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Mihir B. Banerjee ¹, J. R. Gupta ¹, R. G. Shandil ¹, Joginder S. Dhiman ², Vinay Kanwar ³

Affiliations
1 Department of Mathematics, Himachal Pradesh University, Shimla-171005, IN
2 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
3 Department of Mathematics, University Institute of Engineering and Technology, PU, Chandigarh, IN

Abstract

In his masterly contributions to the theory of magnetohydrodynamic thermal convection (or magnetoconvection) Chandrasekhar (1952, 1961) laid down the guiding principles of the subject matter of the following conjectural assertions of which the first two were obtained by making use of the method of Pellew and Southwell (1940) while the rest were speculated through an extremely simple solution of the governing equations that fails to satisfy any plausible set of boundary conditions on the magnetic field (except in the case of the (vi) for which the latter part of the remark does not apply) and pertains, according to him, to the case of dynamically free boundaries which occurs only in real physical situations. These assertions are:

(i) The method of Pellew and South well (1940) is not quite strong enough to establish the validity or otherwise of the principle of exchange of stabilities for the problem.

(ii) There exists a parameter regime for the problem defined by the Chandrasekhar number Q remaining less than or equal to some fixed value Q􀀂 (σ , σ􀀂) in which the total kinetic energy associated with a marginal or unstable disturbance exceeds its total magnetic energy and as a consequence the principle of exchange of stabilities is valid for the problem in this parameter regime.

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Joginder S. Dhiman ¹, Praveen K. Sharma ², Poonam Sharma ¹

Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005, IN
2 UIIT, Himachal Pradesh University, Summerhill, Shimla-171005, IN

Abstract

The aim of the present paper is to investigate the effects of temperature dependent viscosity on the stability of Dufour-Soret driven double-diffusive convection in nonreactive binary fluids, for general boundary conditions. Some analytical results concerning the stability of oscillatory motions and otherwise the complex growth rate of oscillatory motions are derived for this general problem. The expressions for Rayleigh numbers, when instability sets in as stationary modes, for each combination of rigid(slip free) and dynamically free(stress free) boundary conditions are derived numerically using Galerkin's method. The effects of linear and exponential temperature dependent viscosity on the onset stationary convection are studied and computed numerically. Various consequences of the obtained results for different convective problems have been worked out, as special cases.

Keywords

Binary Fluids, Dufour-Soret Effects, Double-Diffusive Convection, Principle of Exchange of Stabilities, Galerkin's Technique, Rayleigh Number.

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