Powder Technology 267 (2014) 256–267

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Powder Technology

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Entropy analysis for an unsteady MHD flow past a stretchingpermeable surface in nano-fluid

Mohammad Hossein Abolbashari a, Navid Freidoonimehr b,⁎, Foad Nazari a, Mohammad Mehdi Rashidi c,d

a Department of Mechanical Engineering, Lean Production Engineering Research Center, Ferdowsi University of Mashhad, P.O. Box 91775-1111, Mashhad, Iranb Young Researchers & Elite Club, Hamedan Branch, Islamic Azad University, Hamedan, Iranc Mechanical Engineering Department, Engineering Faculty of Bu-Ali Sina University, Hamedan, Irand University of Michigan–Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, People's Republic of China

⁎ Corresponding author.E-mail address: nfreidoonimehr@yahoo.com (N. Freid

http://dx.doi.org/10.1016/j.powtec.2014.07.0280032-5910/© 2013 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 9 April 2014Received in revised form 15 July 2014Accepted 17 July 2014Available online 31 July 2014

Keywords:Entropy analysisMHD flowUnsteady boundary-layerNano-fluidStretching surfaceHomotopy simulation

In this article we employ homotopy analysis method (HAM), to study the entropy analysis in an unsteadymagneto-hydrodynamic nano-fluid regime adjacent to an accelerating stretching permeable surface with thewater as the base fluid and four different types of nanoparticles; copper (Cu), copper oxide (CuO), aluminumoxide (Al2O3) and titaniumdioxide (TiO2). The governing partial differential equations are transformed into highlynonlinear coupled ordinary differential equations consisting of themomentumand energy equations via appropri-ate similarity transformations. The current HAM solution demonstrates very good correlation with those of thepreviously published studies in the especial cases. The influences of different flow physical parameters such asthe nanoparticle volume fraction parameter (φ), unsteadiness parameter (A), magnetic parameter (M), suctionparameter (fw), and different types of nanoparticles on the fluid velocity component (f′(η)), the temperaturedistribution (θ(η)), the skin friction coefficient (CfRex1/2), the local Nusselt number (Nux/Rex1/2), and the averagedentropy generation function (NG,av) and also the effects of the Reynolds (Re) number, the Brinkman number (Br)and the Hartmann number (Ha) on the averaged entropy generation function (NG,av) are illustrated graphicallyand discussed in details. This model has important applications in heat transfer enhancement in the renewableenergy systems, industrial thermal management and also materials processing.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Working fluids have great demands placed upon them in terms ofincreasing or decreasing energy release to systems, and their influencesdepend on thermal conductivity, heat capacity and other physical prop-erties in modern thermal and manufacturing processes. A low thermalconductivity is one of the most remarkable parameters that can limitthe heat transfer performance. Suspending the ultrafine solid metallicparticles in technological fluids causes an increase in the thermal con-ductivity. This is one of the most modern and appropriate methods forincreasing the coefficient of heat transfer. It is expected that the ultra-fine solid particle is able to increase the thermal conductivity and heattransfer performance, since the thermal conductivity of solid metals ishigher than that of base fluids. Choi and Eastman [1] were probablythe first to employ a mixture of nanoparticles and base fluid that suchfluids were designated as “nano-fluids”. Experimental studies havedisplayed that with 1–5% volume of solid metallic or metallic oxideparticles, the effective thermal conductivity of the resulting mixturecan be increased by 20% compared to that of the base fluid [2]. A widerange of review papers on nano-fluids and their different applications

oonimehr).

can be found in [3–14]. Hatami et al. [15] displayed asymmetric laminarflow and heat transfer of nano-fluid between contracting rotating disksvia the fourth-order Runge–Kutta–Fehlberg method and least squaremethod (LSM). Their results showed that the temperature profilebecomes more flat near the middle of disks by increasing the injection.Many superior lubricants and thermalworkingfluidsmay be developedfor applications in aerospace, energy systems, medical engineering, etc.A number of theoretical studies have appeared with regard to nano-fluid flows in industrial and medical engineering materials manufac-ture. Rana and Bhargava [16] used the finite element and finite differ-ence methods to study nano-fluid heat and mass transfer past anonlinearly stretching sheet. They concluded that employing differenttypes of nanoparticles can be used effectively for controlling/simulatingthe heat transfer rates in stretching sheet problems. Sheikholeslami andGanji [17] investigated the 3D nano-fluid flow and heat transfer in arotating system in the presence of magnetic field numerically usingthe fourth-order Runge–Kutta method. It can be concluded from theirinvestigations that the skin friction parameter increases with augmentof Reynolds number, rotation parameter and magnetic parameter. Inanother study, Sheikholeslami et al. [18] employed Lattice Boltzmannmethod to study MHD flow utilizing Cu–water nano-fluid in a concen-tric annulus. Their results represented that the enhancement ratioincreases with decrease of Rayleigh number and also it increases with

Nomenclature

constant parameters a, cexternal uniform magnetic field Bconstant magnetic flux density B0skin friction coefficient Cfspecific heat at constant pressure cpself-similar velocity fthermal conductivity kconstant large number mlocal Nusselt number Nuxsurface heat flux qwactual entropy generation rate ˙S ‴

gen

characteristic entropy generation rate S‴0time tfluid temperature Tvelocity component in the x direction uvelocity component in the y direction vuniform suction v0dimensionless axial distance X

