt number for Bi = 1 119877 =1 Ec = 01 and119883 = 1

120593

NuCuO-water(series)

NuCuO-water(numerical)

NuAl2O3-water

(series)

NuAl2O3-water(numerical)

000 0315380 03153802 0315380 03153802001 0326419 03264199 0326493 03264928005 0376286 03762861 0376626 03766263010 0452718 04527175 0453179 04531794015 0547443 05474432 0547555 05475550020 0664319 06643189 0663318 06633184025 0808605 08086051 0805338 08053384030 0987501 09875014 0980249 09802487

08

08

06

06

04

04

02

02

0

0

1

1

CuO-waterAl2O3-water

minus02

minus04

WV

W

V

120578

120593 = 02 R = 30 X = 1

Figure 2 Nanofluids velocity profiles

51 Velocity Profiles with Parameter Variation Figures 2ndash4illustrate the effects of parameter variation on both the axialand normal velocity profiles Generally flow reversal isobserved in the region near the lower fixed plate as repre-sented by negative value of the axial velocity (119882) while thefluid near the upper accelerating plate moves faster forwardin the axial direction Also the normal velocity profiles (119881)are skewed towards the upper plate due to the drag forceexerted on the fluid by the upper plate acceleration Figure 2shows that the intensity of flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate produced by CuO-water are higher thanthat of Al

2O3-water under the same parametric conditions

With CuO-water as the working nanofluid it is observedthat increasing nanoparticles volume fraction concentrationfrom 0 to 30 increases the flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate as shown in Figure 3 With increasingacceleration of the upper plate (ie 119877 increasing) a decrease

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 5

The equations are solved iteratively and the series solutionsfor the velocity and temperature fields are given as

119865 (120578) = 1205782(120578 minus 1) +

11989811198771205782

210(31205783minus 1205782+ 2120578 + 5) (120578 minus 1)

2

minus1198982

111987721205782

5821200(120578 minus 1)

2

times (5041205787minus 840120578

6minus 644120578

5minus 1603120578

4+ 3774120578

3

minus 28611205782minus 256120578 + 2349) + 119874 (119877

3)

120579 (120578) = minus1198983Ec Pr 120578119866

1

1 + 1198984Bi

minus1198983Ec Pr 120578119877119866

2

420 (1 + 1198984Bi)2

+ 119874 (1198772)

119867 (120578) = minus1198663120578

3 (1 + 1198984Bi)2

+ 119874 (119877)

(29)

where the expressions for 119866119894 119894 = 1 2 3 are given in the

appendix Using a computer symbolic algebra package(MAPLE) several terms of the above solution series in (29)are obtained From (29) together with (20) and (23) weobtained the series solutions for the skin friction Nusseltnumber and axial pressure gradient as

119862119891=

119883

(1 minus 120601)25

times (4 +31198981119877

35minus471198982

11198772

323400minus2511198983

11198773

39239200+ 119874 (119877

4))

119860 =1

(1 minus 120601)25

times (6 minus81198981119877

35+2477119898

2

11198772

485100minus43793119898

3

11198773

529729200

+ 119874 (1198774))

Nu = 1

31198984(1 + 119898

4Bi)2

times (minus31198982

4Bi2 + 2119898

3Ec Pr1198982

4Bi2 minus 3119898

4Bi

+ 81198983119864Pr119898

4Bi)

minusPr119877119873

1260 (1 + 1198984Bi)3119898

4

+ 1198832[31198983Ec Pr119898

4Bi

(1 + 1198984Bi)1198984

+1198983Pr Ec119877119872

420 (1 + 1198984Bi)2119898

4

]

+ 119874 (1198772)

(30)

The expressions for119873 and119872 are given in the appendixThesepower series solutions are valid for very small parameter

values of 119877 however using Pade approximation technique[19] that is based on the series summation and improvementmethod the usability of the solution series is extendedbeyond small parameter values of 119877

4 Numerical Procedure

Here we employed Runge-Kutta-Fehlberg method withshooting technique [18] to numerically solve the couplednonlinear ordinary differential equations (14)ndash(16) subjectto the boundary conditions (17)-(18) for different set ofparameter values The procedure involves transforming thenonlinear boundary value problem (BVP) into an initial valueproblem (IVP) as follows let

