a study for mhd boundary layer flow of variable viscosity ......a study for mhd boundary layer flow...

Appl. Math. Inf. Sci.9, No. 3, 1327-1338 (2015) 1327

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/090327

A Study for MHD Boundary Layer Flow of VariableViscosity over a Heated Stretching Sheet via Lie-GroupMethodHossam S. Hassan1,∗, Samar A. Mahrous1, A. Sharara2 and A. Hassan2

1 Department of Basic and Applied Science, College of Engineering and Technology, Arab Academy for Science, Technology andMaritime Transport, P.O. BOX 1029 Alexandria, Egypt

2 Department of Marine Engineering, College of Engineering and Technology, Arab Academy for Science, Technology and MaritimeTransport, P. O. BOX 1029 Alexandria, Egypt

Received: 28 Jul. 2014, Revised: 29 Oct. 2014, Accepted: 30 Oct. 2014Published online: 1 May 2015

Abstract: The present work deals with the study of Magnetohydrodynamic (MHD) boundary layer flow over a heated stretching sheetwith variable fluid viscosity. The fluid viscosity is assumedto vary as a linear function of temperature in the presence ofuniformtransverse magnetic field. The fluid is assumed to be electrically conducting. Lie-group method is applied for determining symmetryreductions for the MHD boundary-layer equations. Lie-group method starts out with a general infinitesimal group of transformationsunder which the given partial differential equations are invariant. The determining equations are a set of linear differential equations,the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the grouphas been determined, a solution to the given partial differential equations may be found from the invariant surface condition such thatits solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the Hartmannnumber (M ), the viscosity parameter (A ) and the Prandtl number (Pr ) on the horizontal and vertical velocities, temperature profiles,wall heat transfer and the wall shear stress (skin friction), have been studied and the results are plotted.

Keywords: Lie-group; Similarity solutions; MHD flow; Stretching sheet; Tempreture-dependent fluid viscosityMSC: 76W05; 76M60; 35Q30

1 Introduction

Flow and heat transfer of an incompressible viscous fluidover a stretching sheet appear in several industrialprocesses such as the extrusion of polymers, the coolingof metallic plates, the aerodynamic extrusion of plasticsheets, etc. In the glass industry, blowing, floating orspinning of fibers are processes, which involve the flowdue to a stretching surface, [1]. The study of heat transferand flow field is necessary for determining the quality ofthe final products of such processes.

Sakiadis [2] presented the pioneering work in thisfield. He investigated the flow induced by a semi-infinitehorizontally moving wall in an ambient fluid.

Crane [3] studied the flow over a linearly stretchingsheet in an ambient fluid and gave a similarity solutionin closed analytical form for the steady two-dimensional

problem. He presented a closed form exponential solutionfor the planar viscous flow of linear stretching case.

Moreover, the study of Magnetohydrodynamic(MHD) flow of an electrically conducting fluid is ofconsiderable interest in modern metallurgical andmetal-working processes. There has been a great interestin the study of magnetohydrodynamic flow and heattransfer in any medium due to the effect of magnetic fieldon the boundary layer flow control and on theperformance of many systems using electricallyconducting fluids, [4].

The problem of flow, heat and mass transfer over astretching sheet in the presence of suction or blowing wasexamined by Gupta and Gupta [5]. Dutta and Gupta [6]extended the pioneering works of Crane [3] to explorevarious aspects of the flow and heat transfer occurring in

∗ Corresponding author e-mail:[email protected]

c© 2015 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.12785/amis/090327

1328 H. S. Hassan et. al. : A Study for MHD Boundary Layer Flow of Variable ...

an infinite domain of the fluid surrounding the stretchingsheet.

The fluid viscosity was assumed uniform in the flowregion in all the pervious mentioned works. Practically,the coefficient of viscosity decreases in case of liquidswhereas it increases in case of gases as the temperatureincreases.

