Research Journal of Engineering and Technology

Joginder Singh Dhiman ¹, Poonam Sharma ¹, Megh Raj Goyal ²

Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla-171005 (H.P.), IN
2 Department of Mathematics, D.A.V. College, Malout-152107, Punjab, IN

Abstract

The present paper extends the analysis of Gupta et al. (2001, J. Math. Anal. Appl., 264, 398) of Veronis and Stern type’s thermohaline convection problems for the case of temperature-dependent viscosity. The stability of the oscillatory motions for both types of problems with variable viscosity is discussed in this paper and the upper bounds for the growth rates for neutral or unstable oscillatory perturbations are also prescribed. The obtained results are uniformly valid for all combination of dynamically free and rigid boundaries and are free from a curious condition on the non-negativity of the second derivative of viscosity parameter. Further, various results for an initially top-heavy as well as an initially bottom heavy configurations follow as consequence.

Keywords

Thermohaline Convection, Oscillatory Motions, Complex Growth Rate,; Temperature-Dependent Viscosity, Eigenvalue Problem.

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Joginder Singh Dhiman ¹, Vijay Kumar ²

Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. College, Sanjauli, Shimla (H.P.)-171006, IN

Abstract

Chandrasekhar proved his famous -law of stationary convection in magnetoconvection problem for the case of both dynamically free boundaries. For the other two cases of boundary conditions, namely; both rigid boundaries and combinations of rigid and dynamically free boundaries he, on the basis of numerical computations, conjectured that the same law must hold true. In the present paper, we have reinvestigated the onset of thermal instability in an electrically conducting fluid layer heated from below in the presence of a uniform magnetic field. In the present analysis the validity of -law for general nature of bounding surfaces is proved using the Galerkin technique. The obtained results are in good agreement with the numerical results of Chandrasekhar and thus validate his claim.

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Joginder Singh Dhiman ¹, Deepak Gupta ²

Affiliations
1 Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla (H.P.)-171005, IN
2 Govt. Post Graduate College, Solan, Distt. Solan (H.P.)-173212, IN

Abstract

The present paper deals with the construction of a generalized setup of eigenvalue problem from the perturbations equations governing Modified Thermohaline convection (derived by Banerjee et al. [1993]) by an appropriate choice of parameters of the fluid. This generalized setup of eigenvalue problem named as; generalized double-diffusive convection problem yields eigenvalue problems for Stern type double-diffusive convection problem and for Dufour-Driven double-diffusive convection problem as a consequence. The stability investigations of this general problem are carried in this paper and some general results concerning the stability or otherwise are derived for the case of both dynamically free boundaries and various consequences of the derived results are also discussed.

Keywords

Modified Thermohaline Convection, Stern Type Convection, Dufour-Driven Convection, Eigenvalue Problem.

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