Dimensionless parametersunsteadiness parameter (c/a) ABrinkman number (μfUw

2 /kfΔΤ) Brfluid friction irreversibility FFIHartmann number B0 L

ffiffiffiffiffiffiffiffiffiffiffiffiσ=μ f

q� �Ha

Heat Transfer Irreversibility HTIjoule dissipation irreversibility JDIentropy generation number NG

magnetic parameter (σB02/aρf) MPrandtl number (νf(ρ cp)f/kf) PrReynolds number (UwL/νf) Relocal Reynolds number (Uwx/νf) Rexsuction parameter −v0=

ffiffiffiffiffiffiffiffiffiν f a

p� �fw

Greek symbolsdimensionless temperature difference (ΔT/T∞) Ωa scaled boundary-layer coordinate ηself-similar temperature θdynamic viscosity μkinematic viscosity νdensity ρelectrical conductivity σstream function ψskin friction τnanoparticle volume fraction φvolume ϑ

Subscriptsaverage condition avfluid phase fnano-fluid nfsolid phase scondition of the wall wcondition of the free steam ∞

257M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

augment of Hartmann number. Their results illustrated that the Nusseltnumber has a direct relationship with nanoparticle volume fraction andit has a reverse relationship with Hartmann number.

Entropy generation minimization method, as a thermodynamicapproach, is employed to optimize the thermal engineering devicesfor higher energy efficiency [19]. The performance of engineeringequipment is reduced in the presence of the irreversibilities. Entropygeneration function is ameasure of the level of the available irreversibil-ities in a process. It is important to emphasize that the second law ofthermodynamics is more reliable than the first law of thermodynamicanalysis, because of the limitation of the first law efficiency in the heattransfer engineering systems [20]. In recent decades, many researchershave been motivated to conduct the applications of the second law ofthermodynamics in the design of thermal engineering systems. Rashidiet al. [21] studied the first and second law analyses of an electricallyconductingfluid past a rotatingdisk in the presence of a uniformverticalmagnetic field analytically and then applied Artificial Neural Networkand Particle Swarm Optimization algorithm to minimize the entropygeneration. In another study, Rashidi et al. [22] investigated the analysisof the second law of thermodynamics applied to an electricallyconducting incompressible nano-fluid flowing over a porous rotatingdisk. They achieved that the disk surface acts as a strong source ofirreversibility.

Some of strongly nonlinear equations used to describe physicalsystems in the form of mathematical modeling do not have the exactsolutions. The numerical or analytical methods can be applied to solvethese nonlinear equations [23–29]. Despite all the benefits, there aresome disadvantages for the numerical methods in comparison withthe analytical methods. One of the most known and reliable techniquesto solve nonlinear problems is HAM. The HAM was employed by Liao,for the first time, to offer a general analytical method for nonlinearproblems [30–32]. Nowadays, HAM has been employed by researchersin different nonlinear problems. Ellahi et al. [33] investigated the effectsof heat and mass transfer on the Couette and generalized Couette flowin a homogeneous and thermodynamically compatible third gradenon-Newtonian viscous fluid analytically using spectral homotopyanalysis method (SHAM). In another study, Shaban et al. [34] employedHAM to study theMHDsqueezeflowbetween twoparallel infinite diskswhere one disk is impermeable and the other is porous in the presenceof an applied magnetic field. Rashidi et al. [35] employed HAM to studythe nano-fluid flow regime adjacent to a nonlinearly porous stretchingsheet. Further, Rashidi et al. [36] examined free convective heat andmass transfer in a steady 2D MHD fluid flow over a stretching verticalsurface in porous medium.

The current study ismainlymotivated by the need to understand thesecond lawof thermodynamic analysis for an unsteadyMHDnano-fluidover a stretching permeable surfacewith thewater as the base fluid andfour different types of nanoparticles. The HAM is employed to solve theordinary differential equations and also to study the effects of flowphysical parameters on the fluid velocity component as well as temper-ature distribution, and the averaged entropy generation function. Thegraphs are plotted and discussed for the variations of different involvedparameters in details.

2. Flow analysis and mathematical formulation

Consider an unsteadyMHD laminar nano-fluid regime over a porousaccelerating stretching surface in a water based incompressible nano-fluid containing different types of nanoparticles, as shown in Fig. 1. Itis assumed that the basefluid and the nanoparticles are in thermal equi-librium and no slip condition exists between them. In addition, for thetime t b 0, the fluid and heat flows are steady. It is also supposed thatthemagnetic Reynolds number is very small. Therefore, it is conceivableto neglect the induced magnetic field in comparison to the appliedmagnetic field. The unsteady fluid and heat flows begin at t = 0 andthe surface being stretched with the velocity Uw(x, t) along the x-axis.The velocity of themass transfer perpendicular to the stretching surfaceis vw(t). The temperature of the surface Tw(x, t) has a linear variationwith x and an inverse square law for its decrease with time, while the

Stretching permeable surface

Momentum and thermal boundary-layers

Diex, u

y, v

Wall Temperature

Ambient temperature

Wall transpiration (suction/injection) Wind-up Roll

Force

Nano-fluid

L

Magnetic field

T∞

Tw (x,t)

0B

vw (t)

Fig. 1. Configuration of the flow above a stretching sheet and geometrical coordinates.

