Nonlinear Convection towards an Exponentially Shrinking Sheet with Magnetic Field

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660 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882

Volume 4, Issue 6, June 2015

Nonlinear Convection towards an Exponentially Shrinking Sheet with Magnetic Field

Rakesh Kumar

Department of Mathematics, Central University of Himachal Pradesh, India E-Mail: [email protected]

ABSTRACT A theoretical analysis is made to investigate the effect of nonlinear convection on the flow of viscous fluid over an exponentially shrinking sheet. The influences of suction and heat source/sink are also considered. Mathematically suitable variables are introduced to convert the governing system of coupled non-linear differential equations into self-similar form. The reduced differential equations are solved to obtain a closed form solution. The physical interpretations of the results have been discussed through the plotted graphs. It is found that the suction parameter, heat source/sink and nonlinear convection considerably affect the flow and heat transfer over the surface of the exponentially shrinking sheet. Keywords: Nonlinear convection, MHD, exponentially shrinking sheet, heat source/sink. I. INTRODUCTION

The extension of boundary layer theory to understand the flow and heat transfer dynamics of viscous fluids over stretching and shrinking surfaces have attracted the researchers from all around the globe due to wide applications in manufacturing processes. The popular examples of these applications are: aerodynamics, extrusion of plastic sheets, extrusion of polymers, process of condensation of metallic plates, cooling of electronic chips or metallic sheets, wire drawing, heat treated materials that travel between a feed and wind-up roll, paper production, glass fiber, crystal growth, filaments spinning and food processing etc. Sakiadis [1, 2] was the first one to extend the classical boundary layer theory to the flows with stretching boundary conditions. Later on, a closed form solution to the two dimensional flow caused by the stretching of the elastic sheet was provided by Crane [3]. Then, Gupta and Gupta [4] extended the above works by considering the effects of suction and blowing on the heat and mass transfer of flow over the stretching sheet. Recently, Rajotia and Jat [5] obtained the dual solutions for the three dimensional hydromagnetic flow over an axisymmetric shrinking sheet considering viscous dissipation and heat source/sink. Although every type of stretching has its own unique significance in the manufacturing processes, but the exponential form of stretching has not been given its berth which it deserves. In two dimensional flows, the topological chaos depend on the periodic motion of the obstacles to form nontrivial braids, and the corresponding motion will produce exponentially stretching of the material lines and will be responsible for proficient mixing. Considering this in mind, Magyari and Keller [6] proposed this new kind of exponential stretching of the sheet in its own plane and presented the heat transfer analysis. Elbashbeshy [7] presented his numerical solutions to analyze the flow and heat transfer dynamics due to the exponentially stretching permeable sheet. In this sequence many research papers on the flow of Newtonian and non-Newtonian viscous fluids over exponentially stretching surfaces under various physical situations appeared in literature viz. Khan and Sanjayanand [8], Partha et al. [9], Sajid and Hayat [10], Mukhopadhyay [11], Mukhopadhyay [12], Hayat et al. [13] and Naramgari and Sulochana [14] etc. Very recently, Nadeem and Hussain [15] presented the heat transfer analysis of Williamson fluid over an exponentially stretching surface.

On the other hand in recent years, the flows due to deforming boundaries have become more popular due to their unusual properties and various applications. These new type of shrinking sheet flows are essentially a backward flow and the physical phenomena are quite distinct from the forward stretching flow (Fang et al. [16]). These flows are still having very less known properties and require more investigations to understand the dynamics of shrinking surfaces. A high quality field coating solution for welded pipe can be provided by a heat shrinking sheet and is also used for wrapping junction (Vyas and Srivastava [17]). Wang [18] in his paper on the investigation of liquid film behavior over an unsteady stretching sheet observed this new type of flow. Then Miklavcic and Wang [19] obtained the existence and uniqueness criteria for the similarity solutions and found that suitable suction is required to control the generated vorticity within the boundary layer. The utility of stagnation point in maintaining the flow over shrinking sheet was proposed by Wang [20].