Pop et al. [7] and Pantokratoras [8] studied the effect ofvariable viscosity on flow and heat transfer to a continuousmoving flat plate. In their work, they assumed that the fluidviscosity varies as an inverse function of temperature.

Abel et al. [9] studied the boundary layer flow andheat transfer of a visco-elastic fluid immersed in a porousover a stretching sheet with variable fluid viscosity. Thefluid viscosity is assumed to vary as an inverse function oftemperature. A numerical shooting algorithm withfourth-order Runge-Kutta integration scheme has beenused to solve the coupled nonlinear boundary valueproblem.

Mukhopadhyay et al. [4] studied the MHD boundarylayer flow over a heated stretching sheet with variableviscosity. The fluid viscosity is assumed to vary as alinear function of temperature. The scaling group oftransformations is applied to the governing equations.The resulting system of non-linear ordinary differentialequations is solved numerically.

Pantokratoras [10] critiqued the work ofMukhopadhyay et al. [4]. He concluded that, in the workof Mukhopadhyay et al. [4], the calculation domain wassmall and the temperature profiles are truncated. Theresults of his work are obtained with the direct numericalsolution of the boundary layer equations taking intoaccount both viscosity and Prandtl number variationacross the boundary layer. The temperature profiles of hiswork are quite different from those of Mukhopadhyay etal. [4].

This paper is concerned with the study of MHDboundary layer flow over a heated stretching sheet withvariable fluid viscosity using Lie-group method.Following Batchelor [11], the fluid viscosity is assumedto vary as a linear function of temperature in the presenceof uniform transverse magnetic field. The fluid is assumedto be electrically conducting. Lie-group theory is appliedto the equations of motion and energy for determiningsymmetry reductions of partial differentialequations, [12–25]. The resulting system of non-lineardifferential equations is then solved numerically usingshooting method coupled with Runge-Kutta scheme. Theobtained results are compared with those ofMukhopadhyay et al. [4], Pantokratoras [10], Chiam [26],Carragher and Crane [27], Grubka and Bobba [28].

2 Mathematical Formulation of the Problem

Consider a steady, two-dimensional flow of a viscous andincompressible electrically conducting fluid over a heatedstretching sheet placed in the region ¯y > 0 of a Cartesian

Fig. 1: Physical model and coordinate system.

system of coordinatesOxy. The stretching surface has auniform temperatureTw and the free stream temperature isT∞ with Tw > T∞. The wall is stretched by applying twoequal and opposite forces along the ¯x− axis, to keep theorigin fixed. A uniform magnetic field of strengthB0 isassumed to be imposed along the ¯y− axis. The viscosityof the fluid is assumed to be temperature-dependent.

Under the above assumptions, the resulting boundary-layer equations are given by:

Continuity Equation:

∂ u∂ x

+∂ v∂ y

= 0, (2.1)

Momentum Equation:

u∂ u∂ x

+ v∂ u∂ y

=1ρ

∂ µ∂ T

∂ T∂ y

∂ u∂ y

+µρ

∂ 2 u∂ y2 − σ B0

2

ρu,

(2.2)Energy Equation:

u∂ T∂ x

+ v∂ T∂ y

= α∂ 2T∂ y2 , (2.3)

and with the following boundary conditions,

(i) u=Cx, v= 0, T = Tw at y= 0, (2.4)

(ii) u→ 0, T → T∞ as y→ ∞,

whereu andv, are the velocity components in the ¯x andydirections, respectively,T is the temperature andµ is thetemperature dependent viscosity of the fluid. Moreover,ρis the fluid density,σ is the electrical conductivity of thefluid, B0 is the strength of uniform magnetic field,α isthe coefficient of the thermal diffusivity,C is a constant,is a constant,Tw is the wall temperature andT∞ is the freestream temperature.