1199111= 119865 119911

2=119889119865

119889120578 119911

3=1198892119865

1198891205782 119911

4=1198893119865

1198891205783

1199115= 120579 119911

6=119889120579

119889120578 119911

7= 119867 119911

8=119889119867

119889120578

(31)

Equation (31) is substituted into (14)ndash(18) and a system of firstorder differential equations is obtained

1198891199111

119889120578= 1199112

1198891199112

119889120578= 1199113

1198891199113

119889120578= 1199114

1198891199114

119889120578= 1198981119877 (11991121199113minus 11991111199114)

1198891199115

119889120578= 1199116

1198891199116

119889120578= minus1198983Ec Pr 1199112

3+ 1198982119877Pr (2119911

51199112minus 11991111199116)

1198891199117

119889120578= 1199118

1198891199118

119889120578= minus2119911

5minus 41198983Ec Pr 1199112

2minus 1198982119877Pr (119911

11199118)

(32)

subject to the initial conditions

1199111(0) = 0 119911

2(0) = 0 119911

3(0) = 119904

1

1199114(0) = 119904

2 119911

5(0) = 0 119911

6(0) = 119904

3

1199117(0) = 0 119911

8(0) = 119904

4

(33)

To determine the unspecified initial conditions 1199041 1199042 1199043 and

1199044in (33) the initial guesses are supplied and resulting IVP

is solved using Runge-Kutta scheme The process is repeateduntil the solution at 120578 = 1 shoots to the boundary conditionsat that point The best way to achieve this is to supply theinitial condition in to the Runge-Kutta scheme as unknownand solve the resulting nonlinear algebraic system of equa-tions iteratively using Newton-Raphson It is important tonote that the resulting nonlinear system of algebraic equationmay not have unique solution therefore a carefully choseninitial guess which is close to the desired solution must besupplied The scheme has quadratic convergence with stepsize taken as Δ120578 = 0001 and tolerance set to 10minus7

6 International Journal of Computational Mathematics

Table 1 Thermophysical properties of the fluid phase (water) andnanoparticles [16 17]

Physicalproperties

Fluid phase(water)

CuO Al2O3

119888119901(JkgK) 4179 6500 765

120588 (kgm3) 9971 5356 3970

119896 (WmK) 0613 20 40

Table 2 Computation showing the skin friction for 119877 = 1119883 = 1

120593

119862119891

CuO-water(series)

119862119891

CuO-water(numerical)

119862119891

Al2O3-water(series)

119862119891

Al2O3-water(numerical)

000 4085563 40855627 4085563 40855627001 4186948 41869479 4189890 41898897005 4630877 46308766 4645589 46455887010 5287037 52870375 5316469 53164695015 6084673 60846727 6128833 61288327020 7065422 70654222 7124319 71243186025 8286944 82869436 8360585 83605848030 9830736 98307360 9919130 99191299

5 Results and Discussion

The Couette flow and heat transfer characteristics of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

with viscous heating and convective cooling have beeninvestigated The nonlinear similarity ordinary differentialequations governing the boundary value problemwere solvedboth analytically using the perturbation method coupledwith series improvement technique and numerically usingRunge-Kutta-Fehlberg integration technique coupled withshooting scheme Thermophysical properties of base fluidand nanoparticles are presented in Table 1 For pure water thePrandtl number is taken as Pr = 62 [12 16 17] and momen-tum diffusivity is dominant within the fluid in comparisonto pure conduction The nanoparticle volume fraction in thebase fluid is taken as 120593 = 0 to 03 (ie ranging from 0 to 30percent) while the case of120593 = 0 corresponds to the absence ofnanoparticle in the based fluid (water) Numerical solutionsare displayed in Tables 2 and 3 together with Figures 2ndash17In Tables 2 and 3 the numerical values of the skin frictionand the Nusselt number are displayed when 119877 = 1 for boththe series solution and the numerical solutionwith increasingnanoparticle volume fraction It is noteworthy that perfectagreement is achieved between the improved series solutionand the numerical solution Moreover it is observed thatboth the skin friction and the Nusselt number increase withincreasing nanoparticles volume fraction Interestingly theskin friction produced by Al

2O3-water nanofluid is higher

than that of CuO-water nanofluid while Nusselt numberproduced by CuO-water nanofluid is higher than that ofAl2O3-water nanofluid