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Later on the dual solutions in the unsteady stagnation point flow were obtained by Bhattacharyya [21] and he found that dual solutions exist for certain range of velocity ratio parameters. He also analyzed that beyond that range the boundary layer separates from the surface and the solutions based on the boundary layer approximations are not possible. Bhattacharyya et al. [22] also investigated the boundary layer flow of Maxwell fluid over a porous shrinking sheet. Recently, the instability of viscous flows over shrinking surface was discussed by Miklavcic [23]. Further, the flow over exponentially shrinking sheet is known for increased rate of heat and mass transfer than the flow over linearly shrinking surface. Considering this in mind, Bhattacharyya [24] studied the boundary layer flow and heat transfer over an exponentially shrinking sheet. In this sequence, the research papers of Bhattacharyya and Vajarvelu [25] and Rohni et al. [26] appeared in literature. These papers were focused on the stagnation point flows over exponentially shrinking surfaces with and without suction. Very recently Kumar [27] analyzed the boundary layer flow and heat transfer over an exponentially shrinking sheet with exponential magnetic field. The analytical solution for the effects of arbitrary suction and porous media over an exponentially shrinking sheet was presented by Kumar [28].

Moreover, the presence of heat source/sink in the fluid is considered to be very important when there are sharp fluctuations in the temperature between the solid surface and the ambient region. The heat source/sink in general is temperature dependent and space dependent and will considerably influence the heat transfer properties of the fluid. Also, the large temperature difference between the surface and the ambient fluid compels the researchers to consider the nonlinear density temperature (NDT) variations in the buoyancy force term due to its significant effect on the flow and heat transfer characteristics. Vajravelu and Sastri [29] studied the flow between two parallel plates by considering the quadratic density temperature (QDT) variation and showed that the flow and heat transfer rates are substantially affected by it. Bhargava and Agarwal [30] investigated the fully developed free convection flow in circular pipe with nonlinear density temperature variations. The nonlinear convection effects on the flow past a flat porous plate has been examined by Vajravelu et al. [31].

Motivated by this, the objective of the present analysis is to investigate the influence of non-linear convection on the flow over an exponentially shrinking sheet. The buoyancy forces which appear due to the heating/cooling of the stretching/shrinking surface are included in this paper as they significantly affect the flow and thermal properties of the physical process. The suction is also considered to be the predominant parameter to control the boundary layer separation and strongly required in aerodynamics to have minimum drag and maximum lift. Therefore, the effects of suction and heat source/sink have also been targeted to achieve here.

II. MATHEMATICAL MODELING

We consider the steady two dimensional mixed convection boundary layer flow of an electrically conducting, viscous and incompressible fluid past a permeable exponentially shrinking sheet. The x -axis is taken along the sheet and y -axis is taken normal to it. The flow is considered to be confined to the region 0y ≥ as shown in Figure 1. The two equal and opposite forces are assumed to be applied along the x -axis towards the origin O of the coordinate system, and these forces are assumed to shrink the sheet while keeping the origin fixed. It is also assumed that a strong magnetic field of the following form ( )0,,0 yBB = exists normal to the plane of the sheet, where ( )LxByB 2/exp0= and 0B is a constant magnetic field. The magnetic Reynolds number is assumed to be small to neglect the induced magnetic field.

