VISCOUS DISSIPATION AND HEAT TRANSFER EFFECTS IN MHD RAYLEIGH PROBLEM

P. T. Hemamalini 1 and M. Deivanayaki 2 . 1. Professor, Department of Science and Humanities, Faculty of Engineering, Karpagam University, Coimbatore. 2. Assistant Professor, Department of Science and Humanities, Faculty of Engineering, Karpagam University, Coimbatore. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

The study of Magnetohydrodynamics of a conducting fluid has applications in a variety of geophysical and astrophysical problems. The unsteady free-convection flow of a electrically conducting, viscous incompressible fluid have gained considerable attention in the presence of applied magnetic field in connection with the theories of fluid motion in the liquid core of the earth, oceanographic and metrological applications.
The viscosity of the fluid in a viscous fluid flow will take energy from the motion of the fluid and transform it into internal energy of the fluid. That means heating up the fluid. This partially irreversible process is referred as dissipation or viscous dissipation. Viscous dissipation is defined as the irreversible process by means of which the work done by a fluid on adjacent layers due to the action of shear forces is transformed into heat.
The effect of viscous dissipation in natural convection was analyzed by Gebhart [1]. Iqbal et.al [2] studied the viscous dissipation effects on combined free and forced convection through vertical circular tubes. Hossian [3] studies the effect of viscous and Joules heating on the flow of an electrically conducting fluid past a semiinfinite plate when temperature varies linearly with the distance from the moving edge and it in the presence of a uniform transverse magnetic field. Vajravelu and Hadjiniwlaou [4] analyzed heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation. The problem of heat transfer on a moving plate with a uniform magnetic field has attracted the attention of many researchers such as Ali [5], Takharet. al. [6] and Zakaria [7].
The simultaneous effects of the heat transfer and Hall current on a MHD flow with a porous medium in a rotating system was investigated by Dileepsing [8]. Loganathan [9] analyzed the viscous dissipation effects on unsteady natural convective flow past an infinite vertical plate with uniform heat and mass flux. In all the above cases either normal or horizontal magnetic field is considered, but this cannot support the entire physical scenario. In Science ISSN: 2320-5407 Int. J. Adv. Res. 5(2), 1314-1320 1315 and Engineering problems oblique magnetic field also play a vital role. Hence this paper mainly deals with the problem involving viscous dissipation and oblique magnetic field.

Author's Contribution:-
The effect of inclined magnetic field with viscous dissipation may become very important in several flow configurations occurring in the Engineering problems. In view of the importance of an inclined magnetic field and dissipation effects, the effect of viscous dissipation in the MHD Rayleigh problem with inclined magnetic field is investigated in this chapter.

Formulation of the Problem:-
An unsteady free convection flow of an electrically conducting viscous incompressible fluid with heat transfer along an infinite flat plate occupying the plane y = 0 has been considered. The -axis is taken in the direction of the motion of the plate.axis lying on the plane normal to both and yaxis. Initially it is assumed that the plate and the fluid rotate in unison with a uniform angular velocity Ω about the y -axis normal to the plate are at the same temperature everywhere in the fluid. At time > 0, the plate starts moving impulsively with the uniform velocity in its own plane along the -axis. Also the temperature of the plate is raised/lowered to ∞ and there after maintained uniform. A uniform magnetic field 0 is applied in a direction which makes an angle with the positive direction of -axis in the − plane. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field can be neglected. (1) Equation of motion Here P is the pressure,  is the density,  is kinematic viscosity and J × B is the Lorentz force. The Energy equation where is the specific heat at constant pressure, is the thermal conductivity and  is the dissipation function due to viscosity.
The generalized Ohm's law is, (is the electrical conductivity). Here J is the current density,e is the electric charge,  is the mean collision time, n is the electron number density and is mass of an electron. As the plate is infinite, all variables in the problem are functions of y and t only. So the term . ∇ q reduces to zero and ∇P vanishes as P is constant in equation (2)

Solution of the Problem
By using (7) and (8)

Shearing Stress and Nusselt number
The shearing stress at the wall along -axis and -axis are given by =   The shear stress along the -axis and -axis with increase of Hartmann number with respect to the Hall parameter are shown clearly in the figure 7. When the strength of the applied magnetic field is increased, the primary skin friction decreases due to the Hall effect. The secondary skin friction increases near the plate. Figure 8 clearly depicts that the shear stress along theaxis, the primary skin friction has a higher influence and it accelerates whereas the shear stress along theaxis, the secondary skin friction retards with increase of Hall parameter with

Conclusion:-
When the magnetic field is increased the velocity profiles also increase but the Hall parameter is not having a notable influence in velocity components. When the rotation parameter is increased the velocity profiles decrease. The primary velocity increase and secondary velocity decrease when the angle of inclination is increased. The temperature profile decreases when time and Prandtl number are increased. The shear stress at the plate decrease and at the plate increase with increase of Hartmann number and with increase of Hall parameter increase and decrease. With increase of Prandtl number, Hartmann number and Hall parameter, the profile of amplitude accelerates.