Table 3 Computation showing the Nusselt number for Bi = 1 119877 =1 Ec = 01 and119883 = 1

120593

NuCuO-water(series)

NuCuO-water(numerical)

NuAl2O3-water

(series)

NuAl2O3-water(numerical)

000 0315380 03153802 0315380 03153802001 0326419 03264199 0326493 03264928005 0376286 03762861 0376626 03766263010 0452718 04527175 0453179 04531794015 0547443 05474432 0547555 05475550020 0664319 06643189 0663318 06633184025 0808605 08086051 0805338 08053384030 0987501 09875014 0980249 09802487

08

08

06

06

04

04

02

02

0

0

1

1

CuO-waterAl2O3-water

minus02

minus04

WV

W

V

120578

120593 = 02 R = 30 X = 1

Figure 2 Nanofluids velocity profiles

51 Velocity Profiles with Parameter Variation Figures 2ndash4illustrate the effects of parameter variation on both the axialand normal velocity profiles Generally flow reversal isobserved in the region near the lower fixed plate as repre-sented by negative value of the axial velocity (119882) while thefluid near the upper accelerating plate moves faster forwardin the axial direction Also the normal velocity profiles (119881)are skewed towards the upper plate due to the drag forceexerted on the fluid by the upper plate acceleration Figure 2shows that the intensity of flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate produced by CuO-water are higher thanthat of Al

2O3-water under the same parametric conditions

With CuO-water as the working nanofluid it is observedthat increasing nanoparticles volume fraction concentrationfrom 0 to 30 increases the flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate as shown in Figure 3 With increasingacceleration of the upper plate (ie 119877 increasing) a decrease

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Computational Mathematics

Table 1 Thermophysical properties of the fluid phase (water) andnanoparticles [16 17]

Physicalproperties

Fluid phase(water)

CuO Al2O3

119888119901(JkgK) 4179 6500 765

120588 (kgm3) 9971 5356 3970

119896 (WmK) 0613 20 40

Table 2 Computation showing the skin friction for 119877 = 1119883 = 1

120593

119862119891

CuO-water(series)

119862119891

CuO-water(numerical)

119862119891

Al2O3-water(series)

119862119891

Al2O3-water(numerical)

000 4085563 40855627 4085563 40855627001 4186948 41869479 4189890 41898897005 4630877 46308766 4645589 46455887010 5287037 52870375 5316469 53164695015 6084673 60846727 6128833 61288327020 7065422 70654222 7124319 71243186025 8286944 82869436 8360585 83605848030 9830736 98307360 9919130 99191299

5 Results and Discussion

The Couette flow and heat transfer characteristics of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

with viscous heating and convective cooling have beeninvestigated The nonlinear similarity ordinary differentialequations governing the boundary value problemwere solvedboth analytically using the perturbation method coupledwith series improvement technique and numerically usingRunge-Kutta-Fehlberg integration technique coupled withshooting scheme Thermophysical properties of base fluidand nanoparticles are presented in Table 1 For pure water thePrandtl number is taken as Pr = 62 [12 16 17] and momen-tum diffusivity is dominant within the fluid in comparisonto pure conduction The nanoparticle volume fraction in thebase fluid is taken as 120593 = 0 to 03 (ie ranging from 0 to 30percent) while the case of120593 = 0 corresponds to the absence ofnanoparticle in the based fluid (water) Numerical solutionsare displayed in Tables 2 and 3 together with Figures 2ndash17In Tables 2 and 3 the numerical values of the skin frictionand the Nusselt number are displayed when 119877 = 1 for boththe series solution and the numerical solutionwith increasingnanoparticle volume fraction It is noteworthy that perfectagreement is achieved between the improved series solutionand the numerical solution Moreover it is observed thatboth the skin friction and the Nusselt number increase withincreasing nanoparticles volume fraction Interestingly theskin friction produced by Al

2O3-water nanofluid is higher

than that of CuO-water nanofluid while Nusselt numberproduced by CuO-water nanofluid is higher than that ofAl2O3-water nanofluid

Table 3 Computation showing the Nusselt number for Bi = 1 119877 =1 Ec = 01 and119883 = 1

120593

NuCuO-water(series)

NuCuO-water(numerical)

NuAl2O3-water

(series)

NuAl2O3-water(numerical)

