MHD MIXED CONVECTION FLOW OF A NON-NEWTONIAN POWELL- ERYING FLUID OVER A PERMEABLE EXPONENTIALLY SHRINKING SHEET

The magnetohydrodynamic (MHD) mixed convection heat transfer in a non-Newtonian Powell-Erying fluid flow due to an exponentially shrinking porous sheet is investigated. Both assisting and opposing flows are considered. After use of the suitable transformations, the governing equations become non-similar ODEs. Numerical computations of resulting equations are obtained by very efficient shooting method for several values of involved parameters. The results exhibit that dual non-similar solutions can be found only when some amount of fluid mass is sucked from the flow field through the porous sheet. Many important results on the effect of external magnetic field on mixed convective flow of Powell-Erying fluid have been explored. Where, it is found that dual non-similar solutions exist for the opposing flow, while for the assisting flow, solution is unique and for steady Powell-Erying fluid flow stronger mass suction is required compared to the Newtonian fluid flow.


INTRODUCTION
The magnetohydrodynamic (MHD) flow of fluid and heat transfer due to a stretching/shrinking sheet is occurred in many problems in geophysics and astrophysics and has several engineering applications such as in MHD generators, geothermal energy extractions, plasma studies, control of boundary layer in aerodynamics and many more. In particular, to control the behavior of the boundary layer, several methods are already developed and out of that, the application of MHD principle is very important method which can alter the structure of boundary layer to the desired direction. The MHD parameter is quite important in controlling the cooling rate and to achieve the best quality product (Hayat et al., 2012a). A comprehensive survey of MHD studies and their applications can be found in the book by Moreau (1990). The study of the MHD boundary layer flow of electrically conducting fluid in presence of a uniform magnetic field was considered by Ariel (1994). Later, Mahapatra and Gupta (2001) studied the MHD stagnation-point flow over a stretching sheet. The radiation effect on MHD flow and heat transfer towards a porous stretching surface was examined by Mukhopadhyay et al. (2010). Ali et al. (2011) investigated the steady MHD mixed convection stagnation-point flow of a electrically conducting fluid over a vertical flat plate. Mukhopadhyay (2013) investigated MHD boundary layer flow with heat transfer effects over a permeable exponential stretching sheet. Tamim et al. (2014) studied the MHD mixed convective stagnation point flow of a nanofluid over a vertical permeable plate.
In the above discussions, Newtonian fluid flows are discussed. But in modern engineering, many fluids show non-Newtonian behavior, therefore many researchers are more interested in those industrial non-Newtonian fluids and their dynamics. A single constitutive equation is not sufficient to describe all physical properties of non-Newtonian fluids and thus several non-Newtonian fluid models (Wilkinson, 1970;Rajagopal, 1980;Rajagopal et al., 1984;Dorier and Tichy, 1992) have been introduced to explain all such behaviors. The one of such non-Newtonian fluid is Powell-Eyring fluid. The Powell-Eyring fluid model has a clear advantage over all the other non-Newtonian fluids because it is derived from kinetic theory of gases rather than from the empirical relations. Moreover, its behavior is like a viscous fluid when shear rates are high (Roşca and Pop, 2014a). The boundary layer flow of a Powell-Eyring fluid past a moving surface with convective boundary condition was discussed by Hayat et al. (2012b). Jalil et al. (2013) studied the flow of Powell-Eyring fluid and heat transfer over a continuously moving permeable surface in the presence of a parallel free stream. Panigrahi et al. (2014) investigated the mixed convective flow of Powell-Eyring fluid past a nonlinear permeable stretching surface with magnetic field. Recently, Akbar et al. (2015) investigated MHD boundary layer flow of a Powell-Eyring fluid past a stretching surface.
Recently, the boundary layer flow developed over a shrinking sheet has attracted the considerable attention of many researchers similar to that of stretching sheet for its increasing technological applications. The flow due to the shrinking sheet is more physically interesting than that of the stretching sheet flow. The existence and uniqueness of the similarity solution for the flow past a shrinking surface were investigated by Miklavčič and Wang (2006). The flow patterns of various Newtonian and non-Newtonian fluids on a shrinking sheet are discussed by many researchers Fang and Zhang, 2009;Lok et al., 2011;. On the other

