On magnetohydrodynamic Veronis thermohaline convection

On magnetohydrodynamic Stern’s thermohaline convection

M. B. Banerjee¹, H. S. Jamwal²

¹Former Professor of Mathematics, Himachal Pradesh University, Shimla-171005, INDIA

²Principal, NSCBM Govt. College, Hamirpur-177005, Himachal Pradesh, INDIA

*Corresponding Author E-mail:

ABSTRACT:

In this paper, the magnetohydrodynamic Stern’s thermohaline convection is studied. We mathematically investigate the appropriate conditions at which the magnetohydrodynamic thermohaline convection motions manifest themselves and derive sufficient conditions for the validity of ‘principle of exchange of stabilities’ for the case of dynamically free boundaries.

KEYWORDS: Convection, magnetohydrodynamic, thermohaline, principle of exchange of stability.

1. INTRODUCTION:

Thermal convection is an important phenomenon that has applications to different areas such as geophysics, food processing, oil reservoir modelling and thermal insulator design etc. The classical theory of Bénard convection in horizontal layers of fluids heated from below has been treated both experimentally and theoretically by Pellew and Southwell [1]. The thermal convection of Newtonian fluid under various assumptions of hydrodynamics and hydromagnetics was discussed in detail by Thompson [2], Linhert and Little [3] and Chandrasekhar [4].

In the past few decades, considerable interest has been evinced in the study of magnetohydrodynamic thermohaline convection because it has various applications in oceanography, astrophysics, limnology and chemical engineering etc. A good account of thermohaline convection problems is as studied by Gupta et al. [5-7], Banerjee et al. [8-10] and Mohan [11]. For the magnetohydrodynamic Bénard convection problem, Banerjee et al. [8] noticed some shortcomings in the solution of problem as derived by Chandrasekhar [4]. These shortcomings have been removed and the correct solution has been constructed for the problem by Banerjee et al. [8] for the case of dynamically free, thermally insulating and electrically perfectly conducting boundaries.

In the present paper, magnetohydrodynamic Stern’s [12] thermohaline convection problem for dynamically free, thermally insulating and electrically perfectly conducting boundaries is studied.

2. The Physical System and the problem:

Consider an electrically conducting viscous Boussinesq fluid confined between two boundaries and of infinite horizontal extension in the presence of uniform vertical magnetic field, acting parallel but opposite to the force field of gravity, and being acted upon by a uniform vertical adverse temperature gradient. Then under appropriate conditions, a phenomenon of more general convective motions, an outcome of hydrodynamic instability, is realised which is termed as magnetohydrodynamic Stern’s thermohaline convection (Stern’s [12], Shirtcliffe [13], Turner [14], Normand et al. [15], Chen and Johnson [16], Rudraih and Shivkumara [17]).

3. Governing equations and boundary conditions:

The governing equations and boundary conditions in their non-dimensional forms for the magnetohydrodynamic Stern’s [12] thermohaline convection problem wherein dynamically free boundaries are thermally insulating and electrically perfectly conducting are given by (c.f. Gupta et al. [5], Banerjee et al. [8])

7. REFERENCES:

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Received on 21.11.2016 Modified on 25.11.2016

DOI: 10.5958/2349-2988.2017.00006.7

Research J. Science and Tech. 2017; 9(1):35-40.