Dufour and radiation effect on MHD boundary layer flow past a wedge through porous medium with heat source and chemical reaction

Magnetohydrodynamics (MHD) boundary layer flow past a wedge with the influence of thermal radiation, heat generation and chemical reaction has been analysed in the present study. This model used for the momentum, temperature and concentration fields. The principal governing equations is based on the velocity ( ) w u x in a nanofluid and with a parallel free stream velocity ( ) e u x and surface temperature and concentration. The governing nonlinear boundary layer equations for momentum, thermal energy and concentration are transformed to a system of nonlinear ordinary coupled differential equations by using suitable similarity transformation with fitting boundary conditions. The transmuted model is shown to be controlled by a number of thermo-physical parameters, viz. the magnetic parameter, buoyancy parameter, radiation conduction parameter, heat generation parameter, Porosity parameter, Dufour number, Prandtl number, Lewis number, Brownian motion parameter, thermophoresis parameter, chemical reaction parameter and pressure gradient parameter. Numerical elucidations are obtained with the legendary Nactsheim-Swigert shooting technique together with Runge–Kutta six order iteration schemes.


INTRODUCTION
It is now well accepted fact that the terms magnetohydrodynamics (MHD), thermal radiation and heat generation extensively appear in various engineering processes. MHD is significant in the control of boundary layer flow and metallurgical processes. Again the thermal radiation and heat generation possessions may arise in high temperature ingredients processing operations. Ingredients may be intelligently designed therefore with judicious implementation of radiative heating to produce the desired characteristics. This recurrently occurs in agriculture, engineering, plasma studies and petroleum industries.  A nanofluid is a fluid containing nanometer-sized particles, called nanoparticles. These fluids are engineered colloidal suspensions of nanoparticles in a base fluid. The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. Common base fluids include water, ethylene glycol and oil. In recent years studies on nanofluid heat and mass transfer boundary layer laminar flow have attracted considerable attention. Nanotechnology [5][6][7][8][9][10][11][12][13][14][15][16][17] has been broadly used in several industrial applications. Nanofluids demonstrate anomalously high thermal conductivity, significant change in properties such as viscosity and specific heat in comparison to the base fluid, features which have attracted many researchers to perform in engineering applications. Kim [18] analyzed the Convective Instability and Heat Transfer Characteristics of Nanofluids. Kang et al. [19] experimentally investigated on nanofluids include thermal conductivity. Jang and Choi [20] reconnoitered nanofluid thermal conductivity parameter effects.Nield and Kuznestov [21] and Kuznestov and Nield [22] considered laminar convective nanofluid boundary layer flow in a porous medium, with Brownian motion and thermophoresis particle deposition effects and simple boundary conditions. Khan and Pop [23,24] studied boundary layer heat-mass transfer free convection flows also in porous media of a nanofluid past a stretched sheet. M. Wahiduzzaman [25] discussed about Viscous Dissipation and Radiation Effects on MHD Boundary Layer Flow of a Nanofluid Past a Rotating Stretching Sheet. Hamad and Pop [26] reported transient hydro magnetic free convection rotating flow of a nanofluid. Md. Shakhaoath Khan et al. [27] analyzed the boundary layer nanofluid flow with MHD radiative possessions. Khan and Pop [28] investigates boundary layer heat and mass transfer analysis past a wedge moving in a nanofluid.The prime objective of the present attempt is to extend the analysis of Khan and Pop [28].
This study finds the effect of thermal radiation, heat generation and chemical reaction on themagneto hydrodynamic convection flow past a wedge moving in a nanofluid. This study also emphasised that Brownian motion and thermophoresis are significant mechanisms in nanofluid performance. This study is encouraged by precise application in materials processing which combines photopyroelectric thermal radiation and magnetic fields simultaneously to modify nanofluid properties. Verification of computations is demonstrated by comparison with previously published literature of Shakhaoath [29]. The present study is applicable to the manufacturing of magnetic nanofluids and chemical engineering operations involving electro-conductive nano fluid suspensions.There are relatively few studies [30][31][32][33][34][35][36][37] also focused on the MHD, convection, radiative heat transfer, heat generation and nanofluid also addressed application for further research.

