On the Characterization of Nonoscillatory Motions in Triply Diffusiveconvection in Porous Medium

Jyoti Prakash*, Shweta Manan and Virender Singh

Department of Mathematics and Statistics, Himachal Pradesh University, Summer Hill, Shimla-171005, India.

*Corresponding Author E-mail: jpsmaths67@gmail.com

ABSTRACT:

The present paper mathematically establishes that ‘the principle of the exchange of stabilities’ for triply diffusive convection in porous medium (Darcy model) is valid in the regime (R_1 σ)/(2〖τ_1〗^2 π^4 )+(R_2 σ)/(2〖τ_2〗^2 π^4)≤1, where R_1 and R_2 are the Rayleigh numbers for the two concentration components, τ_1 and τ_2 are the Lewis numbers for the two concentration components and σ is the thermal Prandtl number. It is further proved that the above result is uniformly valid for any combination of rigid and free boundaries.

KEYWORDS: Triply Diffusive convection, Nonoscillatory motions, principle of the exchange of stabilities, concentration Rayleigh number, Porous medium, Darcy Model.

INTRODUCTION:

The hydrodynamic instability that manifests under appropriate conditions in a static horizontal initially homogeneous viscous and Boussinesq liquid layer of infinite horizontal extension and finite vertical depth which is kept under the simultaneous action of a uniform vertical temperature gradient and a gravitationally opposite uniform vertical concentration gradient in the force field of gravity is known as double diffusive convection or thermohaline convection. The thermohaline convection problem has been extensively studied in the recent past on account of its interesting complexities as a double diffusive phenomenon as well as its direct relevance in many problems of practical interest in the fields of oceanography, astrophysics, limnology and chemical engineering etc. Double diffusive convection is now well known. For a broad view of the subject one may be referred to Stern [1], Veronis [2], Nield [3], Baines and Gill [4] ,Turner [5] and Brandt and Fernando [6] etc. All these researchers have considered the case of two component systems. However, it has been recognized previously (Griffiths [7], Turner [8]) that there are many situations wherein more than two components are present. Examples of such multiple diffusive convection fluid systems include the solidification of molten alloys, Earth core, geothermally heated lakes, and magmas and their laboratory models and seawater etc. Griffiths [7], Lopez et al. [9] etc. have theoretically studied the onset of convection in a triply diffusive fluid layer (where density depends on three independently diffusing agencies with different diffusivities). The essence of the findings of these researchers is that small concentrations of a third component with a smaller diffusivity can have a significant effect upon the nature of diffusive instabilities and ‘oscillatory’ and direct ‘salt finger’ modes are simultaneously unstable under a wide range of conditions when the density gradients due to components with the greatest and smallest diffusivity are of same signs.

The establishment of the non occurrence of any slow oscillatory motions which may be neutral or unstable implies the validity of the principle of the exchange of stabilities (PES). The validity of this principle in stability problems eliminates the unsteady terms from the linearized perturbation equations which results in notable mathematical simplicity since the transition from stability to instability occurs via a marginal state which is characterized by the vanishing of both real and imaginary parts of the complex time eigen value associated with the perturbation. Pellew and Southwell [10] proved the validity of PES (i.e. occurrence of stationary convection) for the classical Rayleigh-Benard instability problem. However no such results existed for other more general hydrodynamic configurations. Banerjee et al. [11] established such a criterion for magnetohydrodynamic Rayleigh – Benard convection problem which has further been extended by Gupta et al. [12] for thermohaline convection problems.

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Received on 20.01.2014 Accepted on 02.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 168-170