Table 1Thermo-physical properties of the base fluid and different nanoparticles.

Physical properties Fluid phase (water) Cu CuO Al2O3 TiO2

cp (J/kg K) 4179 385 531.8 765 686.2ρ (kg/m3) 997.1 8933 6320 3970 4250k (W/m K) 0.613 401 76.5 40 8.9538

258 M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

temperature of the ambient fluid is T∞. In this problem, it is also consid-ered that the viscous dissipation is negligibly small.

The basic unsteady 2D conservation ofmass,momentumand thermalenergy equations for the nano-fluids, using the above assumptions andapplying the Boussinesq and boundary-layer approximations, can bewritten in the form of [37,38]:

∂u∂x þ

∂v∂y ¼ 0; ð1Þ

ρnf∂u∂t þ u

∂u∂x þ v

∂u∂y

� �¼ μnf

∂2u∂y2

−σB2u; ð2Þ

ρcp� �

nf

∂T∂t þ u

∂T∂x þ v

∂T∂y

� �¼ knf

∂2T∂y2

; ð3Þ

where u and v are the velocity components in the x and y directions,respectively, t refers to the time, ρnf and μnf are the density and thedynamic viscosity of the nano-fluid, respectively, where μnf has beenproposed by Brinkman [39], T is the nano-fluid temperature, σ is theelectrical conductivity, B ¼ B0=

ffiffiffiffiffiffiffiffiffiffiffiffi1−c t

pis the magnetic field imposed

along the y-axis, (ρcp)nf is the heat capacitance of nano-fluid and knf isthe effective thermal conductivity of nano-fluid [40]. These nano-fluidconstants are defined by:

ρcp� �

nf¼ 1−φð Þ ρcp

� �fþ φ ρcp

� �s; ρnf ¼ 1−φð Þρ f þ φρs;

knfk f

¼ks þ 2kf

� �−2φ kf−ks

� �ks þ 2kf

� �þ φ kf−ks

� � ; μnf ¼μ f

1−φð Þ2:5 ;ð4Þ

where φ is the nanoparticle volume fraction parameter and f and ssubscripts refer to fluid and solid fraction properties, respectively. Itshould be mentioned that the use of the approximation for knf is

restricted to spherical nanoparticles and does not account for othernanoparticle shapes. The thermo-physical properties of the base fluid(water) and different nanoparticles are given in Table 1 [40].

The appropriate boundary conditions are introduced as

u ¼ Uw x; tð Þ ¼ ax1−ctð Þ ; v ¼ vw tð Þ ¼ v0ffiffiffiffiffiffiffiffiffiffiffiffi

1−ctp ; T ¼ Tw x; tð Þ ¼ T∞ þ ax

1−ctð Þ2 ; at y ¼ 0;

u→0; T→T∞; as y→∞;ð5Þ

where a and c are constants (where a N 0 and c ≥ 0, with ct b 1). Thesetwo constants have “time−1” dimension. The following dimensionlessfunctions f(η) and θ(η), and the similarity variables are employed as:

η ¼ aν f 1−ctð Þ

!1=2

y; ψ x; yð Þ ¼ ν f a1−ct

� �1=2� f ηð Þ; θ ηð Þ ¼ T−T∞

Tw−T∞;

ð6Þ

where ψ(x, y) is the free stream function that satisfies the continuityequation (Eq. (1)) with

u ¼ ∂ψ∂y ¼ ax

1−ctf 0 ηð Þ; v ¼ −∂ψ

∂x ¼ −ν f a1−ct

� �1=2f ηð Þ: ð7Þ

h-0.9 -0.75 -0.6 -0.45 -0.3 -0.15 0

-7

-6

-5

-4

-3

-2

-1

Fig. 2. The ℏ-curves of f″(0) and θ′(0) obtained by the 18th order approximation of theHAM solution when φ = 0.1 and A= M = fw = 1.

Table 2Comparison results of − θ′(0) for different values of Prandtl number (Pr) whenM = A = fw = 0 and φ = 0.0.

Pr Ref. [37] Ref. [38] Ref. [43] Ref. [44] Present

0.72 0.8086 0.80868 0.8086 0.8058 0.808631351.00 1.0000 1.00000 1.0000 0.9961 1.000000003.00 1.9237 1.92368 1.9237 1.9144 1.923682597.00 3.0723 3.07224 3.7207 3.7006 3.0722502110.0 3.7207 3.72067 3.7207 – 3.72067390

259M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

Substituting Eqs. (6)–(7) into Eqs. (2)–(3) and (5), the followingordinary differential equations are obtained:

11−φð Þ2:5 f ‴ ηð Þ− 1−φþ φ

ρs

ρ f

! !f 02 ηð Þ− f ηð Þ f ″ ηð ÞþA f 0 ηð Þ þ 1

2η f ″ ηð Þ

� �8<:

9=;−Mf 0 ηð Þ ¼ 0; ð8Þ

1Pr

knf =kf

1−φþ φ ρcp� �

s= ρcp� �

f

� � θ″ ηð Þ þ f ηð Þ θ ηð Þf 0 ηð Þ θ0 ηð Þ����

����−A 2θ ηð Þ þ 12ηθ0 ηð Þ

� �¼ 0: ð9Þ

Res

idua

l Err

or

0 2 4 6 8 10-0.001

-0.0005

0

0.0005

0.001

Fig. 3. The residual error of Eq. (31) obtained by the 18th order approximation of theHAMsolution when φ = 0.1 and A = M = fw = 1.