Figure 1: Schematic presentation of the problem

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Further the variable nature of heat source/sink parameter is assumed as ( )0 exp /Q Q x L= , where 0Q is the constant

heat source for 0 0Q ≥ and heat sink for 0 0Q ≤ . The general form of the nonlinear density variations with the temperature can be can be written as (Streeter [32])

( ) ( ) ( ) ( )2

22 ...w w w

w w

T T T T T TT Tρ ρρ ρ

∂ ∂ = + − + − + ∂ ∂ If we consider the terms upto second order then, the above

relation can be written as ( ) ( )20 1/ w wT T T Tρ ρ β β∆ = − − − − , where 0β , 1β , are constants. This relation will be called

as nonlinear density temperature (NDT) variations. The governing equations of continuity, motion and energy under the assumption of boundary layer approximation for present problem are Equation of continuity:

0=∂∂

+∂∂

yu

xu

(1)

Equation of momentum:

( ) ( )22

20 12

yBu u uu v g T T g T T ux y y

σν β β

ρ∞ ∞∂ ∂ ∂

+ = + − + − −∂ ∂ ∂

(2)

Equation of energy:

( )2

2p p

T T k T Qu v T Tx y C y Cρ ρ ∞

∂ ∂ ∂+ = + −

∂ ∂ ∂ (3)

The boundary conditions are given by ( ) ( ) ( )0, , exp 2 / at 0

0, 0 asw w wu U x v v T T x T T x L y

u T y∞ = = = = + =

= = →∞

(4)

The shrinking sheet velocity wU is given by ( ) ( )0 exp /wU x U x L= − , where 0 0U > is shrinking constant. Here g ,

Q , σ , ν , ρ , k , pC L , 0T , wT and ∞T are gravitational acceleration, heat source/sink parameter, the electrical conductivity, kinematic viscosity, density, thermal conductivity, specific heat at constant pressure, characteristic length of the sheet, mean temperature, temperature of the sheet and ambient temperature of the fluid respectively. Let us introduce the following similarity variables

( ) ( )02 exp / 2LU f x Lψ ν η= , ( ) ( )wT T T T θ η∞ ∞= + − , (5) whereη is the similarity variable defined by

( )0 exp / 22Uy x L

Lη

ν= (6)

and ψ is the stream function which is defined in the classical form as y

u∂∂

=ψ

and x

v∂∂

−=ψ

. Thus we have the

following expressions as

( ) ( )0 exp / 'u U x L f η= and ( ) ( ) ( )0 exp / 2 '2Uv x L f fL

ν η η η = − + (7)

where prime denotes differentiation with respect to η . This suggests that, we can assume

( ) ( )0 exp / 22wUv x x L SL

ν= − , (8)

where 0>S is the dimensionless suction parameter.

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Using equation (5) to (7) in equations (2) and (3), we obtain the following ordinary differential equations 2 22 2 2 0f ff f Mf λ θ γθ′′′ ′′ ′ ′ + − − + + = (9)

( )Pr 4 2 0f fθ θ θ αθ′′ ′ ′+ − + = (10) The boundary conditions transform to ( ) ( ) ( )( ) ( )

→→→′==−=′=

00,001,1,

ηηθηηηθηη

asfatfSf

(11)

The physical parameters of interest in the present problem, the skin friction coefficient fC and the Nusselt number Nu , are defined by

02 =

∂∂

=yy

u

wUfC

ρ

µ (12)

0=

∂∂

−∞−

=yy

TTwT

LNu (13)

Substituting (5) to (7) into above two equations, we get the following expressions of skin friction and Nusselt number: ( ) ( )2Re exp / 2 '' 0fx L C f= (14)

( ) ( )0'2/expRe/2 θ−=− NuLx (15)

Here20

0 B LMU

σρ

= (Hartmann number), k

pCµ=Pr (Prandtl number), 0

0p

Q LC U

αρ

= (heat source sink parameter),

0Re U Lν

= (Reynolds number), 3

0 02

g T LGr βν

= (Grashoff number), 𝛾 = 𝛽1𝛽0

(𝑇−𝑇∞) (Nonlinear convection parameter)

and 2ReGrλ = (mixed convection parameter) are the dimensionless parameters introduced in the above equations.

The differential equations (9) and (10) under the boundary conditions (11) are solved using the series expansion method as suggested by Singh and Dikshit [33].

