Mixed Convective Magnetohydrodynamic Heat Transfer Flow of Williamson Fluid Over a Porous Wedge

The present article examines the influence of thermal radiation on two-dimensional incompressible magnetohydrodynamic (MHD) mixed convective heat transfer flow of Williamson fluid flowing past a porous wedge. An adequate similarity transformation is adopted to reduce the fundamental boundary layer partial differential equations of Williamson fluid model in to a set of non-linear ordinary differential equations. The solutions of the resulting nonlinear system are obtained numerically using the fifth order numerical scheme the Runge-Kutta-Fehlberg method. The effects of different pertinent physical parameter such as magnetic parameter, Williamson parameter, radiation parameter and Prandtl number on temperature and velocity distributions are observed through graph.


Introduction
The theory of non-Newtonian fluids has attracted several researchers owing to its enormous applications in engineering and industrial sector. In Non-Newtonian fluids, the most frequently encountered fluids are pseudoplastic fluids, and Navier-stokes equations alone are insufficient to describe the rheological properties of these fluids, therefore, to overcome this defect, several rheological model like Ellis model, Power law model, Carreaus model and Cross model are presented, but little attention has been compensated to the Williamson fluid model and estimated to explain the rheological properties of pseudoplastic fluids. In this model both maximum viscosity μ (viscosity as shear rate tends to infinity) and minimum viscosity μ (viscosity as shear rate tends to zero) are to be taken.
Williamson analyzed the flow of pseudoplastic materials and presented model to described the behavior of pseudoplastic material and explain convenient importance of plastic flows, and also recognized that viscous flows are very varied from plastic flows [1]. Nadeem  proposed two-layer model to simulate mixed convection flow in a room [35]. By utilizing Killer box technique numerical solutions of problem of mixed convection axisymmetric flow of air with variable physical properties was obtained by Ramarozara [36]. Bau investigated the thermal convection in a saturated stratified medium bounded between two parallel eccentric cylinders with the help of a regular perturbation expansion along Daarcy-Rayleigh number; it was observed that the appropriate preference of eccentricity values can maximize the heat transfer inside annulus of various thermal insulators [37]. Jackson et al. discussed combine convection in vertical tubes by using constant wall temperature and constant wall heat flux conditions [38]. Fu et al. investigated flow reversal of mixed convection in a three dimensional channel and concluded that an increase in Richardson number, natural convection dominates the flow and thermal field of combine convection [39]. Kaya found nonsimilar solutions of steady laminar mixed convection heat transfer flow from a perpendicular cone in a porous medium with influence of radiation, conduction, interaction and having high porosity [40]. Jafari et al. studied unsteady combined convection flow in a cavity in presence of nanofluid [41]. Bég et al. outlined mixed convection boundary layer flow influenced by thermo-diffusion [42]. Chaudhary and Jain studied the impact of mass transfer, radiation and hall on MHD mixed convection flow of viscoelastic fluid in an infinite plate [43]. Ferdows & Liu obtained the similarity solutions of mixed convection heat convey in parallel surface with internal heat production [44]. Malleswaran & Sivasankaran carried an analysis for mixed convection flow and noticed that the average heat transfer decreases with an increase in Richardson number but in general heat transfer is better at force convection mode than free convection mode [45]. Few other interesting works about convective heat transfer can be found in .
Porous wedge is a very important characteristic in science and engineering that can be illustrated as a material having a minute opening and the opening is almost filled with fluid. The skeletal part of porous wedge is known as Frame and it is frequently a solid, but structures akin to foams. General example of porous wedge is sand, soil, sandstone and foams. Ashraf et al. studied boundary layer flow of fluid in a porous wedge subject to Newtonian heating along heat generation or absorption [69]. Deka and Sharma solved the boundary layer equations of flow over a wedge under variable temperature and chemical reaction [70]. Mukhopadhyay considered the impact of radiation and variable viscosity on the flow through wedge using lie group transformation [71]. Mukhopadhyay and Mandal numerically analyzed flow of a Casson fluid in a symmetric porous wedge along surface heat flux [72]. Hossain et al. considered the case for unsteady boundary layer flow past a wedge [73]. Dalir investigated two dimensional laminar transport of viscous fluid through a permeable wedge [74]. Rostami et al analytically studied laminar viscoelastic fluid flow past a wedge in the presence of Buoyancy forces and discussed the effects of Buoyancy parameter on velocity and temperature profiles [75]. Wedge flow problem under thermal radiation was studied by Rashidi et al. [76].
The main aim of the present paper is to elaborate electrically conducting fluid flow of Williamson fluid over a permeable wedge with thermal radiation. Similarity transformation is used to convert governing partial differential equations of the said phenomenon to couple nonlinear ordinary differential equations. The resulting nondimensional equations are solved numerically using the Runge-Kutta-Fehlberg method. The effects of involved parameters on flow are discussed graphically.
Formulation Consider two dimensional steady incompressible MHD mixed convective heat transfer and electrically directing Williamson fluid past a porous wedge. The Cartesian coordinate plot is assumed to be help out the solution wherein the x-axis & y-axis are together and normal to the wedge. The non-uniform magnetic field = applied to flow and perpendicular to y-axis, the induced magnetic field of flow are supposed to be negligible. It is assumed that the fluid velocity from wedge is = .
Where a and m are constants. The governing partial differential equations of continuity, momentum and energy for the Williamson fluid flow using Boussinesq's and boundary layer assumptions are: where u and v are velocity components in x and y directions respectively, , is the density,is Kinematic viscosity, . is electric conductivity, / is the temperature of the fluid, & is thermal diffusivity,0 1 is heat flux and 2 3 is specific heat.
The boundary conditions associated with the problem are at y = 0, $ = 0, = 6 , T = / 6 (4) as y → ∞ $ = = 0, T = / The assumed wedge surface temperature is / 6 = / + k , where k is constant and / is temperature of the moving fluid. The total wedge angle is equal to ῼ = is wedge angle parameter. The nondimensional form of the given system of partial differential equations is obtained by introducing the following stream function and the similarity variables [76].
Where E 6 is injection/suction parameter.