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Available at www.ThermalFluidsCentral.org 2 hand, very limited attention has been given to study the flow over an exponentially stretching/shrinking sheet though it is very significant in numerous engineering problems. First, Magyari and Keller (1999) investigated the boundary layer flow and heat transfer due to exponentially stretching sheet. The flow and heat transfer over an exponentially stretching sheet with wall mass suction was examined by Elbashbeshy (2001). Al-Odat et al. (2006) investigated the MHD effect on the thermal boundary layer over an exponentially stretching surface with temperature distribution being in exponential form. Later, Sajid and Hayat (2008) examined the effect of thermal radiation on the boundary layer flow over exponentially stretching sheet and reported series solutions by homotopy analysis method (HAM) for velocity and temperature. Ishak (2011) investigated numerically the radiation effect on flow and heat transfer past exponentially stretching surface. On the other hand, the flow due to exponential shrinking sheet with heat transfer was analysed by Bhattacharyya (2011) and the effect of external magnetic field on this flow was illustrated by Bhattacharyya and Pop (2011). Boundary layer flow over an exponentially shrinking sheet near a stagnation-point was discussed by Bhattacharyya and Vajravelu (2012) and same problem with nanoparticle effect was studied by Bachok et al. (2012). Ara et al. (2014) investigated the heat transfer in boundary layer flow of an Eyring-Powell fluid in presence of thermal radiation. Ghosh and Mukhopadhyay (2017) demonstrated the steady nanofluid flow past an exponentially permeable shrinking sheet in presence of slip. In this paper, the properties of MHD mixed convective boundary layer flow and heat transfer of non-Newtonian Powell-Eyring fluid due to a permeable exponentially shrinking sheet are discussed. The effects of external magnetic field, mixed convection parameter, suction and fluid parameters on the skin friction coefficient and the local Nusselt number along with the velocity and temperature profiles are analyzed and discussed. In best of our knowledge, the investigation of this mixed convection flow is not reported so far.

MATHEMATICAL FORMULATION
Let us consider an incompressible steady 2D flow of non-Newtonian Powell-Eyring fluid past a permeable exponentially shrinking sheet. The shrinking velocity and surface temperature are in exponential mode. The velocity of the shrinking sheet is with a being shrinking constant, the variable surface temperature is 2/ xL w T T be   and constant temperature away from the sheet is T  , where w TT   corresponds to heated surface (assisting flow) and w TT   corresponds to cooled surface (opposing flow), respectively. A variable magnetic field is taken and is applied normal to the sheet. The magnetic Reynolds number for the flow is considered to be very small, so that the induced magnetic field can be neglected. The constitutive equations for Eyring-Powell fluid model is given below (Hayat et al., 2012b): (1) and 3 1 1 1 1 1 1 sinh , where  is the viscosity, 1  and c are the material fluid parameters.
Under these conditions boundary layer equations for steady flow of Eyring-Powell fluid and heat transfer over a permeable exponentially shrinking surface may be written in usual notation as: The corresponding boundary conditions are ( ), , where u and v are the velocity components in x and y directions respectively, a is a positive constant,  is the kinematic fluid viscosity,  is the density,  is the thermal diffusivity and p c is the specific heat. The physical sketch of the problem is given in Fig. 1. Let us introduce the following transformation: Using (8), the continuity equation (3) is automatically satisfied and the nonlinear partial differential equations (4) and (5) are converted into the following non-similar ordinary differential equations: and the boundary conditions (6) and (7) become (0) , (0) 1, (0) 1, where prime denotes differentiation with respect to , where w  is shear stress along the exponentially shrinking sheet and w q is heat flux from the sheet and those are defined as Therefore, the wall skin friction coefficient f C and local Nusselt number x Nu are found as follows: where Re xw xu   is local Reynolds number.

RESULTS AND DISCUSSION
The transformed equations (9) & (10) with boundary conditions (11) are solved numerically using shooting method and the detail description of the scheme can be found in the article by Bhattacharyya (2013). The study of the obtained numerical results explores the condition for which steady flow of the non-Newtonian Powell-Eyring fluid is possible. According to study of Miklavčič and Wang (2006) and Fang and Zhang (2009), for Newtonian fluids, the steady two-dimensional flow due to a shrinking surface with wall mass transfer through the porous sheet is possible when the wall mass suction parameter is larger than or equal to 2. However, it is quite different for non-Newtonian Powell-Eyring   as  increases. This is due to the fact that the assisting buoyancy forces produce a favorable pressure gradient which increases the fluid motion. On the other hand, for first solution, skin friction coefficient and the local Nusselt number decrease as S increases, while for the second solution, those values increase. It is evident from these figs that mass suction delays the boundary layer separation. It is also worth mentioning that the computations have been performed until we get the solution with desired accuracy level in asymptotic nature with the boundary conditions and the computations were terminated at this  4 which implies that the fluid exert a drag force on the sheet. It is worth mentioning that the first solution is stable and physically relevant, while the second solution is not, since the first solution is the only solution for the case of assisting flow, and the second solution exists only for certain range of few parameters. However, they are still of interest as the differential equations are concerned, though such solutions are deprived of physical significant. Similar results may arise in other physical situations where the corresponding solutions could have more real meaning (Ridha, 1996). For the similar problems, using a stability analysis, Weidman et al. (2006), Postelnicu and Pop (2011) and very recently by Roşca and Pop (2014b) have shown that first solutions are stable, while second solutions are not.  Fig. 8 shows that velocity profiles decreases with increasing  and consequently it increases the velocity boundary layer thickness for first solution, while the opposite trend is observed for second solution. On the other hand, the opposite way is noted for the fluid parameter  as shown in Fig. 10. It is also clear from these figures that the second solution displays a thicker boundary layer thickness compared to first solution. Fig. 9 shows that both the temperature and thermal boundary layer thickness increase with  for first solution and decrease for second solution. Fig. 11 shows temperature profiles decreases with increasing  for first solution and increases for second solution. It is worth mentioning that first and second solutions satisfied the far field boundary conditions asymptotically, which is supporting the accuracy of obtained numerical results. The suction plays an important role to maintain the steady flow near the surface by delaying the flow separation. So, the effects of suction parameter S on the velocity and temperature profiles are vital in theoretical and practical points of view. Figs. 12 and 13 illustrated the velocity and temperature profiles for various values of suction S. Fig.  12 shows that, for first solution, velocity profiles increases with increasing S and it decreases for second solution. From Fig. 13, it is clear that thermal boundary layer thickness decreases with increasing S for first solution and increases for second solution. Due to suction the fluid mass is brought closer to the sheet and by this it prevents the diffusion of vorticity generated due to shrinking. Figs. 14 and 15 show the effect of magnetic parameter on velocity and temperature profiles. From Fig. 14, it is clear that the velocity of fluid increases and consequently there is reduction in the velocity boundary layer thickness for first solution, whereas the opposite effects are observed in case of second solution. With a rise in strength of magnetic field (i.e., increase of magnetic parameter), the Lorentz force associated with applied magnetic field makes the boundary layer thinner due to its resistance property. The fluid which is decelerated by the viscous action, receives a push due to the magnetic field which works against the viscous effects (Mukhopadhyay and Mandal, 2015). For first solution, the thermal boundary layer thickness reduces with increasing M and increases for second solution as found in Fig. 15. It is also noted that temperature profiles overshoot in the boundary layer region for second solution. The effects of mixed convection parameter  on the dimensionless velocity and temperature profiles are depicted in Fig. 16 -Fig. 19 is the relation between the two magnetic parameters of the two papers) is presented in Fig. 20. Those two are found in excellent agreement and it consequently confirms the accuracy of the obtained results.

CONCLUSIONS
A numerical study has been presented on MHD mixed convective flow and heat transfer of Powell-Eyring fluid past a permeable exponentially shrinking sheet. Non-similar dual solutions of the transformed equations are found using the shooting method for various values of governing parameters. Assisting as well as opposing flows are considered. The study can be summarized as follows: (1) Dual solutions are obtained for a certain range of suction parameter and mixed convection parameter (only for opposing flow).
(2) For the increment of fluid parameter  and suction, skin friction coefficient enhances for first solution and reduces for second solution, whereas the skin friction increases due to magnetic parameter for first solution and decrease for second solution.