III. METHOD OF SOLUTION Let us define

Sηη = , ( ) ( )ηη FSf = and ( ) ( )ηθηθ = (16) The equations (9) to (11) becomes

2 2 22 2 2 0F FF F MF G Gλ γ′′′ ′′ ′ ′ + − − ∈ + ∈ + = (17)

( )Pr 4 2 0G FG F G Gα′′ ′ ′+ − + ∈ = (18) The transformed boundary conditions are

( ) ( ) ( )

( ) ( )

→→→′

===∈−=′=

00,0

01,21,1

ηηη

ηηηη

asGF

atGS

FF (19)

where prime denotes the differentiation with respect to η . For large suction, S assumes large positive values so that ∈ is small. Therefore, F and G can be expanded in terms of small perturbation quantity ∈ as

...33

22

10 +∈+∈+∈+= FFFFF (20)

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...33

22

10 +∈+∈+∈+= GGGGG (21) Substituting (20) and (21) into (17),(18) and (19), we obtain the following sets of ordinary differential equations along with the corresponding boundary conditions : Zeroth Order O(1) :

0202000 =′−′′+′′′ FFFF (22)

( )0 0 0 0 0Pr 4 0G F G F G′′ ′ ′+ − = (23)

( ) ( ) ( )( ) ( )

=∞==∞′=′=

00,100

00,000,100GG

FFF (24)

First-Order ( )∈O :

1 0 1 1 0 0 1 04 2 0F F F F F F F MF′′′ ′′ ′′ ′ ′ ′+ + − − = (25)

( )1 1 0 0 1 0 1 1 0 0Pr 4 4 2 0G FG F G F G FG Gα′′ ′ ′ ′ ′+ + − − + = (26)

( ) ( ) ( )( ) ( )

=∞==∞′=′=

01,001

01,101,001GG

FFF (27)

Second-Order

∈2O :

2 22 2 0 0 2 1 1 0 2 1 1 0 04 2 2 2 2 0F F F F F F F F F F MF G Gλ λγ′′′ ′′ ′′ ′′ ′ ′ ′ ′+ + + − − − + + = (28)

( )2 0 2 1 1 2 0 0 2 1 1 2 0 1Pr 4 4 4 2 0G F G FG F G F G FG F G Gα′′ ′ ′ ′ ′ ′ ′+ + + − − − + = (29)

( ) ( ) ( )( ) ( )

=∞==∞′=′=

02,002

02,002,002GG

FFF (30)

Third-Order

∈3O :

3 0 3 1 2 2 1 3 0 1 2 0 3

2 1 0 1

4 42 2 4 0

F F F F F F F F F F F F FMF G G Gλ λγ′′′ ′′ ′′ ′′ ′′ ′ ′ ′ ′+ + + + − −

′− + + = (31)

( )3 0 3 1 2 2 1 3 0 0 3 1 2 2 1 3 0 2Pr 4 4 4 4 2 0G F G FG F G F G F G FG F G F G Gα′′ ′ ′ ′ ′ ′ ′ ′ ′+ + + + − − − − + = (32)

( ) ( ) ( )( ) ( )

=∞==∞′=′=

03,003

03,103,003GG

FFF (33)

The obtained solutions of the above equations under the corresponding boundary conditions are: 𝐹0(�̅�) = 1 (34) 𝐹1(�̅�) = 1 − exp (−�̅�) (35) 𝐹2(�̅�) = 𝐴8 + 𝐴7 exp(−�̅�) + 𝐴4�̅� exp(−�̅�) − 1

4exp(−2�̅�) + 𝐴5 exp(−Pr�̅�) + 𝐴6 exp(−2Pr�̅�)

(36) 𝐹3(�̅�) = 𝐴38 + 𝐴37exp(−�̅�) − 𝐴29exp(−2�̅�) − 1

24exp(−3�̅�) + 𝐴32exp(−Pr�̅�) + 𝐴33exp(−2Pr�̅�) −𝐴34exp{−(1 + Pr)�̅�} − 𝐴35exp{−(2 + Pr)�̅�} + 𝐴36exp{−(1 + 2Pr)�̅�} + 𝐴28�̅�exp(−�̅�) +𝐴4