Follow Batchelor [11], the temperature dependentviscosity is assume to be in the form

µ = µ∗ (a+b (Tw− T)) , (2.5)


Appl. Math. Inf. Sci.9, No. 3, 1327-1338 (2015) /www.naturalspublishing.com/Journals.asp 1329

whereµ∗ is the reference viscosity, and are constants.The variables in equations (2.1) - (2.4) are

dimensionless according to

x=CxU1

, y=

√

Cν

y, u=u

U1, v=

v√Cν

, T =T −T∞

Tw−T∞,

(2.6)whereU1 is the characteristic velocity andν = µ∗/ρ is thekinematic viscosity.

Substitution from (2.5)-(2.6) into (2.1) - (2.3), gives

∂ u∂ x

+∂ v∂ y

= 0, (2.7)

u∂ u∂ x

+ v∂ u∂ y

=−A∂ T∂ y

∂ u∂ y

+(a+A(1−T))∂ 2u∂ y2 −M2u,

(2.8)

u∂ T∂ x

+ v∂ T∂ y

=1Pr

∂ 2T∂ y2 , (2.9)

where,M2 =σB2

0ρ C , M is the Hartmann number (constant),

A = b(Tw−T∞) is the viscosity parameter (constant) andPr= ν

α is the Prandtl number.The boundary conditions (2.4) will be considered as

follows,

(i) u= x, v= 0, T = 1 at y= 0,

(ii) u→ 0, T → 0 as y→ ∞. (2.10)

From the continuity equation (2.7), a stream functionΨ (x,y) may exist as,

u(x,y) =∂ Ψ∂ y

, v(x,y) =−∂ Ψ∂ x

, (2.11)

which satisfies equation (2.7) identically.Substituting from (2.11) into (2.8)-(2.9) yields

ΨyΨxy−ΨxΨyy+A TyΨyy−(a+A(1−T))Ψyyy+M2Ψy= 0,(2.12)

and

ΨyTx−ΨxTy−1Pr

Tyy = 0, (2.13)

where subscripts denote partial derivatives.The boundary conditions (2.10) will be as follows,

(i) Ψy = x, Ψx = 0, T = 1 at y= 0,

(ii) Ψy → 0, T → 0 as y→ ∞. (2.14)

3 Solution of the Problem

Firstly, we derive the similarity solutions using Lie-groupmethod under which (2.12)-(2.13) and the boundaryconditions (2.14) are invariant, and then we use thesesymmetries to determine the similarity variables.

3.1 Lie Point Symmetries

Consider the one-parameter(ε) Lie group of infinitesimaltransformations in(x,y;Ψ ,T) given by

x∗ = x+ ε ϕ (x,y;Ψ ,T)+O(ε2),y∗ = y+ ε ζ (x,y;Ψ ,T)+O(ε2),Ψ∗ =Ψ + ε η (x,y;Ψ ,T)+O(ε2),T∗ = T + ε F (x,y;Ψ ,T)+O(ε2),

(3.1)

where “ε ” is a small parameter.A system of partial differential equations (2.12)-(2.13)

is said to admit a symmetry generated by the vector field

X ≡ ϕ∂

∂ x+ ζ

∂∂ y

+η∂

∂ Ψ+F

∂∂ T

, (3.2)

if it is left invariant by the transformation(x,y;Ψ ,T)→ (x∗,y∗;Ψ∗,T∗).

The solutionsΨ = Ψ (x,y) and T = T (x,y) , areinvariant under the symmetry (3.2) if

ΦΨ = X (Ψ −Ψ(x,y)) = 0 whenΨ =Ψ (x,y), (3.3)

and

ΦT = X (T −T (x,y)) = 0whenT = T (x,y). (3.4)

Assume,

∆1 =ΨyΨxy−ΨxΨyy+A TyΨyy− (a+A(1−T))Ψyyy+M2 Ψy,(3.5)

∆2 =ΨyTx−ΨxTy−1Pr

Tyy. (3.6)

A vector X given by (3.2), is said to be a Lie pointsymmetry vector field for (2.12)-(2.13) if

X[3] (∆ j)∣

∣

∆ j=0 = 0, j = 1,2, (3.7)

where,

X[3] ≡ ϕ ∂∂ x + ζ ∂

∂ y +η ∂∂ Ψ +F ∂

∂ T +ηx ∂∂ Ψx

+ηy ∂∂ Ψy

+Fx ∂∂ Tx

+Fy ∂∂ Ty

+ηxy ∂∂ Ψxy

+ηyy ∂∂ Ψyy

+Fyy ∂∂ Tyy

+ηyyy ∂∂ Ψyyy

,

(3.8)is the third prolongation ofX.