The transformed boundary conditions become

f ηð Þ ¼ f w; f 0 ηð Þ ¼ 1; θ ηð Þ ¼ 1; at η ¼ 0;

f 0 ηð Þ→0; θ ηð Þ→0; as η→∞;ð10Þ

where primes denote differentiation with respect to η, A = c/a is theunsteadiness parameter, M = σB02/a ρf is the magnetic parameter, Pr =vf/αf is the Prandtl number and f w ¼ −v0=

ffiffiffiffiffiffiffiffiffiν f a

p� �N0 is the suction

parameter.

3. Parameters of engineering interest

3.1. Skin friction coefficient and Nusselt number

The skin friction coefficient Cf and the local Nusselt number Nux arethe physical quantities which are given by

C f ¼τw

ρ f U2w; Nux ¼

xqwkf Tw−T∞ð Þ ; ð11Þ

where τw is the skin friction and qw is the surface heat flux, introduced as

τw ¼ μnf∂u∂y

� �y¼0

; qw ¼ −knf∂T∂y

� �y¼0

: ð12Þ

0 1 2 3 40

0.2

0.4

0.6

0.8

1

A = 0.0A = 0.5A = 1.0A = 2.0A = 3.0

Fig. 4.Effect of unsteadiness parameter on the velocity profilewhenφ=0.1 andM= fw=1.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

A = 0.0A = 0.5A = 1.0A = 2.0A = 3.0

Fig. 5.Effect of unsteadiness parameter on the temperature distributionwhenφ=0.1 andM = fw = 1.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

M = 0M = 1M = 2M = 3M = 4M = 5

Fig. 7. Effect of magnetic parameter on the temperature distribution when φ=0.1 and A=fw = 1.

260 M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

Applying the non-dimensional transformations (6), one obtain

C f Re1=2x ¼ 1

1−φð Þ2:5 f ″ 0ð Þ; Nux=Re1=2x ¼ −

knfk f

θ0 0ð Þ; ð13Þ

where Rex = Uwx/νf is the local Reynolds number.

3.2. Entropy generation analysis

According to [41,42], the volumetric rate of the local entropy gener-ation of the nano-fluid, in the presence of the magnetic field over the

0 1 2 3 40

0.2

0.4

0.6

0.8

1

M = 0M = 1M = 2M = 3M = 4M = 5

Fig. 6. Effect of magnetic parameter on the velocity profile when φ=0.1 and A= fw =1.

stretching surface and by applying the boundary layer approximations,can be described as

˙S ‴gen ¼ knf

T2∞

∂T∂x

� �2

þ ∂T∂y

� �2" #|{z}

Thermal irreversibility

þ μnf

T∞

∂u∂y

� �2

|{z}Fluid frictionirreversibility

þ σB20

T∞u2

|{z}joule dissipationirreversibility

: ð14Þ

The above equation reveals that the entropy generation is due to threeeffects: the first effect, a conduction effect, is the entropy generation due

0 1 2 3 40

0.2

0.4

0.6

0.8

1

ϕ = 0.00ϕ = 0.05ϕ = 0.10ϕ = 0.15ϕ = 0.20

Fig. 8. Effect of nanoparticle volume fraction parameter on the velocity profile when A =M = fw = 1.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

ϕ = 0.20ϕ = 0.15ϕ = 0.10ϕ = 0.05ϕ = 0.00

Fig. 9. Effect of nanoparticle volume fraction parameter on the temperature distributionwhen A = M = fw = 1.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

fw = 0.0fw = 0.5fw = 1.0fw = 1.5fw = 2.0

Fig. 11. Effect of suction parameter on the temperature distributionwhenφ=0.1 andA=M = 1.

261M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

to heat transfer across a finite temperature difference; the second effect isdue to fluid friction irreversibility and the last effect denotes themagneticeffects in the form of joule dissipation irreversibility, caused by themovement of electrically conducting fluid under the magnetic field. Theentropy generation number, dimensionless form of entropy generationrate, represents the ratio between the actual entropy generation rate˙S ‴gen

� and characteristic entropy generation rate ˙S‴

0�

. The similaritytransformation parameters of Eq. (6) are employed to non-

0 1 2 3 40

0.2

0.4

0.6

0.8

1

fw = 0.0fw = 0.5fw = 1.0fw = 1.5fw = 2.0

Fig. 10. Effect of suction parameter on the velocity profile when φ= 0.1 and A=M= 1.

dimensionalize the local entropy generation. Thus the entropygeneration number (NG) becomes:

NG ¼ S‴genS‴0

¼ knfk f

1X2 θ ηð Þ½ �2 þ ReL θ0 ηð Þ �2� �

þ μnf

μ f

Br ReLΩ

f ″ ηð Þh i2 þ Br Ha2

Ωf 0 ηð Þ �2

¼ks þ 2kf

� �−2φ kf−ks

� �ks þ 2kf

� �þ φ kf−ks

� � 1X

θ ηð Þ½ �2 þ ReL θ0 ηð Þ �2� �þ 1

1−φð Þ2:5Br ReL

Ωf ″ ηð Þh i2

þ Br Ha2

Ωf 0 ηð Þ �2

;

ð15Þ

Cf R

e x1/

2

0 0.05 0.1 0.15 0.2-5

-4.5

-4

-3.5

-3

-2.5

-2

CuCuOAl2O3TiO2

Fig. 12. Variation of the skin friction coefficient with respect to nanoparticle volumefraction parameter for different types of nanoparticles when A = M = fw = 1.

Nu x

/Re x

1/2

0 0.05 0.1 0.15 0.28.4

8.6

8.8

9

9.2

9.4

CuCuOAl2O3TiO2

Fig. 13. Variation of the Nusselt number with respect to nanoparticle volume fraction pa-rameter for different types of nanoparticles when A= M = fw = 1.

NG

, av

0 0.05 0.1 0.15 0.23.5

4.5

5.5

6.5

7.5

8.5

A = 0A = 1A = 2A = 3

Fig. 14. Change of NG,avwith respect to nanoparticle volume fraction parameter for differ-ent values of unsteadiness parameter when M = fw = Ha= 1 and ReL = Br= 5.

262 M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

whereS‴0 ¼ kf Δ Tð Þ2=L2 T2∞ is the characteristic entropy generation rate,

X= x/L is the dimensionless axial distance, Re= UwL/νf is the Reynoldsnumber, Br= μ fUw

2 /kfΔT is the Brinkman number, which represents theratio of direct heat conduction from the surface to the viscous heat

generated by shear in the boundary-layer, Ha ¼ B0Lffiffiffiffiffiffiffiffiffiffiffiffiσ=μ f

qis the

Hartmann number, and Ω= ΔT/T∞ is the dimensionless temperaturedifference.

The dimensionless volumetric entropy generation rate, an importantmeasure of total global entropy generation, can be evaluated using thefollowing formula

NG;av ¼1ϑ

Z m

0

Z 1

0NGdXdη; ð16Þ

where ϑ is the considered volume. In order to consider both velocityand thermal boundary-layers, we calculate the volumetric entropygeneration in a large finite domain. Thus, integration of Eq. (16) isobtained in the domain 0≤ X≤ 1 and 0≤ η≤m, wherem is a sufficientlylarge number.

4. HAM solution

We choose the initial approximations such that the boundary condi-tions are satisfied, i.e.:

f 0 ηð Þ ¼ −e−η þ f w þ 1; ð17Þ

Table 3Values of − F″(0) and − θ′(0) for different types of nanoparticles and φ when A = M = fw =

φ Cu CuO

− F″(0) − θ′(0) − F″(0) − θ′(0)

0.00 2.20855243 8.46448124 2.20855243 8.464481240.05 2.42979090 7.46742270 2.30123963 7.486817950.10 2.56567741 6.63060007 2.34086571 6.663514200.15 2.62804243 5.91791249 2.33439501 5.960171460.20 2.62738868 5.30323726 2.28830358 5.35182786

θ0 ηð Þ ¼ e−η: ð18Þ

The linear operators Lf ( f) and Lθ(θ) are introduced as:

L f fð Þ ¼ ∂3 f∂η3

þ ∂2 f∂η2

; ð19Þ

Lθ θð Þ ¼ ∂2θ∂η2

þ ∂θ∂η ; ð20Þ

with the following properties:

L f c1 þ c2ηþ c3e−η� ¼ 0; ð21Þ

Lθ c4 þ c5e−η� ¼ 0; ð22Þ

where ci, i=1− 5 are the arbitrary constants. The nonlinear operators,according to Eqs. (8) and (9), are defined as:

N f f η; qð Þh i

¼ 11−φð Þ2:5

∂3 f η; qð Þ∂η3

− 1−φþ φρs

ρ f

! !

×

∂ f η; qð Þ∂η

!2

− f η; qð Þ ∂2 f η; qð Þ∂η2

þA∂ f η; qð Þ

∂η þ 12η∂2 f η; qð Þ

∂η2

!0BBBB@

1CCCCA−M

∂ f η; qð Þ∂η ;

ð23Þ

1.

Al2O3 TiO2

− F″(0) − θ′(0) − F″(0) − θ′(0)

2.20855243 8.46448124 2.20855243 8.464481242.18454535 7.48944589 2.19850791 7.596801092.13531201 6.66685664 2.15999731 6.849917252.06422048 5.96310009 2.09677766 6.200294471.97448761 5.35371288 2.01243078 5.63010380

NG

, av

0 0.05 0.1 0.15 0.23.5

4.5

5.5

6.5

7.5

8.5

M = 0M = 1M = 2M = 3M = 4M = 5

Fig. 15. Change of NG,av with respect to nanoparticle volume fraction parameter fordifferent values of magnetic parameter when A = fw = Ha= 1 and ReL = Br = 5.