2 𝜂� exp(−2�̅�) − 𝐴42

2 �̅�2exp(−�̅�) − 𝐴30 �̅�exp(−Pr�̅�) − 𝐴31 �̅�exp(−2Pr�̅�) (37)

𝐺0(�̅�) = exp(−Pr�̅�) (38) 𝐺1(�̅�) = −𝐴1 exp(−Pr�̅�) − 𝐴2 �̅�exp(−Pr�̅�) − 𝐴1 exp(−(1 + Pr)�̅�) (39)

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𝐺2(�̅�) = 𝐴18 exp(−Pr�̅�) − 32𝐴5 exp(−2Pr�̅�) − 7

6𝐴6 exp(−3Pr�̅�) −A16exp{−(1 + Pr)�̅�}

−A17exp{−(2 + Pr)�̅�} + A14�̅� exp(−Pr�̅�) +�̅�2

2𝐴12 exp(−Pr�̅�) − A15�̅� exp{−(1 + Pr)�̅�}

(40) The velocity and temperature profiles can be calculated from the following expressions

( ) 21 2 3,f F F Fη′ ′ ′ ′= − +∈ −∈ (41)

( ) .22

10 GGG ∈+∈+=ηθ (42) In order to obtain more accurate results for velocity and temperature profiles, we have evaluated the expression up to the third order.

IV. RESULTS AND DISCUSSION In order to obtain a clear insight of the mathematical modeling of the physical problems, the numerical values of the velocity profiles, temperature profiles, skin friction and rate of heat transfer are computed for various values of the associated parameters. The pertinent parameters are assigned arbitrary values.

Figure 2: Velocity profiles with 1=M , 4=S , 2.0=λ , 2.0=γ and 2=α .

Figure 3: Velocity profiles with 1=M , 4=S , 3Pr = , 2.0=γ and 2=α .

The Figures 2 to 7 are plotted to discuss the influence of Prandtl number, mixed convection parameter, NDT parameter and magnetic field on the velocity profiles. It is inferred from these Figures that the velocity increases and boundary layer thickness decreases with the increase of Prandtl number, mixed convection parameter and NDT parameter. However, the Figure 5 depicts the opposite behaviour of the velocity profiles with respect to the increasing strength of the magnetic field. This is due to the resisting effect of the stronger magnetic field. The Figure 6 shows that both the velocity and boundary layer thickness are reduced with the increasing negative values of the mixed convection parameter.

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Figure 4: Velocity profiles with 1=M , 4=S , 2.0=λ , 3Pr = and 2=α .

Figure 5: Velocity profiles with 3Pr = , 4=S , 2.0=λ , 2.0=γ and 2=α .

Figure 6: Velocity profiles with 1=M , 4=S , 3Pr = , 2.0=γ and 2=α .

Figure 7: Temperature profiles with 1=M , 4=S , 2.0=λ , 2.0=γ and 2=α .

Figure 8: Temperature profiles with 1=M ,

4=S , 3Pr = , 2.0=γ and 2=α .

Figure 9: Temperature profiles with 1=M , 4=S ,

2.0=λ , 3Pr = and 2=α . The influence of Prandtl number, mixed convection parameter, NDT parameter and heat sink parameter have been illustrated through the Figures 7, 8, 9 and 10. It is clear from Figure 7 that both the temperature and thermal boundary layer thickness decreases significantly with the increasing Prandtl number. In general, the thermal boundary layer thickness becomes thinner with the increase in Prandtl number. This is due to the physical fact that the increasing Prandtl number decreases the thermal conductivity of the fluid, hence causes a reduction in the thermal boundary layer thickness.

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The same phenomenon occurs in Figures 8 and 9 with respect to mixed convection parameter and NDT parameter, but the impact is smaller as compared to the decrease with Prandtl number. The heat sink is having an opposite effect on the temperature profiles which is dictated by the Figure 10. It is evident from this Figure that both the temperature and thermal boundary layer thickness are enhanced for 13≤α (app.). The temperature increases and boundary layer becomes thinner for 13>α approximately.

Figure 10: Temperature profiles with 1=M , 4=S , 2.0=λ , 2.0=γ and 3Pr = .

Figure 11: Skin-friction with 1=M , 2=α , 2.0=γ and 3Pr = .

Figure 12: Skin-friction with 1=M , 2=α , 2.0=λ and 3Pr = .

Figure 13: Rate of heat transfer with 1=M , 2=α , 2.0=γ and 3Pr = .

Figure 14: Rate of heat transfer with 1=M ,

2=α , 2.0=λ and 3Pr = .

Figure 15: Rate of heat transfer with 1=M ,

2.0=γ , 2.0=λ and 3Pr = .

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The Figures 11 to 15 presents the impact of mixed convection parameter, NDT parameter and heat source on the skin friction and Nusselt number. Both the skin friction and Nusselt number increases significantly with the mixed convection parameter and NDT parameter. However, the Nusselt number observes the reverse phenomenon with the increasing heat source, that is, it decreases

V. CONCLUSIONS The similarity solutions are obtained for the flow of viscous incompressible fluid over an exponentially shrinking sheet. From this study we conclude that mixed convection parameter and NDT parameter have dominant effect on the skin friction and Nusselt number; heat source significantly reduces the rate of heat transfer; and velocity increased with mixed convection and NDT, but the temperature is reduced with NDT and mixed convection parameter.

VI. APPENDIX

𝐴1 = 4𝑃𝑟−𝑃𝑟2

1+𝑃𝑟, 𝐴2 = 𝑃𝑟 − 2𝛼, 𝐴3 = 𝑃𝑟𝐴1 − 𝐴2, 𝐴4 = 1 + 2𝑀, 𝐴5 = 2𝜆

𝑃𝑟3−𝑃𝑟2 , 𝐴6 = 𝜆𝛾

4𝑃𝑟3−2𝑃𝑟2 , 𝐴7 = 1

2+ 𝐴4 −

𝑃𝑟𝐴5 − 2𝑃𝑟𝐴6, 𝐴8 = −14− 𝐴4 + (𝑃𝑟 − 1)𝐴5 + (2𝑃𝑟 − 1)𝐴6, 𝐴9 = 𝐴3 − 𝑃𝑟𝐴8 − 2𝛼𝐴1, 𝐴10 = 𝐴1(3− 𝐴2) − 𝐴3 −

4𝐴4 + 𝐴7(4 − 𝑃𝑟), 𝐴11 = 𝐴1(𝑃𝑟 − 3) + 𝑃𝑟4− 2, 𝐴12 = 𝐴2(𝑃𝑟 − 2𝛼), 𝐴13 = (𝐴2 + 𝐴4)(4 − Pr),

