SORET AND DUFOUR EFFECTS ON MHD FLOW ABOUT A ROTATING VERTICAL CONE IN PRESENCE OF RADIATION

In this paper a numerical study of Soret and Dufour effect on MHD flow in presence of thermal radiation about a rotating vertical cone has been investigated. The governing equations are nonlinear partial differential equations and so by using similarity transformations they are converted to ordinary differential equations. MATLAB’s built in solver bvp4c is employed to solve numerically the ODE’s. The graph of velocity, temperature and concentration of the fluid are illustrated. To verify the accuracy of the numerical solution, a comparison for wall shear stress in tangential and circumferential direction with the present result and the one available in literature is done and the outcomes are in good match. Also local rate of surface heat transfer and mass transfer for different values of parameters are obtained.


INTRODUCTION
Problems related to transfer of heat and mass are often seen in the field of engineering and 3189 SORET AND DUFOUR EFFECTS ON MHD FLOW geothermal applications because of its various uses in devices like turbines, numerous propulsion devices for aircrafts and missiles, power transformers, metallurgy, satellites and canisters for nuclear disposable wastes etc. The knowledge of heat transport and distribution of temperature is important in aeronautics, dams, multi-storied buildings and petroleum industries. Chemical reaction along with problems of exchange of heat and mass are encountered in chemical as well as metallurgical industries. Chemical reactions are classified into two type's viz. homogeneous and heterogeneous chemical reaction. Homogeneous reaction occurs at a uniform rate in the solution through a given phase whereas heterogeneous reaction happens at the restricted region or at interface of a solution.
Many researchers have studied MHD flow problems with heat transfer as well as mass transfer about a rotating cone. Hartnett and Deland [1] investigated rotating body problems and the consequences of heat flow due to Prandtl number. Sparrow and Cess [2] studied the heat flux on the fluid flow in presence of an axial magnetic field about a rotating disk. Tien and Tsuji [3] did an analytical investigation on exchange of heat on a steady laminar forced flow about a rotating cone. Kafoussias and Williams [4] investigated Soret and Dufour effects with temperature dependent viscosity with mixed convective heat and mass transfer. Postelnicu [5] examined the impact of magnetic field on a vertical surface with heat transfer as well as mass transfer. Anilkumar and Roy [6] investigated numerically the results of mass transfer and thermal diffusion in a rotating fluid due to rotating cone. Afify [7] analysed mass flux on free convective optically dense viscous flow and the influence of radiation and chemical reactions about an erect cone. Chamkha and Al-Mudhaf [8] studied numerically unsteady heat and mass transfer with heat production or absorption effects due to a cone in the existence of magnetic field. Pullepu et al. [9] studied the impact of heat production or absorption and chemical reaction on unsteady flow with free convection and variable temperature about a vertical cone. Sharma and Konwar [10] investigated numerically the transfer of heat and mass on MHD flow under the influence of radiation on a vertical rotating cone.
Siddiqa et al. [11] investigated nanofluid flow with gyrotactic microorganisms through a standing curvy cone with heat and mass transfer and bioconvection. They found that the amplitude 3190 KRISHNANDAN VERMA, DEBOZANI BORGOHAIN, B. R. SHARMA of the cone's curvy surface significantly affect the coefficient of heat and mass transfer as well as microorganisms density. Saleem et al. [12] examined theoretically using Homotopy Analysis Method nanofluid Walter's B flow with magnetic effects. They considered the flow to be time dependent and found notable effect of thermophoresis as well as Brownian motion parameters on heat and mass transfer rate. Verma et al. [13] investigated numerically heat transfer and mass transfer in porous medium considering Forchheimer model with Soret effect through a rotating disk. Theoretical analysis has been carried out to discuss the effect of chemical reaction on heat and mass transfer on magneto-nanofluid that are ionised partially by Nawaz et al. [14]. Nadeem et al. [15] investigated the effects of variable viscosity on an expanding curved body due to carbon nanotubes in the fluid flow containing nanoparticles. Israr-ur-Rehman et al. [16] obtained numerically dual solutions on porous extending/shrinking surface considering anisotropic slip on nanofluid flow near stagnation point under magnetic conditions. Verma et al [17] studied numerically Soret and Dufour effect with heat and mass transfer in porous medium near stagnation point through a stretching sheet considering radiation and chemical effects. Naqvi et al. [18] studied nanofluid flow due to extending/shrinking disk considering heat generation/absorption to be non-uniform. Verma et al. [19] investigated numerically the impact of thermophoresis, chemical reaction and external heat source on MHD micropolar fluid with nanoparticles through a shrinking sheet near stagnation point.
In the field of engineering like nuclear reactors and processes involving liquid metals, the process of separation of binary fluid mixture becomes very significant. The binary fluid mixture composition in any given volume can be expressed by concentration C , which is the ratio of mass of the lighter and rarer constituent to the total mass of the fluid mixture, while composition of the heavier and greater component is given by 1 CC =− . Usually temperature gradient, concentration gradient and pressure gradient are responsible for diffusion of independent species in a fluid mixture. Diffusion flux j stated by Landau and Lifshitz [20] is represented by The present work focuses primarily on Soret and Dufour effects under the influence of 3191 SORET AND DUFOUR EFFECTS ON MHD FLOW radiation on the separation process of the binary fluid mixture. Investigations have been done by many researchers on the process involving transport of heat and mass on a vertical cone and few of them have also worked on Soret and Dufour effect but none of the literatures studied above have discussed on the detachment process of the binary fluid mixture. The main motive of this paper is to discuss numerically Soret and Dufour effects on MHD flow with heat and mass transfer of a binary fluid mixture about a vertical rotating cone along with thermal radiation.