To calculate the prolongation of the giventransformation, we need to differentiate (3.1) with respectto each of the variables, and . To do this, we introduce thefollowing total derivatives

Dx ≡ ∂x+Ψx∂Ψ +Tx∂T +Ψxx∂Ψx +Txx∂Tx +Ψxy∂Ψy + ........,Dy ≡ ∂y+Ψy∂Ψ +Ty∂T +Ψyy∂Ψy +Tyy∂Ty +Ψxy∂Ψx + .........

(3.9)


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Table 1: Table of commutators of the basis operators.

Equation (3.7) gives the following system of linearpartial differential equations

A F Ψyyy −Ψyyηx+(Ψxy+M2)ηy+AΨyy Fy+Ψyηxy+(A Ty−Ψx)ηyy− (a+A(1−T)) ηyyy= 0,

(3.10)and

−Tyηx+Txηy+ΨyFx−ΨxF y− 1

PrFyy = 0. (3.11)

The componentsηx,ηy,Fx,Fy,ηxy,ηyy,Fyy,ηyyy canbe determined from the following expressions

ηS= DSη −ΨxDSϕ −ΨyDSζ ,FS= DSF −TxDSϕ −TyDSζ ,ηJ S= DSηJ −ΨJxDSϕ −ΨJyDSζ , FJS= DSFJ−TJ xDSϕ −TJ yDSζ ,

(3.12)whereSandJ are standing forx,y.

Substitution from (3.12) into (3.10)-(3.11) andsolving the resulting determining equations in view of theinvariance of the boundary conditions (2.14), yields

ϕ =C1 x, ζ =C2, η =C1Ψ +C3, F = 0. (3.13)

So, the nonlinear equations (2.12)-(2.13) have thethree-parameter Lie group of point symmetries generatedby

X1 ≡ x∂

∂ x+Ψ

∂∂ Ψ

, X2 ≡∂

∂ y, and X3 ≡

∂∂ Ψ

. (3.14)

The one-parameter group generated byX1 consists ofscaling, whereasX2 and X3 consists of translation. Thecommutator table of the symmetries is given in Table1,where the entry in theith row and jth column is definedasXi ,Xj ] = XiXj −XjXi .

The finite transformations corresponding to thesymmetriesX1, X2 andX3 are respectively

X1 : x∗ = eε 1x, y∗ = y, Ψ∗ = eε 1Ψ , T∗ = T,X2 : x∗ = x, y∗ = y+ ε2, Ψ ∗ =Ψ , T∗ = T,X3 : x∗ = x, y∗ = y, Ψ∗ =Ψ + ε3, T∗ = T,

(3.15)whereε1, ε2 andε3 are group parameters.

Table 2 shows the solution of the invariant surfaceconditions (3.3)-(3.4).

ForX1 , the characteristic,

Φ = (ΦΨ ,ΦT) , (3.16)

Table 2: Solution of the invariant surface conditions.

has the component

ΦΨ =Ψ − xΨx, ΦT =−Tx. (3.17)

Therefore, the general solutions of the invariant surfaceconditions (3.3)-(3.4) are

Ψ = xG(y) , T = T(y). (3.18)

Substitution from (3.18) into (2.12)-(2.13) yields

(

d Gd y

)2−Gd2G

d y2 +A d Td y

d2Gd y2 − (a+A(1−T)) d3G

d y3 +M2 d Gd y = 0,

(3.19)and

d2Td y2 +Pr G

d Td y

= 0. (3.20)

The boundary conditions (2.14) will be

(i)d Gd y

= 1, G = 0, T = 1 as y= 0,

(ii)d Gd y

→ 0 T → 0 as y→ ∞. (3.21)

ForX2, the characteristic (3.16) has the component

ΦΨ = −Ψy, ΦT =−Ty. (3.22)

Therefore, the general solutions of the invariant surfaceconditions (3.3)-(3.4) areΨ =Ψ(x) andT = T(x), whichcontradict the boundary conditions.

