Applied Mathematics and Mechanics MHD flow and

Appl. Math. Mech. -Engl. Ed., 33(7), 899–910 (2012) DOI 10.1007/s10483-012-1593-8 c

Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Applied Mathematics and Mechanics (English Edition)

MHD flow and mass transfer of chemically reactive upper convected Maxwell fluid past porous surface∗ K. VAJRAVELU1 ,

K. V. PRASAD2 ,

A. SUJATHA2 ,

Chiu-on NG (ÇS)3

(1. Department of Mathematics, University of Central Florida, Orlando 32816, Florida, USA; 2. Department of Mathematics, Bangalore University, Bangalore 560001, lndia; 3. Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, P. R. China)

Abstract The magnetohydrodynamic (MHD) flow and mass transfer of an electrically conducting upper convected Maxwell (UCM) fluid at a porous surface are studied in the presence of a chemically reactive species. The governing nonlinear partial differential equations along with the appropriate boundary conditions are transformed into nonlinear ordinary differential equations and numerically solved by the Keller-box method. The effects of various physical parameters on the flow and mass transfer characteristics are graphically presented and discussed. It is observed that the order of the chemical reaction is to increase the thickness of the diffusion boundary layer. Also, the mass transfer rate strongly depends on the Schmidt number and the reaction rate parameter. Furthermore, available results in the literature are obtained as a special case. Key words chemically reactive species, upper convected Maxwell (UCM) fluid, magnetohydrodynamic (MHD) flow, mass transfer, Keller-box method Chinese Library Classification O343.6 2010 Mathematics Subject Classification

1

74F05

Introduction

The study of transport of heat, mass, and momentum in the boundary layer flow over a continuously moving surface through a quiescent liquid has attracted considerable attention during the past several years. The interest is due to several applications in electrochemistry and polymer processing. Flow due to a continuously moving surface is often encountered in wire drawing, glass fiber and paper production, crystal growth, and drawing of plastic films. In view of these applications, Crane[1] initiated the boundary layer flow caused by an elastic sheet whose velocity varies linearly with the distance from a fixed point on the sheet. Since then, the flow, the heat and mass transfer problems with or without suction (blowing), and the magnetic field are considered by several investigators[2–8] . It is well known that many materials, for example, melts, muds, emulsions, soaps, shampoos, paste, molten plastics, polymeric liquids, food stuffs, condensed milk, sugar solution, etc., do not ∗ Received Jul. 18, 2011 / Revised Jan. 23, 2012 Project supported by the Research Grants Council of the Hong Kong Special Administrative Region of China (No. HKU 715510E) Corresponding author Chiu-on NG, Professor, Ph. D., E-mail: [email protected]

