New Analytical Study of the Effects Thermo-Diffusion, Diffusion-Thermo and Chemical Reaction of Viscous Fluid on Magneto Hydrodynamics Flow in Divergent and Convergent Channels

In this paper, the magneto hydrodynamic (MHD) flow of viscous fluid in a channel with non-parallel plates is studied. The governing partial differential equation was transformed into a system of dimensionless non-similar coupled ordinary differential equation. The transformed conservations equations were solved by using new algorithm. Basically, this new algorithm depends mainly on the Taylor expansion application with the coefficients of power series resulting from integrating the order differential equation. Results obtained from new algorithm are compared with the results of numerical Range-Kutta fourth-order algorithm with help of the shooting algorithm. The comparison revealed that the resulting solutions were excellent agreement. Thermo-diffusion and diffusion-thermo effects were investigated to analyze the behavior of temperature and concentration profile. Also the influences of the first order chemical reaction and the rate of mass and heat transfer were studied. The computed analytical solution result for the velocity, temperature and concentration distribution with the effect of various important dimensionless parameters was analyzed and discussed graphically.


Introduction
The importance of thermal-diffusion and diffusion-thermo effects for various fluid flows has been studied by Eckert and Drake [1]. Olajuwon [2] examined convection heat and mass transfer in a hydromagnetic flow of a second grate fluid past a semi-infinite stretching sheet in the presence of thermal diffusion and thermal radiation. Kumar et al. [3] have investigated thermal diffusion and radiation effects on unsteady magneto hydrodynamics (MHD) flow through porous medium with variable temperature and mass diffusion in the presence of heat source or sink. Magnetohydrodynamics is the study of the interaction between magnetic fields and moving, conducting fluids [4] and the behavior of an electrically conducting fluid in the presence of a magnetic field. In this case, a force is produced inside the fluid which is proportional to fluid velocity and this force always opposes the flow. Another way to produce a force inside a flowing fluid, not known widely, is the application of an externally applied magnetic as well as an externally applied electric field. This force is called Lorentz force and can be generated by a strip wise arrangement of flush mounted electrodes and permanent magnets of alternating polarity and magnetization. The Lorentz force which acts parallel to the plate can either assist or oppose the flow. The idea of using a Lorentz force to stabilize a boundary layer flow over a flat plate belongs probably to Gailitis and Lielausis [5] [6]. It is a known fact that the temperature and concentration gradients present mass and energy fluxes, respectively. Concentration gradients result in Dufuor effect (diffusion-thermo) but Soret effect (thermal-diffusion) is due to temperature gradients. The heat and mass transfer with chemical reaction plays an important role in designing of chemical processing equipment, damage of crops due to frost, formulation and dispersion of fog. The mass transfer can be defined as a phenomenon when there is an escape of vapors into the atmosphere while heat transfer happens when there is heating or cooling of a liquid or fluid. That is, both of these phenomena play an important role in the industry. Because nonlinearity of the equations for these problem exact solutions is known, so many analytical techniques have been studied. Homotopy analysis method [7] [8] and Adomian's decomposition method [9] [10] [11] [12] are also analytical techniques used to solve the nonlinear equations. In this article the governing equations of the problem contain a system of partial differential equations which are transformed by usual transformation into a non-dimensional system of partial coupled non-linear differential equations.

Mathematical Formulation
Consider the flow of an incompressible fluid due to source or sink that is located at the intersection of two rigid plane walls angled 2α apart. Radial and symmetric nature of the flow is taken into consideration. Induced magnetic field is ignored and an applied magnetic field is considered that is applied across the the boundary conditions are, , 0, 0, 0, at , with the use of dimensionless parameters award [6] ( ) Eliminating p from Equations (1) and (2) using Equations (7) and (8), we get a system of nonlinear ordinary differential equation for the normalized velocity where, divergent channel : 0, 0 convergent channel : 0, 0 where, Step (2): We take Taylor series expansion of the function Now, we assume that Step (3): We focus on computing the derivatives of G with respect to η which is the crucial part of the proposed method. Let start calculating  The calculations are more complicated in the second and third derivatives because of the product rules. Consequently, the systematic structure on calculation is extremely important. Fortunately, due to the assumption that the operator G and the solution f are analytic functions, then the mixed derivatives are equivalence.
We note that the derivatives function to f unknown, so we suggest the following hypothesis Therefore Equations (20)-(23) are evaluated by Step (

Application of the New Algorithm to the Magneto Hydrodynamic (MHD) Flow of Viscous Fluid in a Channel with Non-Parallel Plates
The new algorithm described in the previous section can be used as a powerful solver to the nonlinear differential Equations (9)-(10) and to find new an analytical-approximate solution. From step (1) we have rewrite the Equation (29) as follows ( ) ( ) From the boundary conditions the Equation (30) becomes ( ) ( ) and the analytical-approximate solution are ,  3  3  3   3  2   3  3  3   3  2   3  3  3   3  3  3  3   d   d   3  3   3  3   3 3 .

The Analysis of Convergence
Here, the analysis of convergence for the analytical-approximate solution (50) that was resulted from the application of new power series algorithm for solving the problem has been extensively studied.
Definition (1)  Definition (2) The sufficient condition for convergent of the series analytical-approximate solutions n F , n H , n P is given in the following theorems.

Results and Discussions
This section is dedicated to study the influence of various non dimensional physical parameters on velocity field ( ) f η , temperature field ( ) β η and concentration field ( ) φ η . Also the influence of different parameters on rate of heat transfer and rate of mass transfer are under observation for diverging and converging channels. In Table 1 and Table 2 proof convergence the values 3 A , 1 B and 1 C of initial solutions. The stability of these values can be clearly distinguished from the fourth approximation. Table 3 and Table 4 Tables   5-13 are explained to analyze the behavior of Nusselt number and Sherwood number with variation parameters. As for can say that impotent to mention that Nusselt number gives a description of heat transfer rate at the wall, while Sherwood number represents the rate of mass transfer at the wall. In Table 5 and In Figures 2-9 are plotted to show the behavior curves of velocity, temperature and concentration profiles under the impact of different physical parameters. An increasing the opening angle α gives variations in velocity, temperature and concentration profiles as displayed in Figure 2. The influence of parameter α on the velocity field ( ) f η for divergent channel causes more effect at the middle channel as well as it represents as a maximum position at the central line (when 0 η = ). It also has least effect in part near the walls (when  • Channel convergent ( 0 α < ).
For the converging channel, the variations in velocity, temperature and concentration profile due to the varying parameters are depicted in Figures 10-17, the behavior of velocity and temperature for changing angle opening α and Reynolds number Re is quite opposite to the behavior of ( ) f η , ( ) β η and ( ) φ η in diverging channel as seen in Figure 10 and Figure 11. On that other hand, Figures 11-17

Conclusions
In this paper, the unsteady and two-dimensional magneto hydrodynamic ( • Increase in heat transfer rate is observed for increasing r P , c E , r S , c S , f D and γ in both channels. • Increase in Reynolds number and Angle opening gives a drop to mass trans-fer rate for diverging channel and a rise for the converging channel.
• The rate of mass transfer decreased for both channels with an increase in Schmidt, Soret, Prandtl, Eckert, Dufour numbers and chemical reaction parameter.
• Results obtained by new algorithm are in excellent agreement with numerical solution obtained.