ForX3, the characteristic (3.16) has the component

ΦΨ = 1, ΦT = 0. (3.23)

Therefore, no solution invariant underX3.For X1 + βX2, the characteristic (3.16) has the

component

ΦΨ =Ψ − xΨx−β Ψy, ΦT =−xTx−β Ty. (3.24)

Therefore, the general solutions of the invariant surfaceconditions (3.3)-(3.4) areΨ = x L1(ω) and T = T(ω),where ω = y − lnxβ is the similarity variable. Thesesolutions contradict the boundary conditions.



Table 3: Values of the wall heat transfer(−T ′(0)) for different values ofM and Pr atA= 0.0

For X1 + βX3, the characteristic (3.16) has thecomponent

ΦΨ =Ψ +β − xΨx, ΦT =−xTx. (3.25)

Therefore, the general solutions of the invariant surfaceconditions (3.3)-(3.4) are

Ψ = xL2(y)−β , T = T(y). (3.26)

Substitution from (3.26) into (2.12)-(2.13) yields thesame ordinary differential equations (3.19)-(3.20) withthe same boundary conditions (3.21). So, the solutionsinvariant under bothX1 andX1+βX3 are the same.

For X2 + βX3, , the characteristic (3.16) has thecomponent

ΦΨ = β − Ψy, ΦT =−Ty. (3.27)

Therefore, the general solutions of the invariant surfaceconditions (3.3)-(3.4) areΨ = β y+ h(x) andT = T(x),which contradicts the boundary conditions.

For X1+ λX2+ δX3, , the characteristic (3.16) has thecomponent

ΦΨ = Ψ + δ − xΨx−λ Ψy, ΦT =−xTx−λ Ty.(3.28)

Therefore, the general solutions of the invariant surfaceconditions (3.3)-(3.4) areΨ +δ = x L2(π) andT = T(π),where π = y − lnxλ is the similarity variable. Thesesolutions contradict the boundary conditions.

3.2 Numerical Solution

The system of non-linear differential equations(3.19)-(3.20) with the boundary conditions (3.21) issolved numerically using the shooting method, coupledwith Runge-Kutta scheme. We takea = 1, in allcalculations.

From (2.11) and (3.18), we get

ux=

d Gd y

, v=− G(y), T = T(y). (3.29)

4 Results and Discussion

4.1 Horizontal Velocity

4.1.1 The Effect of the Hartmann NumberM

Figure 2 illustrates the behaviour of the horizontalvelocity u/x for Pr= 0.1 and Pr= 1.0 with A= 0, over arange of the Hartmann numberM. As seen, the horizontalvelocity increases by decreasingM, i.e. the transport rateincreases, which indicate that the transverse magneticfield is opposite to the transport phenomena. That isbecause, variation of the Hartmann number leads to thevariation of the Lorentz force due to the transversemagnetic field and accordingly, the Lorentz forceproduces more resistance to transport phenomena, [4].Our profiles are agreed very well with those ofMukhopadhyay et al. [4] and Pantokratoras [10]. As seenfrom Figure 2b, the calculation domain is higher than thatof Figure 2a, that is because the value of the Prandtlnumber is increases.

4.1.2 The Effect of the Viscosity ParameterA

Figure 3 illustrates the behaviour of the horizontalvelocityu/x for Pr= 0.1 and Pr= 1.0 with M = 0, over arange of the viscosity parameterA. As seen, thehorizontal velocity increases by increasingA. That isbecause, with increasingA, the fluid viscosity decreasesresulting in increment of the velocity boundary layerthickness. Unfortunately, in the work of Mukhopadhyayet al. [4] and Pantokratoras [10], the profiles are reversed,i.e. the horizontal velocity decreases asA increases,which is wrong.

4.2 Vertical Velocity


Figure 4 illustrates the behaviour of the vertical velocityvfor Pr= 0.1 and Pr= 1.0 with A= 0 , over a range of theHartmann numberM. As seen, the absolute value of thevertical velocity increases by decreasingM as mentionedbefore. These profiles were not present in both works ofMukhopadhyay et al. [4] and Pantokratoras [10].




Fig. 2: Horizontal velocity profiles over a range ofM with A= 0 at: (a) Pr= 0.1 (b) Pr= 1.0.

Fig. 3: Horizontal velocity profiles over a range ofA with M = 0 at: (a) Pr= 0.1 (b) Pr= 1.0.

Fig. 4: Vertical velocity profiles over a range ofM with A= 0 at: (a) Pr= 0.1 (b) Pr= 1.0.



Fig. 5: Vertical velocity profiles over a range ofA with M = 0 at: (a) Pr= 0.1 (b) Pr= 1.0.


Figure 5 illustrates the behaviour of the vertical velocityv for Pr= 0.1 and Pr= 1.0 with M = 0 , over a range ofthe viscosity parameterA, As seen, the absolute value ofthe vertical velocity increases by increasingA. Also, theseprofiles were not present in both works of Mukhopadhyayet al. [4] and Pantokratoras [10].

4.3 The Temperature Profiles


Figure 6 illustrates the variation of the temperatureprofilesT for Pr= 0.1 with A = 0 , over a range of theHartmann numberM. We notice that, the temperatureincreases asM increases and therefore the thinning of thethermal boundary layer. Our results are in completeagreement with that reported by Mukhopadhyay et al. [4],but in their work, the calculation domain is small whichcauses the temperature profiles appear truncated. Thisdisadvantage was critiqued by Pantokratoras [10].Unfortunately, in the work of Pantokratoras [10], theprofiles are reversed, i.e. the temperature increases asMdecreases, which is wrong.


Figure 7 illustrates the variation of the temperatureprofilesT for Pr= 1.0 with M = 0, over a range of theviscosity parameterA. As seen, the temperature decreasesas A increases. That is because, the increase of theviscosity parameterA causes decrease of the thermalboundary layer thickness which results in decrease of thetemperature. Our results are in complete agreement with

that reported by Mukhopadhyay et al. [4], butunfortunately in their work, the profiles are reversed; seeFigure 2 in their work. The same mistake appeared in thework of Pantokratoras [10]; see Figure 3 in his work.

4.3.3 The Effect of the Prandtl Number Pr

Figure 8 illustrates the variation of the temperatureprofilesT for M = 0 andM = 1 with A= 0, over a rangeof Prandtl number Pr. It is noticed that, as Pr decreases,the thickness of the thermal boundary layer becomesgreater than the thickness of the velocity boundary layeraccording to the well known relationδT

δ ≈ (Pr)−1/2,whereδT is the thickness of the thermal boundary layerandδ is the thickness of the velocity boundary layer. So,the thickness of the thermal boundary layer increases asPr decreases and hence, the temperatureT increases withthe decrease of Pr

4.4 Wall Heat Transfer

When the Prandtl number increases, the thickness ofthermal boundary layer becomes thinner and this causesan increase in the gradient of the temperature. Therefore,the wall heat transfer(−T ′(0)) increases as Pr increases.For different values of the Hartmann numberM andPrandtl number Pr atA = 0.0 , values of the wall heattransfer are computed, Table 3. Also, for fixed value of Pr, the wall heat transfer(−T ′(0)) decreases as theHartmann numberM increases as mentioned before. Thevalue of(−T ′(0)) is positive which is consistent with thefact that the heat flows from the surface to the fluid aslong asTw > T∞ in the absence of viscous dissipation.