NG

, av

0 0.05 0.1 0.15 0.20

6

12

18

24

30

ReL = 1ReL = 3ReL = 5ReL = 10ReL = 15ReL = 20

Fig. 17. Change of NG,av with respect to nanoparticle volume fraction parameter fordifferent values of Reynolds number when A= M = fw = Ha= 1 and Br= 5.

263M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

N θ f η; qð Þ; θ η; qð Þh i

¼ 1Pr

knf =kf

1−φþ φ ρcp� �

s= ρcp� �

f

� � ∂2θ η; qð Þ∂η2

þ f η; qð Þ ∂θ η; qð Þ∂η −∂ f η; qð Þ

∂η θ η; qð Þ !

−A 2θ η; qð Þ þ 12η∂θ η; qð Þ

∂η

!:

ð24ÞThe auxiliary functions become:

Hf ηð Þ ¼ Hθ ηð Þ ¼ e−η: ð25Þ

NG

, av

0 0.05 0.1 0.15 0.22

4

6

8

10

fw = 0.0fw = 0.5fw = 1.0fw = 1.5fw = 2.0

Fig. 16. Change of NG,av with respect to nanoparticle volume fraction parameter fordifferent values of suction parameter when A = M = Ha= 1 and ReL = Br = 5.

The symbolic software MATHEMATICA is employed to solve the ithorder deformation Eqs. (26)–(27).

L f f i ηð Þ−χi f i−1 ηð Þ½ � ¼ ℏH f ηð ÞRf ;i ηð Þ; ð26Þ

Lθ θi ηð Þ−χiθi−1 ηð Þ½ � ¼ ℏHθ ηð ÞRθ;i ηð Þ; ð27Þ

NG

, av

0 0.05 0.1 0.15 0.22

4

6

8

10

12

Br = 1Br = 2Br = 3Br = 4Br = 5Br = 10

Fig. 18. Change of NG,av with respect to nanoparticle volume fraction parameter fordifferent values of Brinkman number when A= M = fw = Ha = 1 and ReL = 5.

NG

, av

0 0.05 0.1 0.15 0.24

5

6

7

8

9

Ha = 1Ha = 2Ha = 3Ha = 4Ha = 5

Fig. 19. Change of NG,avwith respect to nanoparticle volume fraction parameter for differ-ent values of Hartmann number when A= M = fw = 1 and ReL = Br= 5.

264 M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

where ℏ is the auxiliary nonzero parameter and

Rf ;i ηð Þ ¼ 11−φð Þ2:5

∂3 f i−1 ηð Þ∂η3

− 1−φþ φρs

ρ f

! !

×

Xi−1

j¼0

∂ f j ηð Þ∂η

∂ f i−1− j ηð Þ∂η − f j ηð Þ ∂

2 f i−1− j ηð Þ∂η2

!

þ A∂ f i−1 ηð Þ

∂η þ 12η∂2 f i−1 ηð Þ

∂η2

!0BBBBB@

1CCCCCA−M

∂ f i−1 ηð Þ∂η ;

ð28Þ

NG

, av

0 0.05 0.1 0.15 0.24

4.5

5

5.5

6

6.5

7

CuCuOAl2O3TiO2

Fig. 20. Change of NG,av with respect to nanoparticle volume fraction parameter fordifferent types of nanoparticle when A= M = fw = Ha= 1 and ReL = Br= 5.

Rθ;i ηð Þ ¼ 1Pr

knf =kf� � � �� � ∂2θi−1 ηð Þ∂η2

1−φþ φ ρcp

s= ρcp

f

þXi−1

j¼0

f j ηð Þ ∂θi−1− j ηð Þ∂η −θ j ηð Þ ∂ f i−1− j ηð Þ

∂η

!

−A 2θi−1 ηð Þ þ 12η∂θi−1 ηð Þ

∂η

� �;

ð29Þ

and

χi ¼ 0; i≤1;1; iN1;

nð30Þ

are the involved parameters inHAM theory (formore information aboutthe different steps of HAM, see Refs. [30,31]). To control and enhance theconvergence of the approximation series by the help of the so-called ℏ-curve, it is significant to choose a proper value of auxiliary parameter.The ℏ-curves of f‴(0) and θ′(0) obtained by the 18th order of HAMsolution are shown in Fig. 2. To obtain the optimal values of auxiliaryparameters, the averaged residual errors are defined as:

Resf ¼1

1−φð Þ2:5d3 f ηð Þdη3

− 1−φþ φρs

ρ f

! !

×

d f ηð Þdη

� �2− f ηð Þ d

2 f ηð Þdη2

þAd f ηð Þdη

þ 12ηd2 f ηð Þdη2

!0BBBB@

1CCCCA−M

d f ηð Þdη

;

ð31Þ

Resθ ¼1Pr

knf =kf

1−φþ φ ρcp� �

s= ρcp� �

f

� � d2θ ηð Þdη2

þ f ηð Þdθ ηð Þdη

−d f ηð Þdη

θ ηð Þ� �

−A 2θ ηð Þ þ 12ηdθ ηð Þdη

� �:

ð32Þ

In order to survey the accuracy of the present method, the residualerror for the 18th order of HAM solutions of Eq. (31) is illustrated inFig. 3. In addition, we compare some of our results with the results ofthe previously published studies of Refs. [37,38,43,44]. The comparisonsare depicted in Table 2. A very excellent agreement can be observedbetween them.