900

K. VAJRAVELU, K. V. PRASAD, A. SUJATHA, and Chiu-on NG

obey the Newton’s law of viscosity and are therefore called non-Newtonian fluids. The modeling studies dealing with non-Newtonian fluids offer quite a number of challenges to mathematicians, physicists, and engineers. Hence, it is important to understand the heat and mass transfer in non-Newtonian fluids of practical importance. These fluids often obey nonlinear constitutive equations, and the complexity in the equation is the main reason for the lack of exact analytical solutions. For example, visco-elastic fluid models are simple models, such as second-order models and Walters’ model[9–12] , which are known to be good for weakly elastic fluids subjected to slowly varying flows. These two models are known to violate certain rules of thermodynamics. Therefore, significance of the results reported in the above works is limited for the polymer industry. Obviously, for the theoretical results to be of any industrial importance, more general visco-elastic fluid models such as the upper convected Maxwell (UCM) model or the Oldroyd B model would be invoked in the analysis. Indeed, these two fluid models are being used recently to study the visco-elastic fluid flow over a stretching sheet with or without heat transfer[13–17] . By taking the importance of the mathematical equivalence of the thermal boundary layer problem with the concentration analogue, results obtained for heat transfer characteristics can be carried directly to the case of mass transfer by replacing the Prandtl number by the Schmidt number. However, the term introduced by a chemical reaction in the mass diffusion equation prevents us from obtaining analytical solutions, and one has to use numerical techniques. Flow and mass transfer with chemical reaction is of great practical importance, because of its universal occurrence in many branches of science and engineering. Recently, Hayat et al.[18] investigated the mass transfer in magnetohydrodynamic (MHD) flow of the UCM fluid over a porous shrinking sheet in the presence of a chemical reaction. Available literature on Newtonian/nonNewtonian fluid flows in the presence of a chemical reaction shows that the work is not being carried out for UCM fluid flow in the presence of chemical reaction[19–23] . Motivated by these studies, in the present paper, we study the boundary layer flow with chemically reactive species diffusion in a non-Newtonian UCM fluid over a permeable surface. This is an extension over the work of Akyildiz et al.[22] , where we consider the effects of MHD, non-Newtonian Maxwell, and permeable parameters. The presence of the non-Newtonian Maxwell parameter, and the homogeneous nth-order chemical reaction term, respectively, in the momentum and the mass diffusion equation leads to the coupled and nonlinear partial differential equations (PDEs). These PDEs are converted into nonlinear ordinary differential equations by a similarity transformation. To deal with the coupling and nonlinearity in the boundary value problem, a numerical scheme (known as the Keller-box method) is adopted. The available results in the literature are obtained as a special case and are found to be in good agreement.

2

Mathematical formulation

Consider the steady and incompressible MHD boundary layer flow and mass transfer of an electrically conducting fluid obeying UCM model over a porous stretching sheet. The flow is generated due to the stretching of the sheet by applying two equal and opposite forces along the x-axis, keeping the origin fixed and considering the flow to be confined to the region y > 0. The x- and y-axes are taken along and perpendicular to the sheet, respectively. The continuous stretching sheet is assumed to have a linear velocity uw = bx, where b is the stretching rate, and x is the distance from the slit. A uniform magnetic field of strength B0 is imposed along the y-axis. The induced magnetic field is negligible, which is a valid assumption on a laboratory scale under the assumption of small magnetic Reynolds number and the external electric field is zero. An appropriate mass transfer analogous to the problem would be the flow along a flat plate that contains a species A which is slightly soluble

MHD flow and mass transfer of chemically reactive UCM fluid

901

in B. The concentration at the plate surface would be Cw , and the solubility of A in B and the concentration of species A far away from the plate would be C∞ . Let the reaction of the species A with B be an nth-order homogeneous chemical reaction with a rate constant kn . It is desirable to analyze the system by the boundary layer method[24] . Now, the first step would be to derive the boundary layer equations for our fluid of interest in the particular geometry, and this can be done to start with the Cauchy equations of motion in which a source term due to the magnetic field would also be included (see Ref. [25]). In a two-dimensional flow, the equation of continuity, the equations of motion with no pressure gradient present, and the equation of diffusion (mass transfer) can be written as ∂u ∂v + = 0, ∂x ∂y ∂u ∂u ρ u +v = ∂x ∂y ∂v ∂v +v = ρ u ∂x ∂y u

(1) ∂τxx ∂τxy + − σB02 u, ∂x ∂y

(2)

∂τyx ∂τyy + , ∂x ∂y

(3)

∂C ∂2C ∂C +v = D 2 − Kn C n , ∂x ∂y ∂y

(4)

where u and v are the components of the velocity in the x- and y-directions, respectively, ρ is the density, σ is the electrical conductivity, B0 is the uniform magnetic field, C is the concentration of the species diffusion, D is the diffusion coefficient of the diffusing species, and Kn denotes the reaction rate constant of the nth-order homogeneous and irreversible reaction. As mentioned above, the fluid of interest in the present work obeys UCM model. For a Maxwell fluid, the extra tensor τij can be related to the deformation rate tensor dij by an equation of the form as follows: ∆ τij + λ τij = 2δdij , (5) ∆t ∆ where δ is the coefficient of viscosity, and λ is the relaxation time. The time derivative ∆t appearing in the above equation is the so called upper convected time derivative devised to satisfy the requirements of the continuum mechanics (i.e., material objectivity and frame indifference). This time derivative when applied to the stress tensor reads as follows[25] : D ∆ τij = τij − Ljk τik − Lik τkj , ∆t Dt