The computed values of(−T ′(0)) are compared withthose obtained by Chiam [26], Carragher and Crane [27],




Fig. 6: Temperature profiles over a range ofM with Pr= 1.0 andA= 0

Table 4: Comparison between the values of(−T ′(0)) at A= 0.0 andM = 0 for different values of Pr

Table 5: Values of the dimensionless wall shear stressG′′(0) for differentM andA at Pr= 1.0



Fig. 7: Temperature profiles over a range ofA with Pr= 1.0 andM = 0

Fig. 8: Temperature profiles over a range of Pr withA= 0 at: (a) M = 0 (b) M = 1




Table 6: Comparison between the values ofG′′(0) at A= 0.0 andM = 0 for different values of Pr.

Grubka and Bobba [28] and Pantokratoras [10]. Theresults are in very good agreement, Table 4.

4.5 Wall Shear Stress

The dimensionless wall shear stressG′′(0) (skin friction)is computed for different values of the Hartmann numberand the viscosity parameterA at Pr= 1.0. As seen fromTable 5, the absolute value of the dimensionless wallshear stress|G′′(0) | increases asA increases which isconsistent with the fact that, there is progressive thinningof the boundary layer with increasingA. . Also, theabsolute value of the dimensionless wall shear stress|G′′(0) | increases asM increases for fixed value ofviscosity parameterA, Table 5.

The computed values ofG′′(0) are compared withthose obtained by Carragher and Crane [27], Grubka andBobba [28] and Pantokratoras [10] The results are in verygood agreement, Table 6.

5 Conclusion

The steady two-dimensional incompressibleMagnetohydrodynamic (MHD) boundary layer flow ofvariable viscosity over a heated stretching sheet in thepresence of uniform transverse magnetic field has beeninvestigated. The system of non-linear partial differentialequations is solved using Lie-group method. Theresulting ordinary differential equations are solvednumerically using the shooting method coupled withRunge-Kutta scheme. The influences of the HartmannnumberM, the viscosity parameterA, and the Prandtlnumber Pr on the horizontal velocityu/x, verticalvelocity v, temperature profilesT, wall heat transfer−T ′(0) and the wall shear stressG′′(0) (skin friction)were examined. Particular cases of our results arecompared with those of Pantokratoras [10], Chiam [26],Carragher and Crane [27], Grubka and Bobba [28] andwere found a good agreement with their results.Comments on the work of Mukhopadhyay et al. [4] andPantokratoras [10] were discussed and corrected.

Acknowledgements

The authors would like to express their sincere thank thereviewers for suggesting certain changes in the originalmanuscript, for their valuable comments which improvedthe paper and for their great interest in that work.

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[15] P.J. Olver, Applications of Lie Groups to DifferentialEquations, Springer-Verlag, New-York,1986.

[16] Y.Z. Boutros, M.B. Abd-el-Malek, N.A. Badran, H.S.Hassan, Lie-group method for unsteady flows in a semi-infinite expanding or contracting pipe with injection orsuction through a porous wall, J. Comput. Appl. Math.,197(2006) 465-494.

[17] Y.Z. Boutros, M.B. Abd-el-Malek, N.A. Badran, H.S.Hassan, Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towardsa heated stretching sheet placed in a porous medium,MECCANICA, 41, no. 6, (2007) 681-691.

[18] Y.Z. Boutros, M.B. Abd-el-Malek, N.A. Badran, H.S.Hassan, Lie-group method solution for two-dimensionalviscous flow between slowly expanding or contractingwalls with weak permeability, J. Applied MathematicalModelling, 31(2007) 1092-1108.

[19] M.B. Abd-el-Malek, N.A. Badran, H.S. Hassan, Lie-groupmethod for predicting water content for immiscible flowof two fluids in a porous medium, Applied MathematicalSciences, 1, no. 24, (2007) 1169-1180.