000 0315380 03153802 0315380 03153802001 0326419 03264199 0326493 03264928005 0376286 03762861 0376626 03766263010 0452718 04527175 0453179 04531794015 0547443 05474432 0547555 05475550020 0664319 06643189 0663318 06633184025 0808605 08086051 0805338 08053384030 0987501 09875014 0980249 09802487

08

08

06

06

04

04

02

02

0

0

1

1

CuO-waterAl2O3-water

minus02

minus04

WV

W

V

120578

120593 = 02 R = 30 X = 1

Figure 2 Nanofluids velocity profiles

51 Velocity Profiles with Parameter Variation Figures 2ndash4illustrate the effects of parameter variation on both the axialand normal velocity profiles Generally flow reversal isobserved in the region near the lower fixed plate as repre-sented by negative value of the axial velocity (119882) while thefluid near the upper accelerating plate moves faster forwardin the axial direction Also the normal velocity profiles (119881)are skewed towards the upper plate due to the drag forceexerted on the fluid by the upper plate acceleration Figure 2shows that the intensity of flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate produced by CuO-water are higher thanthat of Al

2O3-water under the same parametric conditions

With CuO-water as the working nanofluid it is observedthat increasing nanoparticles volume fraction concentrationfrom 0 to 30 increases the flow reversal near the lower fixedplate and the skewness of the normal velocity towards theupper moving plate as shown in Figure 3 With increasingacceleration of the upper plate (ie 119877 increasing) a decrease

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 7

CuO-water

V

W

080604020 1120578

120593 = 0 01 02 03R = 30 X = 1

08

06

04

02

0

1

minus02

minus04

WV

Figure 3 Axial velocity profiles with increasing 120593

R = 1 10 20 30

V

WCuO-water08

06

04

02

0

1

minus02

minus04

WV

080604020 1120578

120593 = 02 X = 1

Figure 4 Axial velocity profiles with increasing 119877

in the flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper plateis observed as depicted in Figure 4 This decrease in flowreversal can be attributed to an increase in the drag forceexerted on the fluid by the upper plate The intersection ofthe velocities 119882 and 119881 at a region near the upper movingplate as shown in Figures 2 3 and 4 indicates that both theaxial and the normal velocities of nanofluids at this point(ie between eta = 075 and 078) are the same This impliesthat the axial acceleration of the upper plate may not haveany noticeable influence on both 119882 and 119881 velocities at thisintersecting region

52 Temperature Profiles with Parameter Variation The tem-perature profiles of the nanofluids with different parametervariation are displayed in Figures 5ndash10 Generally the fluidtemperature increases within the channel and decreases nearthe uppermoving plate due to convective heat loss to ambient

Ec = 01 Bi = 1

08

08

06

06

04

04

02

02

00

1

1

12

14

16

120578

CuO-waterAl2O3-water

120593 = 02 R = 5 X = 1Φ

Figure 5 Nanofluids temperature profiles

CuO-water

R = 2 X = 1

080604020 1120578

120593 = 0 01 02 03

Ec = 01 Bi = 1

08

06

04

02

0

1

12

14

16

18

Φ

Figure 6 Temperature profiles with increasing 120593

surrounding It is noteworthy that the temperature producedby CuO-water nanofluid is generally higher than that ofAl2O3-water nanofluid under the same flow condition as

shown in Figure 5WithCuO-water as theworking nanofluidthe temperature increases with a rise in nanoparticles volumefraction from 0 to 30 as illustrated in Figure 6 Similar effectof an increase in nanofluid temperature is observed in Figures7 and 8 with an increase in Ec due to viscous dissipationand axial distance along the channel This may be attributedto the fact that as Ec increases the internal heat generationwithin the fluid due velocity gradient increases leading to arise in temperatureThe axial increase in temperature may bejustified by the quadratic expression in axial distance for fluidtemperature in (16) (ie Φ(119883 120578) = 119867(120578) + 1198832120579(120578)) Figures9 and 10 show that the nanofluid temperature decreases with