(6)

where Lij is the velocity gradient tensor. For an incompressible fluid obeying the UCM model, the x-component of the momentum equation and mass transfer equation can be simplified using the usual boundary layer theory approximations[15] as ∂u ∂u ∂2u ∂2u ∂2u u +v + λ u2 2 + v 2 2 + 2uv ∂x ∂y ∂x ∂y ∂x∂y 2 2 ∂ u σB0 u =ν 2 − , (7) ∂y ρ u

∂C ∂C ∂2C +v = D 2 − Kn C n , ∂x ∂y ∂y

where ν is the kinematic viscosity. The relevant boundary conditions for the problem are ( u = uw = bx, v = vw , C = Cw at y = 0, u → 0,

C → C∞

as y → ∞,

(8)

(9)

902


where b is the stretching rate, the second condition v = vw in Eq. (9) is the blowing velocity across the stretching sheet, whereas v = −vw is the suction velocity. To make the problem amenable, we introduce the following non-dimensional quantities: r  b    η= y,   ν      u = bxf ′ (η), (10) √   v = − bνf (η),        φ(η) = C − C∞ ,  Cw − C∞ where η is the similarity variable, and f and φ are the dimensionless stream function and the mass concentration, respectively. In terms of the new variables, the velocity components u and v automatically satisfy Eq. (1). The governing Eqs. (7) and (8) in terms of the new variables f and φ can be written as f ′′′ − Mnf ′ − f ′2 + f f ′′ + β f ′ f ′′′ − f 2 f ′′′ = 0, (11) φ′′ + Scf φ′ = δScφn ,

(12)

where a prime denotes differentiation with respect to η. The dimensionless parameters in Eqs. (10) and (11) are the magnetic parameter Mn, the Maxwell parameter β, the Schmidt number Sc, and the reaction rate parameter δ, respectively, which can be written as follows:  σB02   Mn = ,    ρb       β = λb, ν   Sc = ,   D     n−1    δ = Kn (Cw − C∞ ) . b

It is worth mentioning here that the chemical reaction parameter δ is a real number (δ < 0 indicates the destructive chemical reaction, δ > 0 denotes the generative chemical reaction, and δ = 0 for the non-reactive species). In view of the above transformations, the boundary conditions (9) take the following non-dimensional form, in terms of the stream function f and the dimensionless mass transfer φ: f = fw , ′

f → 0, where

f ′ = 1,

φ=1

φ → 0 as

at η = 0,

η → ∞,

(13) (14)

vw fw = − √ . νb

It follows that for suction fw is positive and fw is negative for blowing, and this parameter is used to control the strength and direction of the normal flow at the boundary. For practical purposes, the functions f (η) and φ(η) allow us to determine the skin friction coefficient Cf =

µ ∂u ρu2w ∂y y=0

(15)


903

and the Sherwood number Shx =

h ∂C (Cw − C∞ ) ∂y y=0 1

= (Rex ) 2 φ′ (0), respectively. Here, Rex =

(16)

uw x ν

is the local Reynolds number.

3

Exact solutions for some special cases

Here, we present exact solutions in certain special cases. Such solutions are useful and serve as a baseline for comparison with the solutions obtained via a numerical scheme. It is worth mentioning here that when β = 0 and n = 1, the boundary layer flow and mass transfer problem degenerates. In this case, the solution for the velocity field is that given by Gupta and Gupta[2] . Evidently, the mass transfer Eq. (12) is coupled with the velocity field through the dimensionless stream function f in the nonlinear mass transfer equation. However, in a special case of reactive species of first-order, the nonlinear term on the right hand side (RHS) of Eq. (12) becomes δ1 Sc φ, and the present mass diffusion boundary layer problem becomes formally equivalent to the analogous thermal boundary layer problem, in which the Prandtl number replaces the Schmidt number. The analytical solution of Eq. (12) for the first-order reaction with respect to the boundary conditions can be written in terms of the usual confluent hypergeometric function, namely, the Kummer’s function M , to wit −α(a + b )η M a , b , − Sc exp(−αη) 1 1 0 0 α2 φ(η) = exp , (17) Sc 2 M a1 , b 1 , − α 2

where

 Sc Sc   a0 = 2 + 2 f w ,   α α    r     b = a2 + 4δSc , 0 0 α2    a0 + b 0 − 2   a1 = ,    2    b1 = 1 + b0 .

(18)

The exact analytical solution of the complete mass transfer Eq. (12) for the case when β = δ = fw = 0 is expressed in terms of incomplete gamma function as noted in Ref. [26]. In general, for n > 1 and β 6= 0, the presence of nonlinearity in Eqs. (11) and (12) makes numerical solution inevitable. To deal with the coupling and nonlinearity, we use the following numerical procedure to solve the coupled nonlinear boundary value problem.

4

Numerical procedure

The transformed nonlinear coupled ordinary differential Eqs. (11) and (12) with the boundary conditions (13) and (14) are solved numerically by the Keller-box method[27–29] . The numerical solutions are obtained in the following four steps. (i) Reduce Eqs. (11) and (12) to a system of first-order equations.

904


(ii) Write the difference equations by use of central differences. (iii) Linearize the algebraic equations by Newton’s method, and write them in a matrixvector form. (iv) Solve the linear system by the block tridiagonal elimination technique. For the sake of brevity, further details of the solution process are not presented here. It is also important to note that the computational time for each set of input parametric values should be short. Because physical domain in this problem is unbounded, whereas the computational domain has to be finite, we apply the far field boundary conditions for the similarity variable η at a finite value denoted by ηmax . We run our bulk of computations with the value ηmax = 10, which is sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered. For numerical calculations, a uniform step size of ∆η = 0.01 is found to be satisfactory, and the solutions are obtained with an error tolerance of 10−6 in all the cases. To assess the accuracy of the present method, comparison of the skin friction and the mass transfer gradient between the present results and previously published results is made for a special case and is shown in Table 1. Table 1

Comparison of skin friction and wall mass transfer gradient for different values of parameters Mn=0.0

f ′′ (0) with fw = β = 0.0

5

Mn=1.0

Mn=1.5

Mn=2.0

−1.000 174 −1.224 751 −1.414 214 −1.581 139 −1.732 051

Andersson et al.[26]

−1.000 000 −1.224 900 −1.414 400 −1.581 000 −1.732 000 Pr =0.01 Sc

φ′ (0) with δ = fw = β = Mn = 0.0

Mn=0.5

Present results

Pr =0.72 Sc

Pr =1.00 Sc

Pr =3.00 Sc

Pr =10.00 Sc

Present results

−0.009 933 −0.495 977 −0.582 789 −1.159 972 −2.308 010

Grubka and Bobba[4]

−0.009 900 −0.463 100 −0.582 000 −1.165 200 −2.308 000

Chen[5]

−0.009 100 −0.463 150 −0.581 990 −1.165 230 −2.307 960

Results and discussion

Numerical computation is carried out for several sets of values of the Maxwell parameter β, the magnetic parameter Mn, the injection parameter fw , the Schmidt number Sc, the reaction rate parameter δ, and the order of the chemical reaction n. In order to analyze the salient features of the problem, the numerical results are presented in Figs. 1–5. These figures depict the changes in the horizontal velocity and the fluid concentration. Changes in the skin friction and the wall mass transfer gradient for several sets of the pertinent parameters are recorded in Tables 2 and 3. Figures 1(a)–1(c) are the graphical representation of the horizontal velocity component f ′ (η) for different values of the Maxwell parameter β, the magnetic parameter Mn, and the injection parameter fw . From Fig. 1(a), we see that an increase in Mn leads to a decrease in the horizontal velocity. This is because of the fact that the introduction of tensile stress due to elasticity causes a transverse contraction of the boundary layer and the increase of magnetic parameter leads to the enhanced deceleration of the flow. Hence, the velocity decreases. In Figs. 1(b)–1(c), the horizontal velocity profiles f ′ (η) are shown for different values of fw and Mn. It can be seen that the suction parameter, namely, fw > 0 reduces the horizontal velocity boundary layer thickness, where as the blowing fw < 0 has quite the opposite effect on the velocity boundary layer. These results are consistent with the physical situation. In Fig. 1(c), the velocity profiles are presented for the same set of physical parameters for non-zero values of Mn. Comparison of these figures reveals that the effect of Mn is to decrease the horizontal velocity for all values of fw . This is due to the fact that the induction of transverse magnetic field normal to the flow direction has the tendency to create drag known as the Lorenz force, which tends to resist the flow. Hence, the horizontal velocity f ′ (η) profile decreases as Mn increases.


Fig. 1

905

Horizontal velocity profiles for f ′ vs. η

Figures 2(a) and 2(b) depict the concentration profiles φ(η), respectively, for the first (n = 1) and third (n = 3) order chemical reactions, for different values of Mn and when the other parameters are fixed. The effect of the increasing values of β is to enhance the species distribution for different values of pertinent parameters encountered in the flow and mass transfer. However, the species distribution tends asymptotically to zero as the distance increases from the boundary. This behavior is even true for zero values of Mn. This is due to the fact that the addition of a transverse magnetic filed to an electrically conducting fluid will give rise to a resistive type of force known as the Lorenz force. This force makes the fluid experience a resistance by increasing the friction between its layers, and due to this, there is an increase in the species distribution. This trend holds good for nonlinear (higher-order) chemical reaction, as shown in Fig. 2(b). Comparison of these figures reveals that the effect of increasing the order of the chemical reaction n is to enhance the wall concentration gradient (shown in Table 2). Hence, the thickness of the species distribution increases as n increases. Concentration profiles φ(η) across the flow field for the first and higher-order reaction rates are, respectively, plotted in Figs. 3(a) and 3(b) for different values of the parameter δ and the injection parameter fw . From these graphs, we observe that the effect of an increase in the value of δ is to decrease the thickness of the concentration boundary layer in the flow field. However, the concentration thickness is higher for negative values of δ as compared to the zero value of δ. Physically, δ < 0 means the generative chemical reaction, i.e., the species which diffuses from

906


the stretching sheet is produced by the chemical reaction in the free stream, and δ > 0 means the destructive chemical reaction, i.e., to reduce the thickness of the concentration layer and to increase the magnitude of the wall mass transfer, as shown in these figures. This behavior is true for all values of fw , namely, suction fw > 0, impermeability fw = 0, and blowing fw < 0 at the wall. From the graphical representation, we also observe that an increase in fw leads to a decrease in the concentration profile φ(η) and the magnitude of the wall concentration gradient increases with fw . This is due to the fact that the species distribution boundary layer is thicker in the case of suction as compared to the case of impermeability, but is thinner in the case of blowing.

Fig. 2

Concentration profiles φ(η) vs. η for different values of β and Mn with δ=0.5, Sc=1.0, and fw =0.0

Fig. 3

Concentration profiles φ(η) vs. η for different values of fw and δ with Mn=0.5, Sc=1.0, and β=0.2

In Figs. 4(a) and 4(b), the profiles for the concentration species distribution φ are plotted for different values of the Schmidt number Sc and the reaction rate parameter δ for suction when the other parameters are fixed. The effect of increasing values of Sc leads to a decrease in the thickness of the concentration boundary layer. This is due to the thinning of the concentration boundary layer with the introduction of species diffusion. This phenomenon is even true in the cases of destructive chemical reaction δ > 0 and generative chemical reaction δ < 0. By


907

comparing this figure with that for a higher-order chemical reaction, as shown in Fig. 4(b), we see that the concentration boundary layer thickness is higher for a third-order chemical reaction as compared to the first or second-order chemical reaction. This phenomenon is clearly noticeable in Figs. 5(a) and 5(b).

Fig. 4

Concentration profiles φ(η) vs. η for different values of Sc and δ with Mn=0.5, β=0.2, and fw =0.5

Fig. 5

Concentration profiles φ(η) vs. η for different values of β and n with δ=0.5, fw =0.5, and Sc=1.0

The values of f ′′ (0) and φ′ (0) are given in Tables 2 and 3 for different values of the physical parameters, namely, fw , β, n, and Mn. From these tables, we observe that f ′′ (0) decreases monotonically with the increases in the Maxwell parameter, the magnetic parameter, and the injection parameter; whereas the reverse trend is seen with the mass transfer parameter. It is interesting to note that the magnitude of the wall mass transfer increases with the Schmidt number and the reaction rate parameter. It is further noted that the effect of a higher-order chemical reaction is to enhance the mass transfer gradient for destructive chemical reaction. However, the reverse trend is observed with generative chemical reaction. This is true for all values of the injection, the Maxwell, and the magnetic parameters.

908

K. VAJRAVELU, K. V. PRASAD, A. SUJATHA, and Chiu-on NG Skin friction f ′′ (0) and mass transfer gradient ϕ′ (0) for different values of parameters when Sc=1.0 and δ=0.0

Table 2

f ′′ (0) Mn

β

ϕ′ (0)

fw =−0.1

fw =0.0

fw =0.5

fw =−0.1

fw =0.0

fw =0.5

0.0

0.0 0.2 0.4 0.6 0.8

−0.952 761 −0.984 720 −1.014 978 −1.043 609 −1.070 715

−1.000 000 −1.052 725 −1.102 408 −1.150 449 −1.196 909

−1.281 524 −1.514 256 −1.775 909 −2.070 067 −2.401 952

−0.526 856 −0.515 647 −0.505 219 −0.495 543 −0.486 572

−0.587 277 −0.573 263 −0.560 204 −0.548 089 −0.536 867

−0.931 586 −0.899 486 −0.868 124 −0.838 099 −0.809 719

0.5

0.0 0.2 0.4 0.6 0.8

−1.175 941 −1.198 799 −1.220 729 −1.241 766 −1.261 951

−1.224 751 −1.266 705 −1.307 627 −1.347 669 −1.386 841

−1.500 092 −1.723 246 −1.975 395 −2.260 587 −2.584 186

−0.486 423 −0.478 577 −0.471 199 −0.464 263 −0.457 740

−0.548 546 −0.538 091 −0.528 263 −0.519 036 −0.510 378

−0.903 952 −0.876 268 −0.849 022 −0.822 620 −0.797 322

1.0

0.0 0.2 0.4 0.6 0.8

−1.365 123 −1.382 501 −1.399 297 −1.415 531 −1.431 221

−1.414 238 −1.450 259 −1.485 741 −1.520 676 −1.555 062

−1.686 155 −1.904 575 −2.151 651 −2.431 686 −2.750 195

−0.455 745 −0.449 985 −0.444 519 −0.439 329 −0.434 400

−0.519 017 −0.520 853 −0.503 120 −0.495 798 −0.488 865

−0.882 428 −0.857 872 −0.833 541 −0.809 786 −0.786 820

Wall concentration gradient φ′ (0) for different values of physical parameters

Table 3

Mn=0.0 Sc

fw

δ

β

−0.2

n=2

n=3

n=1

n=2

n=3

0.0 0.2

−0.264 804 −0.229 341

−0.390 104 −0.370 848

−0.428 219 −0.411 655