METHODS: MATHEMATICAL MODEL
Here we consider the two dimensional MHD laminar boundary layer heat and mass transfer flow past an impermeable stretching wedge with the influence of thermal radiation, heat generation and chemical reaction and moving with the velocity () w ux in a nanofluid, and the free stream velocity is () e ux, where x is the coordinate measured along the surface of the wedge. The pattern of the physical configuration and coordinate system are shown in Figure 1 by following Khan and Pop [28]. S e p t e m b e r 1 6 , 2 0 1 5 (1) With the boundary conditions In equation (2) the 3rd term on the right hand side is the convection due to thermal expansion and gravitational acceleration, the 4th term on the right hand side is the convection due to mass expansion and gravitational acceleration and the 5th term generated by the magnetic field strength because a strong magnetic field 0 (0, ,0) BB  is applied in the y-direction. Again in equation (3) the 2nd term on the right hand side is the effect of heat generation on temperature flow and thermophoresis diffusion term due to nanofluid flow, 3rd term on the right hand side expressed the radiative [30] heat transfer flow, and is the rate of chemical reaction on the net mass flows, the last term indicates the Brownian motion due to nanofluid heat and mass transfer flow.
In order to conquers a similarity solution to equations (1) to (4) with the boundary conditions (5) the following similarity transformations, dimensionless variables are adopted in the rest of the analysis; S e p t e m b e r 1 6 , 2 0 1 5 For the similarity solution of equations (1) to (4) with considering the value (from the properties of wedge,( [28]) Therefore, the constant moving parameter λ is defined as λ = c/a, whereas λ < 0 relates to a stretching wedge, λ > 0 relates to a contracting wedge, and λ = 0 corresponds to a fixed wedge, respectively.
From the above transformations the non-dimensional, nonlinear, coupled ordinary differential equations are obtained as; The transformed boundary conditions are as follows; where the notation primes denote differentiation with respect to η and the parameters are defined as:

NUMERICAL (SHOOTING QUADRATURE) SIMULATIONS
The non-dimensional, nonlinear, coupled ordinary differential equations (7) to (9) with boundary condition (10) are solved numerically using standard initially value solver the shooting method. For the purpose of this method, the Nactsheim-Swigert shooting iteration technique together with Runge-Kutta six order iteration scheme is taken which determines the temperature and concentration as a function of the coordinate η.
The boundary conditions equation (10) associated with the ordinary nonlinear differential equations of the boundary layer type is of the two-point asymptotic class. Two point boundary conditions have values of the independent variable specified at two different values of the independent variable. Specification of an asymptotic boundary condition implies the value of velocity approaches to unity and the value of temperature approaches to zero as the outer specified value of the independent variable is approached. The method of numerically integrating two-point asymptotic boundary value problem of the boundary layer type, the initial value method, requires that the problem be recast as an initial value problem. Thus it is necessary to set up as many boundary conditions at the surface as they are at infinity. The governing differential equations are then integrated with these assumed surface boundary conditions. If the required outer boundary condition is satisfied, a solution has been achieved. However, this is not generally the case. Hence a method must be devised to logically estimate the new surface boundary conditions for the next trial integration.
Asymptotic boundary value problems such as those governing the boundary layer equations are further complicated by the fact that the outer boundary condition is specified at infinity. In the trial integration infinity is numerically approximated by some large value of the independent variable. There is no a priori general method of estimating this value. Selection of too small a maximum value for the independent variable may not allow the solution to asymptotically converge to the required accuracy. Selecting a large value may result in divergence of the trial integration or in slow convergence of surface boundary conditions required satisfying the asymptotic outer boundary condition. Selecting too large a value of the S e p t e m b e r 1 6 , 2 0 1 5 independent variable is expensive in terms of computer time. Nachtsheim-Swigert developed an iteration method, which overcomes these difficulties. Extension of the iteration shell to above equation system of differential equations (10) is straightforward, there are three asymptotic boundary condition and hence three unknown surface conditions

RESULTS AND DISCUSSION
In order to get the physical insight to the system of ordinary differential equation (7) It is seen the increase velocity of M leads to increase in velocity distribution. This is due to the fact of that applied transverse magnetic field produces a Lorenz force which beneficial to increase in profile of all points.     Du  and in the absence of ( 0) RR the result is good agreement with that of Shakhaoath [29]. It is observed that presence of Nb are illustrated in Fig-10. It interesting to note that increasing value of Nt and Nb increases the temperature profile significantly. The thermophoresis force generated by the temperature gradient creates a fast flow from the surface. In this way more fluid is heated away from the surface and consequently, as Nt increases, temperature increases in the boundary layer. Also both in increase in the Brownian motion parameter the random motion of particle increase with result in on enhancement in the temperature profile.