A Characterization Theorem in Magnetohydrodynamic Triply Diffusive Convection with Viscosity Variations

Jyoti Prakash, Rajeev Kumar*

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

*Corresponding Author Email : rajeevkumar2012math@gmail.com

ABSTRACT:

The paper mathematically establishes that magnetohydrodynamic triply diffusive convection, with variable viscosity and with one of the components as heat with diffusivity , cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the two concentration Rayleigh numbers and , the Lewis numbers and for the two concentrations with diffusivities and respectively (with no loss of generality ), (the minimum value of viscosity in the closed interval ) and the Prandtl number satisfy the inequality provided is positive everywhere. It is further proved that this result is uniformly valid for any combination of rigid and/or free perfectly conducting boundaries.

KEYWORDS: Triply diffusive convection, variable viscosity, concentration Rayleigh number, oscillatory motion, initially bottom heavy configuration and Chandrasekhar number.

1. INTRODUCTION

The problem of thermosolutal instability, aside from its various applications in the fields of geophysics, astrophysics, oceanography, chemical engineering etc. has received considerable attention due to its complexities as double-diffusive phenomenon. For a broad view of the subject one may be referred to Turner (1974), Brandt and Fernando (1996), Radko (2013).

These researchers have considered only the case of two component systems. However, it has been recognized later on (Griffiths (1979), Pearlstein et al. (1989), Lopez et al. (1990), Terrones (1993), Turner (1985)) 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 sea water etc. The presence of more than one salt in fluid mixtures is very often requested for describing natural phenomena such as underground water flow, contaminant transport, warming of the stratosphere, acid rain effects. Further a number of technologically important alloys such as nickel- based alloys (Pearlstein et al. (1989)) used in turbine blades and another high-strength applications, containing significant mass fraction of as many as seven metallic elements.

The recently established characterization theorem of Prakash et al. (2015), which states that oscillatory motions (neutral or unstable) of growing amplitude cannot manifest in an initially bottom heavy magnetohydrodynamic triply diffusive convection whenever the sum of the concentration Rayleigh numbers is less than a critical value, has brought a fresh outlook to the subject matter of triply diffusive convection and paved the way for further theoretical and experimental investigations in this field of enquiry. The summary of Prakash et al.’s (2015) characterization theorem is that it provides a classification of the neutral and unstable magnetohydrodynamic triply diffusive convection classes namely the bottom heavy class and the top heavy class and strikes a distinction between them by means of characterization theorems which disallow the existence of oscillatory motions in the former class. For the field of applications of the Prakash et al.’s (2015) theorem in flows that are of interest in certain fields like geophysics, oceanography, astrophysics etc. it is necessary to extend the classical analysis where in the fluid viscosity is a function of temperature and /or depth because the effects of viscosity variation play an important role in several physical situations in these fields (Banerjee et al. (1977), Hooman and Gurgenci (2008), Korenga and Jordhan (2002), Sunil and Choudhary (2013), Torrance, and Turcotte (1971) ). Since the variation of viscosity of liquids with temperature is extremely rapid, the inclusion of variation effects certainly extends the domain of validity of the existing results in the literature.

The considerations of a temperature dependent viscosity on the pattern of density in the triply convection problems has the limitations that viscosity is a linear function of vertical coordinate which need not necessarily be so in a real physical situation. Therefore, in the governing equations of the problem, we consider viscosity as an arbitrary function of the vertical coordinate which is in accordance with the formulation regarding the role of viscosity in Rayleigh-Taylor instability problem. From the 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 context of Prakash et al.’s (2015) theorem and extend the domain of validity of the earlier results in the literature.

2. Mathematical Formulation and analysis

An infinite horizontal layer filled with a Boussinesq viscous fluid is statically confined between two horizontal boundaries and (kept under the influence of a uniform vertical magnetic field), maintained at constant temperatures and () and uniform solute concentrations, and , () at the lower and upper boundaries respectively. Let the origin be taken on the lower boundary with z-axis perpendicular to it. It is further assumed that cross diffusion effects may be neglected

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Received on 17.08.2016 Accepted on 02.09.2016

Int. J. Tech. 2016; 6(2): 81-86.

DOI: 10.5958/2231-3915.2016.00012.2