−0.130 296 −0.095 684

−0.319 136 −0.303 563

−0.367 819 −0.355 190

0.5

0.0 0.2

−0.877 700 −0.872 759

−0.764 900 −0.757 764

−0.708 486 −0.699 946

−0.858 928 −0.855 604

−0.739 367 −0.734 704

−0.678 276 −0.672 628

−0.2

0.0 0.2

−0.341 557 −0.302 692

−0.459 967 −0.437 725

−0.496 092 −0.476 600

−0.230 064 −0.192 585

−0.397 534 −0.379 168

−0.441 736 −0.426 286

0.5

0.0 0.2

−0.932 211 −0.925 764

−0.817 280 −0.808 086

−0.760 947 −0.750 061

−0.913 731 −0.909 069

−0.791 992 −0.785 502

−0.731 212 −0.723 436

−0.2

0.0 0.2

−0.753 276 −0.697 736

−0.840 855 −0.800 907

−0.868 457 −0.831 295

−0.705 948 −0.655 301

−0.806 522 −0.771 434

−0.836 472 −0.804 065

0.5

0.0 0.2

−1.237 919 −1.220 170

−1.120 349 −1.096 442

−1.068 162 −1.041 184

−1.222 463 −1.207 301

−1.099 655 −1.079 387

−1.044 860 −1.021 873

δ

Sc

n=1

n=2

n=3

n=1

n=2

n=3

−0.2

1.0 2.0 3.0

−0.229 341 −0.483 304 −0.651 335

−0.370 848 −0.609 898 −0.777 659

−0.411 655 −0.654 109 −0.826 441

−0.095 684 −0.365 487 −0.554 774

−0.303 562 −0.541 915 −0.711 089

−0.355 190 −0.593 486 −0.765 772

1.0 2.0 3.0 1.0 2.0 3.0

−0.872 759 −1.240 781 −1.513 017 −0.302 692 −0.623 190 −0.855 970

−0.757 763 −1.093 749 −1.342 682 −0.437 724 −0.744 091 −0.978 146

−0.699 946 −1.020 779 −1.258 014 −0.476 600 −0.786 138 −1.024 462

−0.855 604 −1.219 742 −1.490 150 −0.192 585 −0.529 834 −0.776 454

−0.734 704 −1.064 541 −1.311 366 −0.379 168 −0.684 944 −0.920 228

−0.672 628 −0.986 774 −1.222 110 −0.426 286 −0.732 501 −0.970 947

1.0 2.0 3.0 1.0 2.0 3.0

−0.925 764 −1.350 814 −1.679 943 −0.697 736 −1.401 269 −2.029 406

−0.808 085 −1.200 179 −1.504 981 −0.800 906 −1.495 245 −2.125 250

−0.750 061 −1.127 326 −1.420 587 −0.831 295 −1.527 121 −2.159 236

−0.909 069 −1.330 199 −1.657 492 −0.655 301 −1.364 146 −1.995 585

−0.785 501 −1.171 610 −1.474 338 −0.771 434 −1.465 283 −2.096 416

−0.723 436 −1.094 379 −1.385 794 −0.804 065 −1.498 576 −2.131 466

1.0 2.0 3.0

−1.220 170 −1.985 098 −2.666 593

−1.096 443 −1.832 242 −2.493 400

−1.041 184 −1.766 812 −2.420 548

−1.207 301 −1.969 211 −2.649 710

−1.079 387 −1.811 471 −2.471 875

−1.021 873 −1.744 013 −2.397 295

−0.1

1.0

0.0

0.5

β

fw

−0.1 0.5

−0.2 0.2

Mn=0.5

n=1

0.0 0.5

−0.2 0.5 0.5


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Acknowledgements The authors thank for the comments of the reviewers, which lead to the definite improvement of the paper. K. V. PRASAD is thankful to the Department of Science and Technology, New Delhi, India, for providing him with the financial support through BOYSCAST fellowship.