5. Results and discussion

The nonlinear ordinary differential Eqs. (8)–(9) subject to the bound-ary conditions (10) are solved via HAM, a semi-analytical/numericalmethod, for some values of the nanoparticle volume fraction parameter(φ), unsteadiness parameter (A), magnetic parameter (M), and suctionparameter (fw). We consider four types of nanoparticles; copper (Cu),copper oxide (CuO), aluminum oxide (Al2O3), and titanium dioxide(TiO2) with water as the base fluid. The copper nanoparticle is used inall figures in this section except those which focus on the influence ofthe type of applied nanoparticles on the engineering parameters suchas the skin friction coefficient, the local Nusselt number and the averagedentropy generation function. For the present investigation, we assumethat the value of the Prandtl number (Pr) is equal to 6.2 (for water). Inaddition, the value of the nanoparticle volume fraction parameter (φ)varies from 0 (regular Newtonian fluid) to 0.2.

Figs. 4–5 display the effect of unsteadiness parameter (A) on thefluid velocity profile and temperature distribution in both steady mo-tion (A = 0) and acceleration case (A N 0). The velocity profile reduces

265M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

for the higher acceleration. This states an accompanying reduction ofthe thickness of themomentumboundary layer. Thismentioned behav-ior changes by crossing away from the vertical surface. This behaviorchanges by crossing away from the vertical surface. This means thatthe velocity boundary-layer thickness becomes thicker for the largeramplitude of unsteadiness parameter. In addition, the temperaturedistribution enhances for the lower acceleration.

Figs. 6–7 illustrate the effect of magnetic parameter (M) on thevelocity profile and temperature distribution. A drag-like force thatcalled Lorentz force is created by the infliction of the vertical magneticfield to the electrically conducting fluid. This force has the tendency toslow down the flow over the stretching surface. According to the aboveexplanation, the velocity boundary-layer thickness gets depressed andthe temperature distribution increases slightly with the increase in themagnetic parameter. It clearly demonstrates that the transversemagnet-icfield opposes the transport phenomena. It is important tomention thatthe large resistances on the fluid particles, which cause heat to begenerated in the fluid, apply as the vertical magnetic field increases.

The effect of the nanoparticle volume fraction parameter on thevelocity profile and temperature distribution is shown in Figs. 8–9.The velocity profile decreases as the value of the nanoparticle volume

a)

b)

c)

d)

N - HTI) ave44%

N - FFI) ave54%

N - JDI) ave2%

N - FFI) ave63%

N - JDI) ave2%

N - HTI) ave34%

N - FFI) ave64%

N - JDI) ave2%

N - FFI) ave63%

N - JDI) ave2%

N - HTI) ave37%

N - FFI) ave61%

N - JDI) ave2%

N - FFI) ave63%

N - JDI) ave2%

N - HTI) ave30%

N - FFI) ave66%

N - JDI) ave4%

N - FFI) ave63%

N - JDI) ave2%

Fig. 21. Contributions of heat transfer,fluid friction and joule dissipation in the averaged entropyfw = Ha=1 and ReL = Br=5. b) Unsteadiness parameter whenM= fw = Ha=1, φ=0.1 and) Suction parameterwhen A=M=Ha=1,φ=0.1 and ReL= Br=5. e) Reynolds numberwHa= 1, φ = 0.1 and ReL = 5. g) Hartmann number when A = M = fw = 1, φ = 0.1 and ReL =

fraction parameter increases. This is because the presence of solid nano-particles leads to further thinning of the velocity boundary-layer thick-ness. In addition, the thermal conductivity enhances and consequentlythe thermal boundary-layer thickness increases, as the nanoparticlevolume fraction parameter increases. This issue is in compliance withthe primary proposes of employing nano-fluids [45]. This also agreeswith the physical behavior, when the volume of nanoparticles enhancesthe thermal conductivity increases, and then the thermal boundarylayer thickness increases.

The effect of suction parameter on the velocity profile and tempera-ture distribution is presented in Figs. 10–11. In the current investigation,the suction parameter has been applied, because the primary assumptionin boundary-layer definition says that the boundary-layer thickness issupposed to be practically very thin. Applying suction at the stretchingsurface causes to draw the amount of the fluid into the surface and conse-quently the hydrodynamic boundary-layer gets thinner and also the ther-mal boundary-layer gets depressed by increasing the suction parameter.

