Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid

Appl. Math. Mech. -Engl. Ed., 33(10), 1301 1312 (2012) DOI 10.1007/s10483-012-1623-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2012 Applied Mathematics and Mechanics (English Edition) Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid T. HAYAT 1,2, S. A. SHEHZAD 1, A. ALSAEDI 2 (1. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan; 2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia) Abstract This article studies the Soret and Dufour effects on the magnetohydrodynamic (MHD) flow of the Casson fluid over a stretched surface. The relevant equations are first derived, and the series solution is constructed by the homotopic procedure. The results for velocities, temperature, and concentration fields are displayed and discussed. Numerical values of the skin friction coefficient, the Nusselt number, and the Sherwood number for different values of physical parameters are constructed and analyzed. The convergence of the series solutions is examined. Key words Soret and Dufour effects, magnetohydrodynamic (MHD) flow, Casson fluid, heat and mass transfer Chinese Library Classification O373 2010 Mathematics Subject Classification 76A05 1 Introduction Non-Newtonian fluid flows generated by a stretching sheet have been widely analyzed for the importance in several manufacturing processes such as extrusion of molten polymers through a slit die for the production of plastic sheets, processing of food stuffs, paper production, and wire and fiber coating. The quality of the final product in such processes greatly depends on the rate of cooling in the heat transfer process. The magnetohydrodynamic (MHD) parameter is one of the important parameters by which the cooling rate can be controlled and the product of the desired quality can be achieved. Crane [1] provided the closed form solution for steady and two-dimensional incompressible boundary layer flows of viscous fluids generated by a stretching surface. This flow problem has been extended under diverse physical aspects. At the present, we only refer some recent representative studies on stretched flows. Hassani et al. [2] investigated the analytical solutions for the boundary layer flow of a nanofluid past a stretching surface. Kazem et al. [3] studied the stagnation point flow of a viscous fluid over a porous stretching surface. Hayat et al. [4] discussed the slip effects on the flow of a second grade fluid past a streched surface in a porous space. Rahman [5] studied the Hiemenz flow of a viscous fluid over a linearly stretching sheet. He analyzed the heat and mass transfer characteristics in the presence of Soret and Dufour effects. Yao and Chen [6] analytically discussed the Falkner-Skan equation with the Received May 18, 2011 / Revised Dec. 28, 2011 Project supported by the Deanship of Scientific Research (DSR) of King Abdulaziz University of Saudi Arabia Corresponding author T. HAYAT, Professor, Ph. D., E-mail: pensy t@yahoo.com

1302 T. HAYAT, S. A. SHEHZAD, and A. ALSAEDI stretching boundary. Fang et al. [7] carried out a study on the unsteady boundary layer flow over a stretched surface. Yao et al. [8] analyzed the effects of the convective surface boundary condition on the boundary layer flow of a viscous fluid. Hayat et al. [9] presented an analysis for the flow of a Maxwell fluid with heat and mass transfer in the presence of a chemical reaction. Hayat et al. [10] preseted a study on the radiation effects on the MHD flow induced by a streching sheet. Ahmad and Asghar [11] analyzed the boundary layer flow of a second grade fluid over a sheet stretched with arbitrary velocities. Muhaimina et al. [12] discussed the boundary layer flow of a viscous fluid over a porous shrinking sheet in the presence of suction. Having in mind the above reported studies on the boundary layer flow, we venture further in the regime of two-dimensional flows of the Casson fluid. In addition, the Soret and Dufour effects are considered. The fluid is taken to be electrically conducting and the flow is induced by a stretching surface. The Soret effect (thermal diffusion) is the occurrence of a diffusion flux because of a temperature gradient, whereas the Dufour effect is the occurrence of a heat flux due to a chemical potential gradient. Such effects have worth in the areas of geoscience and chemical engineering. The thermal diffusion effect has been utilized for isotope separation and in mixtures between gases with very light molecular weight (e.g., H 2 and He) and of medium molecular weight (e.g., N 2 and air), and the diffusion-thermo effect is significant. Moreover, the coupled heat and mass transfer have several applications in engineering problems. Such specific applications include the migration of moisture through air contained in fibrous insulation and grain storage insulations and chemical pollutants spreading into soil and medicine diffusion in blood veins [13]. Moreover, most of the performed experiments about blood suggested the blood as a Casson fluid [14 15]. Having all these in mind, the layout of this article is as follows. The problem formulation is presented in Section 2. Sections 3 and 4 develop the solutions and the related convergence analysis by the homotopy analysis method [16]. Vosughi et al. [17] also presented an optimal homotopy analysis method for the calculation of strong nonlinear problems. Homotopy solutions for interesting problems have been developed in Refs. [18 26]. Section 5 in this paper presents the graphical illustrations. Section 6 contains the main conclusions. 2 Mathematical model Let us consider the steady MHD flow of an incompressible Casson fluid over a heated stretching surface at y = 0. The surface is elastic. The x-axis is chosen to be parallel to the surface and the y-axis is normal to the surface. The motion in an incompressible fluid is induced because of the stretching property. This occurs in view of the elastic properties of the surface parallel to the x-axis through equal and opposite forces when the origin is fixed (see Refs. [27 28]). A constant magnetic field B 0 is exerted in the transverse direction to the surface. The induced magnetic field is negligible due to small magnetic Reynolds numbers. We also considered the heat and mass transfer processes in the presence of Soret and Dufour effects. The rheological equation of state for an isotropic and incompressible flow of the Casson fluid is ( τ ij = 2 µ B + p y )e ij, π π c, (1) 2π where π = e ij e ij with e ij being the (i, j)th component of the deformation rate, π depicts the product of the component of the deformation rate with itself, π c denotes a critical value of this product based on the non-newtonian model, µ B indicates the plastic dynamic viscosity of non-newtonian fluids, and p y is the yield stress of the fluid. The resulting boundary layer equations in the MHD flow under consideration are u x + v = 0, (2) y

Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid 1303 u u (1 x + v u y = ν + 1 ) 2 u β y 2 σb2 0 ρ u, (3) u T x + v T y = α 2 T T y 2 + D mk T 2 C C s c p y 2, (4) u C x + v C y = D 2 C m y 2 + D mk T 2 T T m y 2, (5) where u and v denote the velocity components in the x- and y-directions, respectively, β = µ B 2πc /p y is the Casson parameter, T the fluid temperature, C is the concentration field, ν is the kinematic viscosity, ρ is the fluid density, D m is the mass diffusivity, α T is the thermal conductivity, k T is the thermal-diffusion ratio, c p is the specific heat, C s is the concentration susceptibility, T m is the fluid mean temperature, and σ is the fluid electrical conductivity. The boundary conditions for the present analysis can be written as { u = uw (x) = cx, T = T w (x) = T + ax, (6) v = 0, C = C w (x) = C + bx at y = 0, and u 0, T T, C C (7) as y, where c, a, and b are all positive constants and have the dimension (time) 1. For the stretching flow c > 0. Here, T w and C w are the variable temperature and concentration, respectively. T is the uniform ambient temperature, and C is the uniform ambient concentration. Using the following transformations: c η = y ν, u = cxf (η), v = cνf(η), (8) θ(η) = T T T w T, φ(η) = C C C w C, the continuity equation (1) is identically satisfied, and the other equations become ( 1 + 1 ) f + ff f 2 Ha 2 f = 0, (9) β 1 Pr θ + fθ f θ + Dfφ = 0, (10) φ + PrLefφ PrLef φ + SrLeθ = 0, (11) f(0) = 0, f (0) = 1, f ( ) = 0, (12) θ(0) = 1, θ( ) = 0, (13) φ(0) = 1, φ( ) = 0. (14) In these expressions, the prime indicates the differentiation with respect to η, Ha, Pr, and Le are the Hartman number, the Prandtl number, and the Lewis number expressed by Ha 2 = σ ρc B2 0, Pr = υ α T, Le = α T D e,