Solution of the Problem
The system of ordinary differential equations 7 and 8 subject to the boundary conditions 9 and 10 is first reduced to a system of first order ordinary differential equations using the substitutions E K =`, $ K = a, F K = b. This With the boundary conditions E 0 E 6 , $ 0 0, F 0 1, $ ∞ 1, F ∞ 0 (13) The resulting system in Eq. (11-13) is solved numerically with the help of 5 th order Runge-Kutta-Fehlberg method. Further details about the obtained numerical solutions are presented in the next section.   The transformed governing equations (11-12) subjected to boundary conditions (13) are solved numerically by employing the fifth order Runge-Kutta-Fehlberg method. The influence of all pertinent parameters on flow and heat transfer are graphed and discussed in Figures (1-8). To examine accuracy of our work a comparison has been made with the available works of Ishak et al. [78], Yih [79] and Rashidi et al. [76] in Table. 1. The agreement of our work with the prior results is stable. Figures (1-2) illustrate the influence of wedge angle parameter < with on velocity and temperature profile. It is observed that velocity increases by increasing the wedge angle parameter < , but the thermal boundary layer thickness is decreased. Since the wedge angle parameter < is a dependent over the pressure gradient, and its values may be positive or negative.       Figure 3 displays the velocity profile for various values of the magnetic parameter M, the ratio of electromagnetic force to the viscous force that quantifies the intensity of applied magnetic field. It is observed from graph that with an increase in magnetic parameter M, there is a decline in the velocity distribution. This is due to the appliance of the transverse magnetic field which declines the fluid speed. This eminent phenomenon is known as Lorentz force that squeezes the momentum boundary layer. Furthermore an increase in λ may cause increase in temperature of flow. Figure 6 drafts the non-dimensional velocity E′ for different values of suction parameter E 6 . From figure it is observed that an increase in the value of E 6 , results in an increase in velocity. Figures 7-8 illustrate the behavior of thermal radiation and Prandtl number on fluid flow region with M= E 6 = O = < =1. It is clear from graph that an increase in thermal radiation parameter leads to increase in temperature and thermal boundary layer thickness. The influence of thermal radiation is to enhance the amount of heat, while in other hand an increase in values of Prandtl number causes to decline the temperature distribution. Because the prandtl number is the relation of momentum diffusivity to thermal diffusivity, when it increases then it decreases the thermal boundary layer thickness and temperature but increases thermal capacity of fluid. Generally prandtl number is applicable in heat transform problem in order to decrease the thickness of the boundary layer and momentum.

Conclusion
The steady, incompressible two dimensional boundary layer flow of Williamson fluid past a porous wedge is analyzed numerically using the 5 th order Fehlberg technique. The important conclusions of the analysis are 1. The non-dimensional velocity profile increases by increasing the wedge angle parameter <. 2. The non-dimensional temperature profile decreases by increasing the wedge angle parameter <.  [23] Reddy, G. MHD boundary layer flow of a rotating fluid past a porous plate. Int. j. of comput. and appl. Maths. 2017, 12 (2), 579-593.