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Computational Mathematics

0

5

10

15CuO-water

Ec = 01 05 07 1

080604020 1120578

Φ

R = 5 X = 1 120593 = 02 Bi = 1

Figure 7 Temperature profiles with increasing Ec

0

2

4

6

8

10

12

14

16

18

X = 1 2 3 4

CuO-water

R = 5 120593 = 02 Bi = 1 Ec = 01

080604020 1120578

Φ

Figure 8 Temperature profiles with increasing119883

Bi = 01 1 5 10

R = 5 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

08

06

04

02

0

1

12

14

16

18

Φ

Figure 9 Temperature profiles with increasing Bi

0

05

15

1

2

25

R = 01 1 2 3

Bi = 1 120593 = 02 X = 1 Ec = 01

080604020 1120578

CuO-water

Φ

Figure 10 Temperature profiles with increasing 119877

0 005 01 02 03015 0255

6

7

8

9

10

11

R = 10 X = 1

CuO-waterAl2O3-water

120593

Cf

Figure 11 Skin friction with increasing 120593

increasing Biot number (Bi) and upper plate acceleration (119877)This is expected since an increase in Biot number indicatesa rise in convective cooling due to heat loss to the ambientsurrounding from the upper plate Moreover an increase inthe upper plate acceleration also enhanced convective heatloss consequently the nanofluid temperature decreases

53 Skin Friction Pressure Gradient and Nusselt NumberFigures 11 and 12 depict the skin friction profiles for bothCuO-water and Al

2O3-water nanofluids at the upper moving

plateThe skin friction generally increases with an increase innanoparticles volume fraction however it is noticed that theskin friction produced by Al

2O3-water is more intense than

the one produced by CuO-water as shown in Figure 11 Thisis expected since the axial velocity gradient of Al

2O3-water

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 9

4

5

6

7

8

9

10

11

12

X = 1

R = 1 10 20 30

CuO-water

Cf

0 005 01 02 03015 025120593

Figure 12 Skin friction with increasing 119877

6

7

8

9

10

11

12

13

14

A

R = 1

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 13 Axial pressure gradient with increasing 120593

at the upper moving plate is higher than that of CuO-waterWith CuO-water as the working nanofluid it is observedthat the skin friction increases with an increase in upperplate acceleration as indicated by increasing values of 119877 inFigure 12 The pressure drop along the channel is illustratedin Figures 13 and 14 For both CuO-water and Al

2O3-water

nanofluids it is seen in Figure 13 that the pressure dropincreases with increasing nanoparticles volume fraction andCuO-water produced higher pressure drop as compared toAl2O3-water With CuO-water as the working nanofluid an

increase in the upper plate acceleration causes a decrease inpressure drop as shown in Figure 14 Figures 15ndash17 illustratethe effects of parameter variation on the rate of heat transferat the moving upper plate For both nanofluids the Nusseltnumber increases with an increase in nanoparticles volume

6

7

8

9

10

11

12

13

14

A

R = 01 1 2 3

CuO-water

0 005 01 02 03015 025120593

Figure 14 Axial pressure gradient with increasing 119877

13

14

15

16

17

18

19

20

21

22

Bi = 1 X = 1 Ec = 1 R = 01

0 005 01 02 03015 025120593

CuO-waterAl2O3-water

Figure 15 Nusselt number with increasing 120593

fraction and the Nusselt number produced by CuO-waternanofluid is higher than that of Al

2O3-water nanofluid (see

Figure 15) In Figures 16 and 17 it is observed that thestrength of the upper plate heat transfer rate is enhanced withincreasing axial distance viscous dissipation and convectivecooling This may be attributed to a rise in the temperaturegradient due to convective heat exchange with the ambientalong the upper plate However a decrease in the Nusseltnumber is observed with increasing upper plate accelerationThe closeness of the profiles for the Nusselt number valuesin Figure 15 with small increase in nanoparticles volumefraction shows that at very small values of nanoparticlesvolume fraction (120593 lt 012) the rate of heat transferacross the upper moving plate surface for both nanofluidsis the same however as the nanoparticles volume fraction

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Computational Mathematics

0

5

10

15

20

25

30

35

R = 01 1 2Bi = 3 5 10

120593 = 02 X = 1

080604020 1

CuO-water

minus5

Ec

Figure 16 Nusselt number with increasing Ec 119877 and Bi

0

10

20

30

40

50

60

70

80

90

100

X = 1 2 3 4120593 = 02 R = 2 Bi = 1

080604020 1Ec

CuO-water

Figure 17 Nusselt number with increasing119883

increases the CuO-water produces higher heat transfer rateas compared to Al

2O3-water

6 Conclusions

We analysed the combined effects of viscous dissipation andconvective cooling on Couette flow and heat transfer of waterbase nanofluids containing CuO and Al

2O3as nanoparticles

The nonlinear boundary value problem is solved usingboth perturbation series improvement method and Runge-Kutta-Fehlberg integration numerical technique coupledwith shooting scheme The summary of essential features ofphysical interest from the above analysis is given below

The flow reversal near the lower fixed plate and theskewness of the normal velocity towards the upper moving

plate produced by CuO-water are higher than that of Al2O3-

water which increases with 120593 and decreases with 119877 CuO-water produced higher temperature as compared to Al

2O3-

water The temperature increases with 120593 Ec and 119883 butdecreases with Bi and 119877 The coefficient of skin frictionincreases with 119877 and 120593 while Nusselt number increaseswith increasing 120593 Ec Bi and 119883 and decreases with 119877Moreover Al

2O3-water produced higher skin friction while

CuO-water produced higher Nusselt number The pressuredrop produced by CuO-water is higher than that of Al

2O3-

water and it is enhanced by 120593 but decreases with 119877

Appendix

Consider

1198661= 3 (1 + 119898

4Bi) 1205783 minus 4 (1 + 119898

4Bi) 1205782

+ 2 (1 + 1198984Bi) 120578 minus 4 minus 119898

4Bi

1198662= (54119898

1+ 45119898

2

4Bi21198982Pr+ 541198982

4Bi1198981+ 108119898

4Bi1198981

+ 901198982Pr1198984Bi + 45119898

2Pr) 1205787

+ (minus1441198982

4Bi21198981minus 240119898

2Pr1198984Bi minus 1201198982

4Bi21198982Pr

minus 1441198981minus 120119898

2Pr minus 288119898

4Bi1198981) 1205786

+ (1681198981+ 168119898

2

4Bi21198982Pr + 168119898

2Pr

+ 1681198982

4Bi21198981+ 336119898

4Bi1198981+ 336119898

2Pr1198984Bi) 1205785

+ (minus1891198982

4Bi2119898119890Pr minus 112119898

4Bi1198981minus 693119898

119890Pr1198984Bi

minus 561198981minus 56119898

2

4Bi21198981minus 504119898

119890Pr) 1205784

+ (1051198982

4Bi2119898119890Pr + 420119898

2Pr 525119898

2Pr1198984Bi

minus 1921198984Bi1198981minus 96119898

1minus 96119898

2

4Bi21198981) 1205783

+ (2081198984Bi1198981+ 104119898

1+ 104119898

2

4Bi21198981) 1205782

+ (minus801198984Bi1198981minus 40119898

1minus 40119898

2

4Bi21198981) 120578

+ 101198984Bi1198981+ 10119898

2

4Bi21198981minus 91198982

4Bi21198982Pr +312119898

2Pr

minus 121198982Pr1198984Bi

1198663= 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205785

+ 61198983Ec Pr (minus2119898

4Bi minus 1198982

4Bi2) 1205784

+ 31198983Ec Pr (1 + 1198982

4Bi2 + 2119898

4Bi) 1205783

+ 1198983Ec Pr (4 + 1198982

4Bi2 + 5119898

4Bi) 1205782

minus 31198982

4Bi2 minus 12119898

3Ec Prminus3119898

4Bi minus 119898

3Ec Pr1198982

4Bi2

minus 71198983Ec Pr119898

4Bi

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Computational Mathematics 11

119873 = 631198982

4Bi21198982minus 10119898

3Ec1198984Bi1198981

+ 8181198983Ec1198982Pr1198984Bi + 52119898

3Ec11989824Bi21198982Pr

+ 101198983

4Bi31198983Ec1198981minus 10119898

3

4Bi31198983Ec1198982Pr

+ 631198983

4Bi31198982

119872 = 101198982

4Bi21198981+ 10119898

41198981Bi minus 321119898

2Pr1198984Bi

minus 61198982

4Bi21198982Pr

(A1)

Nomenclature

(119906 V) Velocity components(119909 119910) Coordinates119896nf Nanofluid thermal conductivityPr Prandtl numberBi Local Biot number119879119886 Ambient temperature