1304 T. HAYAT, S. A. SHEHZAD, and A. ALSAEDI and Df and Sr are the Dufour number and the Soret number defined by Df = D mk T C s c p (C w C ) (T w T )υ, Sr = D mk T T m α T (T w T ) (C w C ). (15) The dimensionless expressions of the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are where Re x = u w (x)x/ν is the local Reynolds number. 3 Series solutions Re 1 2 x Cf = (1 + 1/β)f (0), (16) Nu(Re x ) 1 2 = θ (0), (17) Sh(Re x ) 1 2 = φ (0), (18) In order to proceed the desired solutions, we select the following initial guesses and auxiliary linear operators: f 0 (η) = (1 exp( η)), θ 0 (η) = exp( η), φ 0 (η) = exp( η), (19) L f = d3 f dη 3 df dη, L θ = d2 θ dη 2 θ, L φ = d2 φ φ, (20) dη2 L f (C 1 + C 2 exp(η) + C 3 exp( η)) = 0, (21) { Lθ (C 4 exp(η) + C 5 exp( η)) = 0, L φ (C 6 exp(η) + C 7 exp( η)) = 0, (22) in which C i (i = 1, 2,, 7) denote the arbitrary constants. 3.1 Zeroth- and mth-order deformation problems Let the non-linear operators N f, N θ and N φ be in the forms N f ( ˆf(η, p)) = ( 1 + 1 ) 3 ˆf(η, p) β η 3 + ˆf(η, p) 2 ˆf(η, p) η 2 ( ˆf(η, p) ) 2 Ha 2 ˆf(η, p), (23) η η N θ ( ˆf(η, p), ˆθ(η, p), ˆφ(η, p)) = 1 2ˆθ(η, p) Pr η 2 + ˆf(η, p) ˆθ(η, p) η ˆθ(η, p) ˆf(η, p) η N φ ( ˆf(η, p), ˆθ(η, p), ˆφ(η, p)) = 2 ˆφ(η, p) η 2 + PrLe ˆf(η, p) ˆφ(η, p) η PrLeˆφ(η, p) ˆf(η, p) η + Df 2 ˆφ(η, p) η 2, (24) + SrLe 2ˆθ(η, p) η 2. (25) Let p [0, 1] and f, θ, and φ be non-zero auxiliary parameters. Then, the zeroth-order

Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid 1305 problems have the forms (1 p)l f ( ˆf(η, p) f 0 (η)) = pħ f N f ( ˆf(η, p), ˆθ(η, p), ˆφ(η, p)), (26) (1 p)l θ (ˆθ(η, p) θ 0 (η)) = pħ θ N θ ( ˆf(η, p), ˆθ(η, p), ˆφ(η, p)), (27) (1 p)l φ (ˆφ(η, p) φ 0 (η)) = pħ φ N φ ( ˆf(η, p), ˆθ(η, p), ˆφ(η, p)), (28) ˆf(η; p) = 0, p) = 1, p) = 0, η=0 η η η=0 η= (29) η= ˆθ(η; p) = 1, ˆθ(η; p) = 0, (30) η=0 η= ˆφ(η; p) = 1, ˆφ(η; p) = 0. (31) η=0 For the mth-order deformation problems, we first differentiate Eqs.(26) (31) m times with respect to p, divide them by m!, and then set p = 0. Then, we have L f (f m (η) χ m f m 1 (η)) = ħ f R f m(η), (32) L θ (θ m (η) χ m θ m 1 (η)) = ħ θ R θ m(η), (33) L φ (φ m (η) χ m φ m 1 (η)) = ħ φ R φ m(η), (34) f m (0) = 0, f m (0) = 0, f m ( ) = 0, θ m (0) = 0, θ m ( ) = 0, (35) φ m (0) = 0, φ m ( ) = 0, m 1 R f m(η) = (1 + 1/β)f m 1(η) + (f m 1 k f k f m 1 kf k) Ha 2 f m 1(η), (36) k=0 R θ m(η) = 1 m 1 Pr θ m 1(η) + (f m 1 k θ k f m 1 kθ k ) + Dfφ m 1(η), (37) k=0 m 1 R φ m (η) = φ m 1 (η) + PrLe (f m 1 k φ k f m 1 k φ k) + SrLeθ m 1 (η), (38) k=0 χ m = { 0, m 1, 1, m > 1. By using Taylor s series, we have ˆf(η; p) = f 0 (η) + f m (η) = 1 m! f m (η)p m, m=1 m ˆf(η; p) p m, p=0 (39) (40)

1306 T. HAYAT, S. A. SHEHZAD, and A. ALSAEDI ˆθ(η; p) = θ 0 (η) + θ m (η)p m, m=1 θ m (η) = 1 mˆθ(η; p) m! p m, p=0 ˆφ(η; p) = φ 0 (η) + φ m (η) = 1 m! φ m (η)p m, m=1 m ˆφ(η; p) p m. p=0 (41) (42) For p = 0 and p = 1, one may write ˆf(η; 0) = f 0 (η), ˆf(η; 1) = f(η), (43) ˆθ(η; 0) = θ 0 (η), ˆθ(η; 1) = θ(η), (44) ˆφ(η; 0) = φ 0 (η), ˆφ(η; 1) = φ(η). (45) The auxiliary parameters are selected so that the series solutions converge for p = 1. Therefore, f(η) = f 0 (η) + θ(η) = θ 0 (η) + φ(η) = φ 0 (η) + f m (η), (46) m=1 θ m (η), (47) m=1 φ m (η). (48) The general solutions (f m, θ m, and φ m ) in terms of the special solutions (f m, θ m, and φ m ) are given by m=1 f m (η) = f m (η) + C 1 + C 2 exp(η) + C 3 exp( η), (49) θ m (η) = θ m (η) + C 4 exp(η) + C 5 exp( η), (50) φ m (η) = φ m(η) + C 6 exp(η) + C 7 exp( η). (51) 4 Convergence of homotopy solutions The convergence of the series solutions (46) (48) is examined by the parameters f, θ, and φ. For this aim, we draw the -curves for the 20th-order approximation in Fig.1. From these curves, it is found that the admissible ranges of f, θ, and φ are 1.1 f 0.15, 1.3 θ 0.4, 1.2 φ 0.4. The series (32) (34) converge in the whole region of η when f = 0.6 and θ = φ = 0.8. Table 1 indicates that how many terms for each physical quantity are necessary in the convergent solution. It is noticed that less terms are required in the convergent expression of the velocities.

Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid 1307 Fig. 1 ħ-curves for functions f (0), θ (0), and φ (0) at 20th-order approximation with β = 1.5, Ha = 0.5, Pr = 0.7,, Df = 0.5, and Le = 1.0 Table 1 Convergence of homotopy solutions for different orders of approximations when β = 1.5, Ha = 0.5, Pr = 0.7, Df = 0.5,, Le = 1.3, f = 0.6, and θ = φ = 0.8 Order 1 6 15 24 30 37 46 50 f (0) 0.854 17 0.866 03 0.86603 0.866 03 0.866 03 0.86603 0.866 03 0.866 03 θ (0) 0.680 00 0.581 25 0.56529 0.564 31 0.654 18 0.56411 0.564 11 0.564 11 φ (0) 0.756 00 0.814 40 0.81994 0.819 96 0.819 96 0.81996 0.819 96 0.819 96 5 Discussion In this section, we aim to discuss the effects of several important parameters on the velocity, temperature, concentration, the skin-friction coefficient, the local Nusselt number, and the local Sherwood number. The results are plotted in Figs. 2 15 and listed in Tables 2 and 3. The effects of the Casson parameter β and the Hartman number Ha on the velocity f (η) are shown in Figs.2 and 3. Figure 2 shows that the Casson parameter decreases both the velocity and the boundary layer thickness. Figure 3 shows that the velocity decreases when Ha increases. The Hartman number is due to the Lorentz force, which opposes the flow. That is why the velocity is a decreasing function of the Hartman number. Fig. 2 Effects of β on f (η) with Ha = 0.5 Fig. 3 Effects of Ha on f (η) with β = 1.0 Figures 4 9 are plotted to see the effects of the Casson parameter β, the Hartman number Ha, the Prandtl number Pr, the Lewis number Le, the Dufour number Df, and the Soret number Sr on the velocity θ(η). From Figs. 4 and 5, we can see that the temperature and the thermal boundary layer thickness are increasing functions of β and Ha. Figure 6 shows that both the temperature and the thermal boundary layer thickness decrease when the Prandtl number

1308 T. HAYAT, S. A. SHEHZAD, and A. ALSAEDI increases the thermal diffusivity, and the temperature increases by increasing the Lewis and Dufour numbers. Figures 7 and 8 show that an increase in the temperature due to the Dufour number is larger than that due to the Lewis number. Figure 9 depicts that an increase in the Soret number shows decreases in the temperature and the associated boundary layer thickness. Fig. 4 Effects of β on θ(η) with Ha = 0.5, Pr = 0.7, Le = 1.0, Df = 0.5, and Fig. 5 Effects of Ha on θ(η) with β = 1.0, Pr = 0.7, Le = 1.0, Df = 0.5, and Fig. 6 Effects of Pr on θ(η) with β = 1.0, Ha = 0.5, Le = 1.0, Df = 0.5, and Fig. 7 Effects of Le on θ(η) with β = 1.0, Ha = 0.5, Pr = 0.7, Df = 0.5, and Fig. 8 Effects of Df on θ(η) with β = 1.0, Ha = 0.5, Pr = 0.7, Le = 1.0, and Fig. 9 Effects of Sr on θ(η) with β = 1.0, Ha = 0.5, Pr = 0.7, Le = 1.0, and Df = 0.5 The variation of several parameters on the concentration field φ(η) are presented in Figs. 10 15. The effects of the Casson parameter and the Hartman number on the concentration profile

Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid 1309 are qualitatively similar. Such observations in a qualitative sense are similar to those of the temperature (see Figs. 10 and 11). The concentration field decrease when the Prandtl number increases. The increase in the Lewis number leads to decreases in the concentration profile and the associated boundary layer thickness. It is interesting to note from Figs. 7 and 13 that the Lewis number has opposite effects on the temperature and concentration. Here, we also observed Fig. 10 Effects of β on φ(η) with Ha = 0.5, Pr = 0.7, Le = 1.0, Df = 0.5, and Fig. 11 Effects of Ha on φ(η) with β = 1.0, Pr = 0.7, Le = 1.0, Df = 0.5, and Fig. 12 Effects of Pr on φ(η) with β = 1.0, Ha = 0.5, Le = 1.0, Df = 0.5, and Fig. 13 Effects of Le on φ(η) with β = 1.0, Ha = 0.5, Pr = 0.7, Df = 0.5, and Fig. 14 Effects of Df on φ(η) with β = 1.0, Ha = 0.5, Pr = 0.7, Le = 1.0, and Fig. 15 Effects of Sr on φ(η) with β = 0.1, Ha = 0.5, Pr = 0.7, Le = 1.0, Df = 0.5

1310 T. HAYAT, S. A. SHEHZAD, and A. ALSAEDI that the variation of the concentration profile is larger when compared with the temperature profile. Figure 14 depicts that an increase in the Dufour number decreases the concentration field. An increase in the Soret number increases the concentration and the boundary layer thickness. The effects of the Soret number on the temperature and concentration fields are quite opposite. Table 2 shows the numerical values of the skin-friction coefficient for different values of β and Ha. An increase in β decreases the skin-friction coefficient whereas an increase in Ha increases the skin-friction coefficient. The numerical values of the local Nusselt number and the local Sherwood numbers are presented in Table 3 for different values of β, Ha, Pr, Sr, Df, and Le. The values of the local Nusselt number and the local Sherwood number increase by increasing Pr, whereas the local Nusselt number increases by increasing Sr. The local Sherwood number decreases when Sr increases. Table 2 Numerical values of skin friction coefficient (1 + 1/β)f (0) for different values of β and Ha β 0.8 1.4 2.0 3.0 0.8 0.8 0.8 0.8 Ha 0.5 0.5 0.5 0.5 0.0 0.6 1.2 1.7 (1 + 1/β)f (0) 1.677 05 1.463 85 1.36931 1.290 99 1.224 75 1.42829 1.91312 2.415 57 Table 3 Numerical values of local Nusselt number θ (0) and local Sherwood number φ (0) for different values of β, Ha, Pr, Sr, Df, and Le β Ha Pr Sr Df Le θ (0) φ (0) 0.8 0.5 0.7 0.4 0.5 1.0 0.650 27 0.710 96 1.4 0.5 0.7 0.4 0.5 1.0 0.621 82 0.682 52 3.0 0.5 0.7 0.4 0.5 1.0 0.592 41 0.633 05 2.0 0.0 0.7 0.4 0.5 1.0 0.631 57 0.692 27 2.0 0.6 0.7 0.4 0.5 1.0 0.596 48 0.657 14 2.0 1.2 0.7 0.4 0.5 1.0 0.519 23 0.579 03 2.0 0.5 0.5 0.4 0.5 1.0 0.477 51 0.528 59 2.0 0.5 1.0 0.4 0.5 1.0 0.771 42 0.843 93 2.0 0.5 1.5 0.4 0.5 1.0 0.999 05 1.087 74 2.0 0.5 0.7 0.0 0.5 1.0 0.552 31 0.816 55 2.0 0.5 0.7 0.5 0.5 1.0 0.641 17 0.575 30 2.0 0.5 0.7 1.0 0.5 1.0 0.736 79 0.332 97 2.0 0.5 0.7 0.4 0.0 1.0 0.816 54 0.605 13 2.0 0.5 0.7 0.4 0.7 1.0 0.504 25 0.698 48 2.0 0.5 0.7 0.4 1.5 1.0 0.321 68 0.755 79 2.0 0.5 0.7 0.4 0.5 0.0 0.694 82 0.509 57 2.0 0.5 0.7 0.4 0.5 1.3 0.612 94 0.784 12 2.0 0.5 0.7 0.4 0.5 2.0 0.518 21 1.072 66 6 Conclusions The MHD flow of the Casson fluid with the Soret and Dufour effects is examined. The results are as follows. (i) The increases in the Casson parameter β and the Hartman number Ha decrease the velocity f (η) while increase the temperature and concentration profiles. (ii) The temperature, the concentration, and the boundary layer thickness decrease by increasing the Prandtl number Pr.