MATHEMATICAL FORMULATION
Consider a heated, permeable vertical rotating cone in an incompressible, boundary layer steady flow of a binary fluid mixture. The fluid is viscous, hydromagnetic, laminar, electrically conducting and chemically reacting and the cone is rotating around its axis with uniform angular velocity Ω. Curvilinear coordinate system ( , , ) is considered where the x-axis, y-axis and zaxis are taken along the tangential, circumferential and normal direction respectively to the cone.
The components of velocity along tangential, circumferential and normal to the cone are , and respectively. Along z-axis, which is perpendicular to the surface of the cone, a uniform magnetic field 0 is applied. The same magnetic field is acting all the points on the circular section on the surface of the rotating cone. The applied magnetic field is axisymmetric. The surface of the cone is experiencing a uniform suction/injection of the fluid with velocity 0 .
The fluid is viscous and because of the rotation of the cone a velocity in the y-direction originates and due to centrifugal force, a velocity develops in the z-direction due to which fluid is thrown in the z-direction. To fill up the vacant place fluid moves from the vertex in x-direction and in this way a fully developed three-dimensional flow generates. At the surface of the cone, the fluid temperature is and species concentration is and both, temperature as well as concentration, are supposed to change linearly with distance x. Away from the surface, let the temperature of the fluid be ∞ and species concentration be ∞ .
The following assumptions are considered in the problem: 1. The cone is considered symmetric about the axis of rotation. 4. The medium is considered to be optically thin with comparatively low density and the order of chemical reaction is one.
To make equations (2) to (6) where represents the characteristics length along the cone's slant height.
The radiation heat flux is given by Within the fluid flow region, the temperature differences are sufficiently small and so 4 can be expanded using Taylor's series about ∞ as By the above transformations, equation of continuity (2) is identically satisfied and equations from (3) to (6) transform to the set of ordinary differential equations: The modified boundary conditions for (7) and (8) The suction or injection velocity is where in case of (i) Impermeable rotating cone = 0 For heat absorption < 0 and heat generation > 0.

METHOD OF SOLUTION
We have used MATLAB'S built in solver bvp4c to solve numerically the non-linear equations (12) to (15) subject to the boundary conditions represented by equations (16) and (17).

RESULTS AND DISCUSSIONS
The outcomes for velocity, temperature and concentration profiles are represented graphically for different values of parameters , and . To verify the precision of the numerical approach, the current result of wall shear stresses in tangential and circumferential direction are compared with the results obtained by Sparrow and Cess [2] in Table 1 and the present outcomes are in good match with the previous one. Local rate of heat and mass transfer of the present problem is calculated and is presented in Table 2.

ACKNOWLEDGMENT
The authors are very much thankful to Dibrugarh University for providing academic facilities throughout the preparation of the research paper.