References [1] Crane, L. J. Flow past a stretching plate. ZAMP, 21, 645–647 (1970) [2] Gupta, P. S. and Gupta, A. S. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng., 55, 744–746 (1977) [3] Chakrabarti, A. and Gupta, A. S. Hydro-magnetic flow and heat transfer over a stretching sheet. Quart. Appl. Math., 37, 73–78 (1979) [4] Grubka, L. J. and Bobba, K. M. Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME J. Heat Trans., 107, 248–250 (1985) [5] Chen, C. H. Laminar mixed convection adjacent to vertical continuously stretching sheets. Heat Mass Transfer, 33, 471–476 (1998) [6] Cortell, R. Viscous flow and heat transfer over a nonlinearly stretching sheet. Appl. Math. Comp., 184, 864–873 (2007) [7] Ishak, A., Jafar, K., Nazar, R., and Pop, I. MHD stagnation point towards a stretching sheet. Physica A, 388, 3377–3383 (2009) [8] Abbasbandy, S. and Hayat, T. Solution of the MHD Falkner-Skan flow by homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul., 14, 3591–3598 (2009) [9] Rajagopal, K. R., Na, T. Y., and Gupta, A. S. Flow of a visco-elastic fluid over a stretching sheet. Rheol. Acta, 23, 213–215 (1984) [10] Andersson, H. I. MHD flow of a visco-elastic fluid past a stretching surface. Acta Mech., 95, 227–232 (1992) [11] Cortell, R. A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet. Int. J. Non-Linear Mech., 41, 78–85 (2006) [12] Subhas Abel, M. and Mahesha, N. Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl. Math. Model., 32, 1965–1983 (2008) [13] Bhatnagar, R. K., Gupta, G., and Rajagopal, K. R. Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity. Int. J. Non-Linear Mech., 30, 391–405 (1995) [14] Renardy, M. High Weissenberg number boundary layers for upper convected Maxwell fluid. J. Non-Newton. Fluid Mech., 68, 125–132 (1997) [15] Sadeghy, K., Najafi, A. H., and Saffaripour, M. Sakiadis flow of an upper-convected Maxwell fluid. Int. J. Non-Linear Mech., 40, 1220–1228 (2005) [16] Hayat, T., Abbas, Z., and Sajid, M. Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys. Lett. A, 358, 396–403 (2006) [17] Aliakbar, V., Alizadeh-Pahlavan, A., and Sadeghy, K. The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets. Commun. Nonlinear Sci. Numer. Simul., 14, 779–794 (2009) [18] Hayat, T., Abbas, Z., and Ali, N. MHD flow and mass transfer of an upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species. Phys. Lett. A, 372, 4698–4704 (2008) [19] Chambre, P. L. and Young, J. D. On the diffusion of a chemically reactive species in a laminar boundary layer flow. Phys. Fluids, 1, 48–54 (1958) [20] Anjali Devi, S. P. and Kandaswamy, R. Effects of chemical reaction heat and mass transfer on MHD flow past a semi infinite plate. ZAMM, 80, 697–701 (2000) [21] Takhar, H. S., Chamkha, A. J., and Nath, G. Flow and mass transfer on a stretching sheet with a magnetic field and chemically reactive species. Int. J. Eng. Sci., 38, 1303–1314 (2000) [22] Akyildiz, F. T., Bellout, H., and Vajravelu, K. Diffusion of chemically reactive species in a porous medium over a stretching sheet. J. Math. Anal. Appl., 320, 322–339 (2006)

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[23] Cortell, R. Toward an understanding of the motion and mass transfer with chemically reactive species for two classes of viscoelastic fluid over a porous stretching sheet. Chem. Eng. Proc., 46, 982–989 (2007) [24] Gupta, A. S. and Wineman, A. S. On the boundary layer theory for non-Newtonian fluids. Lett. Appl. Eng. Sci., 18, 875–883 (1980) [25] Bird, R. B., Armstrong, R. C., and Hassager, O. Dynamics of Polymeric Liquids, John Wiley and Sons, New York (1987) [26] Andersson, H. I., Hansen, O. R., and Holmedal, B. Diffusion of a chemically reactive species from a stretching sheet. Int. J. Heat Mass Transf., 37, 659–664 (1994) [27] Cebeci, T. and Bradshaw, P. Physical and Computational Aspects of Convective Heat Transfer, Springer-Verlag, New York (1984) [28] Keller, H. B. Numerical Methods for Two-Point Boundary Value Problems, Dover Publications, New York (1992) [29] Andersson, H. I., Bech, K. H., and Dandapat, B. S. Magnetohydrodynamic flow of a power law fluid over a stretching sheet. Int. J. Non-Linear Mech., 27, 929–936 (1992)