Soret and Dufour effects on magnetohydrodynamic (MHD) flow of Casson fluid 1311 (iii) The effects of the Soret number Sr on θ(η) and φ(η) are reverse. (iv) The thermal boundary layer thickness and the temperature field increase when Df increases. (v) The impacts of β and Ha on the skin-friction coefficient are opposite. References [1] Crane, L. J. Flow past a stretching plate. Journal of Applied Mathematics and Physics (ZAMP), 21, 645 647 (1970) [2] Hassani, M., Tabar, M. M., Nemati, H., Domairry, G., and Noori, F. An analytical solution for boundary layer flow of a nanofluid past a stretching sheet. International Journal of Thermal Sciences, 50, 2256 2263 (2011) [3] Kazem, S., Shaban, M., and Abbasbandy, S. Improved analytical solutions to a stagnation-point flow past a porous stretching sheet with heat generation. Journal of the Franklin Institute, 348, 2044 2058 (2011) [4] Hayat, T., Javed, T., and Abbas, Z. Slip flow and heat transfer of a second grade fluid past a stretching sheet through a porous space. International Journal of Heat and Mass Transfer, 51, 4528 4534 (2008) [5] Rahman, G. M. A. Thermal-diffusion and MHD for Soret and Dufour s effects on Hiemenz flow and mass transfer of fluid flow through porous medium onto a stretching surface. Physica B, 405, 2560 2569 (2010) [6] Yao, B. and Chen, J. Series solution to the Falkner-Skan equation with stretching boundary. Applied Mathematics and Computation, 208, 156 164 (2009) [7] Fang, T., Zhang, J., and Yao, S. A new family of unsteady boundary layers over a stretching surface. Applied Mathematics and Computation, 217, 3747 3755 (2010) [8] Yao, S., Fang, T., and Zhang, J. Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Communications in Nonlinear Science and Numerical Simulation, 16, 752 760 (2011) [9] Hayat, T., Awais, M., Qasim, M., and Hendi, A. A. Effects of mass transfer on the stagnation point flow of an upper-convected Maxwell (UCM) fluid. International Journal of Heat and Mass Transfer, 54, 3777 3782 (2011) [10] Hayat, T., Qasim, M., and Abbas, Z. Radiation and mass transfer effects on the magnetohydrodynamic unsteady flow induced by a stretching sheet. Zeitschrift für Naturforschung, 65a, 231 239 (2010) [11] Ahmad, A. and Asghar, S. Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field. Applied Mathematics Letters, 24, 1905 1909 (2011) [12] Muhaimina, Kandasamy, R., and Hashim, I. Effect of chemical reaction, heat and mass transfer on nonlinear boundary layer past a porous shrinking sheet in the presence of suction. Nuclear Engineering and Design, 240, 933 939 (2010) [13] Kandasamy, R., Periasamy, K., and Prabhu, K. K. S. Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects. International Journal of Heat and Mass Transfer, 48, 4751 4761 (2005) [14] Mrill, E. W., Benis, A. M., Gilliland, E. R., Sherwood, T. K., and Salzman, E. W. Pressure flow relations of human blood hollow fibers at low flow rates. Journal of Applied Physiology, 20, 954 967 (1965) [15] McDonald, D. A. Blood Flows in Arteries, 2nd ed., Arnold, London (1974) [16] Liao, S. J. Beyond Perturbation: Introduction to Homotopy Analysis Method, CRC Press, Boca Raton (2003) [17] Vosughi, H., Shivanian, E., and Abbasbandy, S. A new analytical technique to solve Volterra s integral equations. Mathematical Methods in the Applied Sciences, 34, 1243 1253 (2011) [18] Abbasbandy, S. and Shirzadi, A. Homotopy analysis method for a nonlinear chemistry problem. Studies in Nonlinear Sciences, 1, 127 132 (2010)

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