Hydromagnetic Steady Flow of Maxwell Fluid over a Bidirectional Stretching Surface with Prescribed Surface Temperature and Prescribed Surface Heat Flux

This paper investigates the steady hydromagnetic three-dimensional boundary layer flow of Maxwell fluid over a bidirectional stretching surface. Both cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF) are considered. Computations are made for the velocities and temperatures. Results are plotted and analyzed for PST and PHF cases. Convergence analysis is presented for the velocities and temperatures. Comparison of PST and PHF cases is given and examined.


Introduction
Interest of recent researchers in analysis of boundary layer flows over a continuously moving surface with prescribed surface temperature or heat flux has increased substantially during the last few decades. These flows have abundant applications in many metallurgical and industrial processes. Specific examples of such industrial and technological processes include wire-drawing, glassfiber and paper production, the extrusion of polymer sheets, the cooling of a metallic plate in a cooling bath, drawing of plastic films etc. Such situations occur in the class of flow problems relevant to the polymer extrusion in which the flow is generated by stretching of plastic surface [1,2]. In addition, internal heat generation/absorption has key role in the heat transfer from a heated sheet in several practical aspects. The heat generation/ absorption effects are also important in the flow problems dealing with the dissociating fluids. Influences of heat generation/ absorption may change the temperature distribution which corresponds to the particle deposition rate in electronic chips, nuclear reactors, semiconductor wafers etc. The idea of boundary layer flow over a moving surface was introduced by Sakiadis [3]. He discussed the boundary layer flow of viscous fluid over a solid surface. This analysis was extended by Crane [4] for a linearly stretched surface. He provided the closed form solutions of twodimensional boundary layer flow of viscous fluid over a surface. Numerous literature now exists on the boundary layer flow with heat transfer and in the presence of heat generation/absorption effects (see [5][6][7][8][9][10] and many refs. therein).
A large number of industrial fluids like polymers, soaps, molten plastics, sugar solutions pulps, apple sauce, drilling muds etc. behave as the non-Newtonian fluids [11]. The Navier-Stokes equations cannot explore the properties of such materials. In the literature, different types of fluids models are developed according to the nature of fluids. The non-Newtonian fluids are mainly divided into three categories which are known as the differential, rate and integral types. The fluid considered here is called the Maxwell fluid. It is subclass of rate type fluids predicting the characteristics of relaxation time. The properties of polymeric fluids can be explored by Maxwell model for small relaxation time. Zierep and Fectecau [12] discussed the energetic balance for the Rayleigh-Stokes problem involving Maxwell fluid. Closed form solutions of unsteady flow of Maxwell fluid due to the sudden movement of the plate was described by Hayat et al. [13]. Fetecau et al. [14] provided the exact solutions for the unsteady flow of Maxwell fluid. Here they considered that the flow is generated due to the constantly accelerating plate. Flow of Maxwell fluid with fractional derivative model between two coaxial cylinders was also addressed by Fetecau et al. [15]. Here the inner cylinder is subjected to the time-dependent longitudinal shear stress generating the fluid motion. Helical unidirectional flows of Maxwell fluid due to shear stresses on the boundary have been studied by Jamil and Fetecau [16]. They provided the exact solution by Hankel transform method. Stability analysis for the flow of Maxwell fluid under soret-driven double-diffusive convection in a porous medium was examined by Wang and Tan [17]. Twodimensional boundary layer flow of Maxwell fluid over a linearly stretching surface was analyzed by Hayat et al. [18]. Mukhopadhyay [19] presented an analysis for the unsteady flow of Maxwell fluid in a porous medium with suction/injection. Falkner-Skan flow of Maxwell fluid with mixed convection over a surface was analytically discussed by Hayat et al. [20].
The main theme of present analysis is to discuss the steady three-dimensional boundary layer flow of Maxwell fluid over a bidirectional stretching surface subject to prescribed surface temperature and prescribed surface heat flux. The effects of applied magnetic field are also included in this analysis. To our knowledge, not much is known about flows induced by a bidirectional stretching surface. Wang [21] discussed the threedimensional flow of viscous fluid over a bidirectional stretching surface. Ariel [22] provided the exact and homotopy perturbation solution for ref. [21]. Liu and Andersson [23] discussed the heat transfer analysis over a bidirectional stretching surface with variable thermal conditions. Ahmed et al. [24] extended the analysis of ref. [23] for hydromagnetic flow in a porous medium. They presented the series solutions. Hayat et al. and Shehzad et al. [25,26] studied the boundary layer flows of Maxwell and Jeffery fluids over a bidirectional stretching surface. The present analysis is arranged as follows. The next section contains the mathematical formulation of the problem. Sections three and four are for the homotopy solutions (HAM) [27][28][29][30][31][32][33][34], convergence study and discussion. Both cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF) are given due attention in the discussion section. The main observations of this research are listed in the last section. Further, the correct modelling for magnetohydrodynamic case of Maxwell fluid is given.