Figs. 12–13 present the results for the skin friction coefficient CfRex1/2

and the local Nusselt numberNux/Rex1/2 for a wide range of the nanopar-ticle volume fraction parameters and four types of nanoparticles. It canbe observed that the value of skin friction coefficient reduces and local

N - HTI) ave35%

N - HTI) ave30%

N - FFI) ave69%

N - JDI) ave1%

N - HTI) ave35%

N - HTI) ave36%

N - FFI) ave63%

N - JDI) ave1%

N - HTI) ave35%

N - HTI) ave29%

N - FFI) ave70%

N - JDI) ave1%

N - HTI) ave35%

N - HTI) ave38%

N - FFI) ave61%

N - JDI) ave1%

generation for different values of a) nanoparticle volume fraction parameterwhen A=M=d ReL = Br=5. c) Magnetic parameter when A= fw = Ha=1, φ=0.1 and ReL = Br=5.hen A=M= fw=Ha=1,φ=0.1 and Br=5. f) Brinkman number when A=M= fw=Br = 5.

e)

f)

g)

N - HTI) ave34%

N - FFI) ave59%

N - JDI) ave7% N - HTI)

ave35%

N - FFI) ave63%

N - JDI) ave2% N - HTI)

ave35%

N - FFI) ave64%

N - JDI) ave1%

N - HTI) ave73%

N - FFI) ave26%

N - JDI) ave1%

N - HTI) ave35%

N - FFI) ave63%

N - JDI) ave2%

N - HTI) ave21%

N - FFI) ave77%

N - JDI) ave2%

N - HTI) ave35%

N - FFI) ave63%

N - JDI) ave2%

N - HTI) ave31%

N - FFI) ave56%

N - JDI) ave13% N - HTI)

ave25%

N - FFI) ave46%

N - JDI) ave29%

Fig. 21 (continued).

266 M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

Nusselt number enhances almost linearly with increasing the nanoparti-cle volume fraction parameter. From Table 1, Cu and Al2O3 have themaximum andminimum densities between the different types of nano-particles which are considered in this article. Therefore, selecting Al2O3

as the nanoparticle leads to the minimum absolute value of skin frictioncoefficient, while choosing the Cu gives the maximum absolute value ofit. Because of the largest thermal conductivity value, Cu has the largestlocal Nusselt number. In addition, it is obvious that the lowest heat trans-fer rate is obtained for the TiO2 nanoparticles due to domination of con-duction mode of heat transfer. As can be observed from Table 1, this isbecause TiO2 has the lowest thermal conductivity compared to othernanoparticles. This behavior is similar to that reported by Oztop andAbu-Nada [40] and Bachok et al. [46]. The results of this figure illustratethat the nanoparticle type is an important factor in the coolingand heating processes. In addition, Table 3 presents the valuesof − F″(0) and − θ′(0) for different types of nanoparticles and nano-particle volume fraction parameters.

Figs. 14–20 demonstrate the results of the averaged entropy genera-tion number as a function of the unsteadiness parameter (A), magneticparameter (M), and suction parameter (fw), Reynolds number (Re),Brinkman number (Br), and Hartmann number (Ha) for a wide rangeof the nanoparticle volume fraction parameter (φ). The results showthat the increasing of the nanoparticle volume fraction parameter,unsteadiness parameter, magnetic parameter, suction parameter, Reyn-olds number, Brinkman number, and Hartmann number cause anincrease of the entropy generation number. It can be concluded fromEq. (15) that the entropy generation produced by the heat transfer andfluid friction irreversibilities and consequently the averaged entropygeneration number increase with increasing of the Reynolds number.An increase in the entropy generation produced by fluid friction and

joule dissipation irreversibilities occurs with increasing the value ofthe Brinkman number. In addition, joule dissipation irreversibility andthe averaged entropy generation number are the increasing functionof the Hartmann number. It is obvious from Fig. 20 that the Cu nanopar-ticle has the largest averaged entropy generation number and Al2O3 hasthe smallest one. The contributions of heat transfer, fluid friction andjoule dissipation irreversibility in the averaged entropy generationnumber for different values of the flow parameters, i.e. nanoparticlevolume fractionparameter, unsteadiness parameter,magnetic parameter,suction parameter, Reynolds number, Brinkman number, and Hartmannnumber are shown in Figs. 21(a)–(g).

6. Conclusion

In the current perusal, a mathematical formulation has been derivedfor the entropy analysis for an unsteady MHD nano-fluid over an accel-erating stretching permeable surface. We have considered the water asthe base fluid and four different types of nanoparticles; copper, copperoxide, aluminum oxide and titanium dioxide. HAM is successfully ap-plied to solve the system of ordinary differential equations. The presentsemi-numerical/analytical simulations agree closely with the previousstudies in the especial cases. The averaged entropy generation numberis derived as a function of the velocity and temperature gradients. Theinfluences of the seven key thermo-physical parameters governing theflow i.e. unsteadiness parameter, magnetic parameter, nanoparticlevolume fraction parameter, suction parameter, Reynolds number,Brinkman number, and Hartmann number on the longitudinal veloc-ity and temperature distributions, skin friction coefficient, averagedentropy generation function, and local Nusselt number have beenpresented graphically and interpreted in details. The main goal of the

267M.H. Abolbashari et al. / Powder Technology 267 (2014) 256–267

second law of thermodynamics that constitute minimizing the entropygeneration is reached by decreasing all the involved thermo-physicalparameters. Because of the great important role of the second law ofthermodynamics in the thermal engineering equipment, one can studythe first and second laws of thermodynamics for various geometries,assumptions and different boundary conditions, as a suggestion forfurther investigations in these similar research areas.

Acknowledgment

The authorswish to express their appreciation to the ResearchDeputyof Ferdowsi University of Mashhad for supporting this project throughgrant no. 26954.

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