[20] M.B. Abd-el-Malek, H.S. Hassan, Internal ow through aconducting thin duct via symmetry analysis, in: Symmetryand Perturbation Theory, Proceedings of the InternationalConference on SPT 2007, Otranto, Italy, 2007, pp. 233234.

[21] M.B. Abd-el-Malek, H.S. Hassan, Symmetry analysisfor steady boundary-layer stagnation-point flow ofRivlinEricksen fluid of second grade subject to suction,Nonlinear Analysis: Modelling and Control, 15, no.4,(2010) 379396.

[22] M.B. Abd-el-Malek, H.S. Hassan, Solution of Burgersequation with time-dependent kinematic viscosity via Lie-group analysis, Proceedings of the Fifth InternationalWorkshop ”Group Analysis of Differential Equations &Integrable Systems”, Protaras- Cyprus, 2010, pp. 6-14.

[23] M.B. Abd-el-Malek, H.S. Hassan, Lie group method forsolving the problem of fission product behaviour in nuclearfuel, Proceedings of the ninth International Conference ofNumerical Analysis and Applied Mathematics, ICNAAM2011, Halkidiki, 2011, pp. 1367-1371.

[24] M.B. Abd-el-Malek, N.A. Badran, H.S. Hassan, H.H.Abbas, New solutions for solving the problem of particletrajectories in linear deep-water waves via Lie-groupmethod, Appl. Math. Comput. 219 (2013) 11365-11375.

[25] Abd-el-Malek, and Hassan, H.S., Lie group method forsolving the problem of fission product behaviour in nuclearfuel, Math. Methods Appl. Sci., 37, no.3, (2014) 420-427.

[26] T.C. Chiam, Heat transfer in a fluid with variable thermalconductivity over a linearly stretching sheet, Acta Mech.129 (1998) 6372.

[27] P. Carragher, L.J. Crane, Heat transfer on a continuousstretching sheet, ZAMM 62 (1982) 564565.

[28] L.J. Grubka, K.M. Bobba, Heat transfer characteristics ofa continuous stretching surface with variable temperature,ASME J. Heat Transfer 107 (1985) 248250.

Hossam S. Hassanholds a PhD in EngineeringMathematics from Facultyof Engineering-AlexandriaUniversity-Egypt. Hecurrently works as anAssociate Professor ofEngineering Mathematicsat College of Engineering andTechnology, Arab Academyfor Science, Technology and

Maritime Transport, Alexandria, Egypt. His main fields ofinterest are Magnetohydroddynamics (MHD), WaterWaves, Nonlinear Partial Differential Equations andLie-Group. He has published more than 20 researchpapers in reputed international journals of Mathematics.He is referee of several international journals ofMathematics.

Samar A. Mahrousreceived the M.Sc degree inMechanical Engineering fromthe College of Engineeringand Technology, ArabAcademy for Science,Technology and MaritimeTransport, Alexandria,Egypt. She currently worksas an Assistant lecturer ofEngineering Mathematics at

College of Engineering and Technology, Arab Academyfor Science, Technology and Maritime Transport,Alexandria, Egypt.

A. Sharara receivedthe PhD degree in MechanicalEngineering from theAin Shams university-Egyptin 2010. He currently worksas an Assistant Professorat Marine Engineeringdepartment, College ofEngineering and Technology,Arab Academy for Science,Technology and Maritime

Transport, Alexandria, Egypt.




A. Hassan receivedPhD degree in ComputionalFluid Dynamics fromUniversity of Nottingham-UKin July 2002. He has served asHead of Marine EngineeringDepartment at College ofEngineering and Technology,Arab Academy for Science,Technology and MaritimeTransport, Alexandria, Egypt,

from 2008-2014. He currently works as the Deanof college of Engineering and Technology, ArabAcademy for Science, Technology and MaritimeTransport, Alexandria, Egypt. His main field of interest isComputional Fluid Dynamics. He is referee of severalinternational journals of Marine and MechanicalEngineering.


a study for mhd boundary layer flow of variable viscosity ......a study for mhd boundary layer flow...

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