On exchange Principle in Multicomponent Convection with Viscosity Variations

On exchange Principle in Multicomponent Convection with Viscosity Variations

Chitresh Kumari*, Jitender Kumar, Harjinder Singh, Jyoti Prakash

Department of Mathematics & Statistics, Himachal Pradesh University, Shimla-171005.

Abstract:

In the present paper, the multicomponent convection problem has been studied by considering variable viscosity. A sufficient condition has been derived for the validity of the occurrence of stationary convection. Further the results derived herein are uniformly valid for all combinations of the bounding surfaces.

INTRODUCTION:

In many fluid systems, density depends on more than three independently stratifying agencies having different diffusivities which is known as multicomponent diffusive convection. It is an interesting topic for the researchers due to its manifold applications. Examples of multicomponent diffusive convection fluids includes magmas and their laboratory models, earth’s core and sea water (Griffiths et. al. (1979) and Terrones (1993)), solidification of molten alloys and exothermally heated lakes.

Several researchers have shown their interest in this field of enquiry. Griffiths (1979) and Pearlstein et al.(1989) studied the onset of convection in a horizontally infinite layer of a triply diffusive fluid. The general case of n independently diffusing stratifying agencies has been considered by Terrones and Pearlstein (1989). Ryzhkov and Shevtsova (2007) investigated the case of multicomponent mixture with application to the thermogravitational column. Ryzhkov and Shevtsova (2009) also investigated the longwave instability of a multicomponent fluid layer with Soret effect. Rionero (2013) examined the multicomponent diffusive-convective fluid motions in porous layer. For a broad view of the subject one may also be referred to Turner (1985), Tracey (1996), Straughan and Tracey (1999).

Since the effect of viscosity variation plays an important role in several physical situations. Therefore, it is necessary to extend the classical analysis wherein the fluid viscosity is a function of temperature and/or depth (Banerjee et.al.(1977), Prakash et al.(2015)). Further, the change of viscosity with temperature is extremely rapid, the inclusion of variation effects certainly extends the domain of validity of the existing results in the literature. From a mathematical point of view, the resulting differential equations have variable coefficients contrary to the case wherein viscosity is constant and therefore these more general problems introduce extra analytical complexities. In the present paper, we make an attempt to mathematically handle these more complex problems in the case of multicomponent convection.

The “principle of the exchange of stabilities”, which implies that convection occurs initially as stationary convection or we can say that all non-decaying disturbances are non-oscillatory in time which further implies that the first unstable eigen value of the linearized system has imaginary part equal to zero (Herron (2000)) due to which mathematical handling of the governing equations becomes easy. Therefore, the aim of the present paper is to establish a sufficient condition for the validity of “principle of the exchange of stabilities” in multicomponent convection problem with variable viscosity.

The Physical Problem:

The physical problem consists of an infinite horizontal layer of a Boussinesq viscous fluid which is statically confined between two horizontal boundaries  and  maintained at constant temperatures  and  and uniform solute concentrations   and  … … …  at the lower and upper boundaries respectively.