119865 Dimensionless stream function1198790 Lower wall temperature

119879 Temperature119867 Dimensionless temperature119877 Reynolds number119888119901 Specific heat at constant pressure

119896119904 Solid fraction thermal conductivity

119896119891 Base fluid thermal conductivity

119882 Dimensionless axial velocity119881 Dimensionless normal velocityEc Eckert number119864 Upper wall acceleration parameter119883 Dimensionless axial coordinate119860 Axial pressure gradient coefficient119886 Channel widthNu Nusselt number119862119891 Skin friction coefficient

Greek Symbols

120595 Stream function120579 Dimensionless temperature120583nf Nanofluid dynamic viscosity120572nf Nanofluid thermal diffusivity120578 Dimensionless normal coordinate120588nf Nanofluid density120588119904 Solid fraction density

120592119891 Base fluid kinematic viscosity

120583119891 Base fluid dynamic viscosity

120593 Solid volume fraction parameter120588119891 Base fluid density

Ω Vorticity120595 Dimensionless stream functionΩ Dimensionless vorticityΦ Dimensionless temperature

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere thanks to anony-mous reviewers for their valuable suggestions and comments

References

[1] B R Munson D F Young and T H Okiishi Fundamentals ofFluidMechanics JohnWiley amp Sons New York NY USA 2002

[2] R N Jana and N Datta ldquoCouette flow and heat transfer in arotating systemrdquo Acta Mechanica vol 26 no 1ndash4 pp 301ndash3061977

[3] A K Singh ldquoNatural convection in unsteady Couette motionrdquoDefence Science Journal vol 38 no 1 pp 35ndash41 1988

[4] A J Kearsley ldquoA steady state model of couette flow with viscousheatingrdquo International Journal of Engineering Science vol 32 no1 pp 179ndash186 1994

[5] B K Jha ldquoNatural convection in unsteady MHD couette flowrdquoHeat and Mass Transfer vol 37 no 4-5 pp 329ndash331 2001

[6] O D Makinde and O O Onyejekwe ldquoA numerical study ofMHD generalized Couette flow and heat transfer with variableviscosity and electrical conductivityrdquo Journal of Magnetism andMagnetic Materials vol 323 no 22 pp 2757ndash2763 2011

[7] G S Seth M S Ansari and R Nandkeolyar ldquoUnsteady hydro-magnetic couette flow within a porous channelrdquo TamkangJournal of Science and Engineering vol 14 no 1 pp 7ndash14 2011

[8] R K Deka and A Bhattacharya ldquoUnsteady free convectiveCouette flow of heat generationabsorbing fluid in porousmediumrdquo International Journal of Mathematical Archive vol 2no 6 pp 853ndash863 2011

[9] S U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME InternationalMechanical Engineering Congress and Exposition FED 231MD66 pp 99ndash105 ASME San Francisco Calif USA 1995

[10] J Lee and I Mudawar ldquoAssessment of the effectiveness ofnanofluids for single-phase and two-phase heat transfer inmicro-channelsrdquo International Journal of Heat and Mass Trans-fer vol 50 no 3-4 pp 452ndash463 2007

[11] J Niu C Fu and W Tan ldquoSlip-flow and heat transfer of a non-newtonian nanofluid in a microtuberdquo PLoS ONE vol 7 no 5Article ID e37274 2012

[12] T GMotsumi andO DMakinde ldquoEffects of thermal radiationand viscous dissipation on boundary layer flow of nanofluidsover a permeable moving flat platerdquo Physica Scripta vol 86 no4 Article ID 045003 2012

[13] C J Choi S P Jang and S U S Choi ldquoElectrokinetic effects ofcharged nanoparticles in microfluidic Couette flowrdquo Journal ofColloid and Interface Science vol 363 no 1 pp 59ndash63 2011

[14] J C Maxwell Electricity and Magnetism Clarendon PressOxford UK 3rd edition 1904

[15] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquo The Journal of Chemical Physics vol 20 no 4pp 571ndash581 1952

[16] H F Oztop and E Abu-Nada ldquoNumerical study of naturalconvection in partially heated rectangular enclosures filled with

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Journal of Computational Mathematics

nanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[17] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[18] T YNaComputationalMethods in Engineering BoundaryValueProblems Academic Press New York NY USA 1979

[19] O D Makinde ldquoThermal criticality in viscous reactive flowsthrough channels with a sliding wall an exploitation ofthe Hermite-Pade approximation methodrdquo Mathematical andComputer Modelling vol 47 no 3-4 pp 312ndash317 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of