𝐴14 = 𝐴9 + 𝐴12𝑃𝑟

, 𝐴15 = 𝐴13𝑃𝑟1+𝑃𝑟

, 𝐴16 = 𝑃𝑟𝐴10+(𝑃𝑟+2)𝐴15 1+𝑃𝑟

, 𝐴17 = 𝑃𝑟𝐴174+2𝑃𝑟

, 𝐴18 = 32𝐴5 + 7

6𝐴6 + 𝐴16 + 𝐴17 , 𝐴19 = 𝐴14 −

𝑃𝑟𝐴18, 𝐴20 = 𝐴16(1 + 𝑃𝑟) − 𝐴15 , 𝐴21 = 𝐴12 − 𝑃𝑟𝐴14 , 𝐴22 = 2(𝐴4 − 𝐴7) + 2𝑀(𝐴4 − 𝐴7) + 𝐴7 + 𝐴8, 𝐴23 = 1 +2(𝐴4 − 𝐴7) + 𝑀 , 𝐴24 = 𝑃𝑟𝐴5(𝑃𝑟 + 2𝑀) − 2𝐴1𝜆, 𝐴25 = 𝑃𝑟𝐴6(𝑃𝑟 + 𝑀) − 𝐴1𝜆𝛾, 𝐴26 = 𝐴5(𝑃𝑟2 − 4𝑃𝑟 + 1) −2𝜆𝐴1, 𝐴27 = 𝐴6(4𝑃𝑟2 − 8𝑃𝑟 + 1), 𝐴28 = 𝐴22 − 2𝐴42, 𝐴29 = 𝐴23

4− 𝐴4, 𝐴30 = 2𝐴2𝜆

𝑃𝑟3−𝑃𝑟2 , 𝐴31 =

𝐴2𝜆𝛾2𝑃𝑟3−𝑃𝑟2

, 𝐴32 = 𝐴24𝑃𝑟3−𝑃𝑟2

− 3𝑃𝑟−2𝑃𝑟2−𝑃𝑟

𝐴30 , 𝐴33 = 𝐴252𝑃𝑟3−𝑃𝑟2

− 3𝑃𝑟−1𝑃𝑟2−𝑃𝑟

𝐴31 , 𝐴34 = 𝐴26(1+𝑃𝑟)3−(1+Pr)2

,

𝐴35 = 𝐴27(2+𝑃𝑟)3−(2+𝑃𝑟)2

, 𝐴36 = 4𝐴1𝜆𝛾(1+2𝑃𝑟)3−(1+2Pr)2

, 𝐴37 = 18

+ 𝐴42

+ 𝐴28 + 2𝐴29 − 𝐴30 − 𝐴31 − 𝑃𝑟𝐴32 − 2𝑃𝑟𝐴33 +

(1 + 𝑃𝑟)𝐴34 + (2 + 𝑃𝑟)𝐴35 − (1 + 2𝑃𝑟)𝐴36, 𝐴38 = − 112− 𝐴4

2− 𝐴28 − 3𝐴29 + 𝐴30 + 𝐴31 + (𝑃𝑟 − 1)𝐴32 +

(2𝑃𝑟 − 1)𝐴33 − 𝑃𝑟𝐴34 − (1 + 𝑃𝑟)𝐴35 + 2𝑃𝑟𝐴36 , REFERENCES [1] Sakiadis, B. C., (1961). Boundary layer behavior on continuous solid surface: I. Boundary layer equations for two

dimensional and axisymmetric flow. AIChE Journal, 7, 26-28. [2] Sakiadis, B. C., (1961). Boundary layer behavior on continuous solid surface: II. Boundary layer equations for

two dimensional and axisymmetric flow. AIChE Journal, 7, 221-225. [3] Crane, L. J., (1970). Flow past a stretching plate. Zeitschrift four Angewandte Mathematik und physik, 21, 645-

647. [4] Gupta P. S.; Gupta, A. S., (1977). Heat and mass transfer on a stretching sheet with suction and blowing. The

Can. J. Chem. Engg., 55, 744-746. [5] Rajotia, D; Jat, R. N., (2014). Dual solutions of three dimensional MHD boundary layer flow and heat transfer

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[6] Magyari E.; Keller, B., (1999). Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J. Phys. D. Appl. Phys., 32, 577-585.

[7] Elbashbeshy, E. M. A., (2001). Heat transfer over an exponentially stretching continuous surface with suction. Arch. Mech., 53, 643-651.

[8] Khan, S.K. Sanjayanand, E., (2005). Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet. Int. J. Heat Mass Trans., 48, 1534-1542.

[9] Partha, M. K.; Murthy, P. V. S. N.; Rajasekhar, G. P., (2005). Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Heat and Mass Transfer, 41, 360366.

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Nonlinear Convection towards an Exponentially Shrinking Sheet with Magnetic Field