Flow Model
Consider three-dimensional magnetohydrodynamic (MHD) boundary layer flow of an incompressible Maxwell fluid. The flow is induced by bidirectional stretching surface (at z~0) with PST and PHF. Steady flow of an incompressible Maxwell fluid is considered for zw0: Flow analysis is carried out in the presence of heat generation/absorption parameter. The fluid is electrically conducting in the presence of applied magnetic field with constant strength B 0 : No electric field contribution is taken into account. Induced magnetic field effects are ignored through large magnetic Reynolds number consideration. The geometry of considered flow is shown in Fig. 1. The conservation of mass, momentum and energy for steady flow in presence of magnetic field and heat source/sink can be expressed as divV~0, ð1Þ in which r depicts the density, J the current density, B the magnetic field in the z{ direction, c p the specific heat, k the thermal conductivity and Q the heat generation/absorption parameter with Qw0 (heat generation) and Qv0 (heat absorption). B~B 0k k (k k is a unit vector parallel to the z{ axis). The definition of J for present flow consideration is where V denotes the fluid velocity and s the electrical conductivity. The Lorentz force thus reduces to Expressions of Cauchy (T) and extra stress (S) tensors in Maxwell fluid are [11]: where D=Dt is the Covariant differentiation and l 1 is the relaxation time. The first Rivilin Ericksen tensor A 1 is defined as where * indicates the matrix transpose and the velocity field V here is taken as The definition of D=Dt is [11] Da i Dt~L Following the procedure of ref. [11] at pages 221-223 and using above equations, we have the following scalar expressions After employing the boundary layer assumptions [35], the above equations in the absence of pressure gradient yield The associated boundary conditions are defined as follows.
For temperature, the boundary conditions are specified as [23,24]: Type i. Prescribed surface temperature (PST) T~T w (x, y)~T ? zCx r y s atz~0, Type ii.
Here k is the thermal conductivity of the fluid, T ? the constant temperature outside the thermal boundary layer, C and D the positive constants. The power indices r and s determine how the temperature or the heat flux varies in the xy{ plane.

Homotopy Analysis Solutions
In this section, we solve the problem consisting of Eqs.
The general solutions are arranged as follows

Convergence of Series Solutions and Discussion
It is well known fact that the homotopy analysis method has a great freedom to choose the auxiliary parameters h f , h g, h h and h w for adjusting and controlling the convergence of series solutions.                   Table 1 has been prepared to analyze the convergent values of the velocities, h(g) and w(g): We have seen that our solutions for velocities converge from 16th order of approximations whereas one needs 25th order of deformations for h(g) and w(g): Hence we need less deformations for the velocities in comparison to temperatures for a convergent solution. Table 2 provides the values of temperature gradient h 0 (0) for different values of a, r and s when b~M~0 and Pr~1:0: One can see that our solutions has an excellent agreement with the previous results in a limiting case [20,21]. Further, it is observed that the temperature gradient at surface h 0 (0) becomes positive and reduces for r~{2:0 and s~0 and negative for r~0 and s~{2:0: Table 3 presents the numerical values of h 0 (0) and w(0) for different values of Pr and B when b~M~0, r~s~1:0 and a~0:25: From this Table we noted that our series solutions have very good agreement with the previous results available in the literature.

Concluding Remarks
In this study, the three-dimensional MHD flow of Maxwell fluid generated by bidirectional stretching surface is investigated for two cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The effects of applied magnetic field B 0 are also taken into account. Interesting observations of this study can be mentioned below: