On linear growth rates in thermohaline convection with viscosity variations

Jyoti Prakash* and Kanu Vaid

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

*Corresponding Author: jpsmaths67@gmail.com

ABSTRACT:

In the present paper it is proved that the complex growth rate (where and are real and imaginary parts of p ) of an arbitrary oscillatory motions of growing amplitude, neutral or unstable, for thermohaline convection configuration of Veronis type (Veronis, G., J. Mar. Res., 23(1965)1), with the viscosity variations must lie inside a semicircle in the right half of the prpi- plane whose centre is at the origin and radius equals . A similar theorem is also proved for thermohaline convection of Stern type (Stern, M.E., Tellus 12(1960)172). Furthermore the above results are uniformly valid for all combinations of rigid and free bounding surfaces. The results obtain herein, in particular, also yield sufficient conditions for the validity of the ‘principle of the exchange of the stabilities’ for the respective configurations.

KEY WORDS: Thermohaline instability; oscillatory motions; Veronis type; stern type; variable viscosity.

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 thermohaline instability. The thermohaline instability problem has been extensively studied in the recent past on account of its interesting complexities as a double diffusive phenomena as well as its direct relevance in many problems of practical interest in the fields of oceanography, astrophysics, limnology and chemical engineering etc. (Turner (1974)). For a broad view of the subject one may be referred to Brandt and Fernando (1996). Two fundamental configurations have been studied in the context of thermohaline instability problem, one of Veronis (1965), wherein the temperature gradient is destabilizing and the concentration gradient is stabilizing and another by Stern (1960), wherein the temperature gradient is stabilizing and concentration gradient is destabilizing. The main results derived by Veronis and Stern for their respective configurations are that both allow the occurrence of a stationary convection or an oscillatory convection of growing amplitude, provided the destabilizing temperature gradient or concentration gradient is sufficiently large.

However, oscillatory convection is the preferred mode of onset of instability in case of Veronis’ configuration, whereas stationary convection is the preferred mode of instability in case of Stern’s configuration. Furthermore, these results are independent of the initially gravitationally stable or unstable character of the two configurations.

The problem of obtaining bounds for the linear growth rate of an arbitrary oscillatory motions of growing amplitude, which may be neutral or unstable, in thermohaline configurations of Veronis and Stern type is an important problem especially when both the boundaries are not dynamically free so that exact solutions in the closed form are not obtainable and one has to depend on numerical solutions which rather laborious. Bannerjee.et.al (1981) formulated a noble way of combining the governing equations and boundary conditions and obtained such bounds for the complex growth rate of thermohaline convection configuration.

The present analysis is primarily motivated by the consideration of the viscosity variation effects in thermohaline instability problem. Since the variation of viscosity of liquids with temperature is extremely rapid (Lighthill (1963)), therefore the effects of viscosity variations play an important role in several physical situations in the field of oceanography, astrophysics, geophysics etc. (Toramce and Turcotte (1971), Bannerjee.et.al (1977)) and the inclusion of viscosity variation effects certainly extend the domain of validity of the existing results in the literature.

The following result is obtained in this direction:

The complex growth rate of an arbitrary oscillatory motions of growing amplitude, neutral or unstable, for thermohaline convection configuration of Veronis type, with the viscosity variations must lie inside a semicircle in the right half of the p_rp_i- plane whose centre is at the origin and radius equals . A similar theorem is also proved for thermohaline convection of stern type. Furthermore the above results are uniformly valid for all combinations of rigid and free bounding surfaces. The results obtain herein, in particular, also yield sufficient conditions for the validity of the ‘principle of the exchange of the stabilities’ for the respective configurations.

Mathematical Formulation and Analysis

An infinite horizontal layer of fluid of thickness ‘d’ is statically confined between two horizontal boundaries at and with respective temperature and concentration, T₀ and T₁(< T₀) and S₀ and S₁(< S₀). Let the origin be taken on lower boundary with the positive direction of the z – axis along the vertically upward direction. The problem is to investigate the stability of this initially stationary state when viscosity variations due to thermal effects are taken into account. The considerations of a temperature dependent viscosity on the pattern of density in the thermohaline convection problems has the limitation that the viscosity is a linear function of vertical coordinate which need not necessarily be so in a real physical situation (Jyoti Prakash (1995)).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.

REFERENCE:

1. Banerjee M. B., Katoch D. C., Dube G. S. and Banerjee K., Bounds for growth rate of a perturbation in thermohaline convection, Proc. Roy. Soc. London. A 387(1981)301.

2. Banerjee M. B., Gupta J. R. and Shandil R.G., Generalized thermal convection with viscosity variations, J. Math. Phys. Sc. 11(5)(1977)421.

3. Brandt A. and Fernando H. J. S., Double Diffusive Convection, Am. Geophys. Union, Washington, (1996).

4. Lighthill M. J., Introduction to Boundary Layer Theory in Laminar Boundary Layers, (Ed.: L. Rosenhead), Clarendon Press, Oxford, (1963).

5. Prakash J., A Mathematical theorem for thermohaline convection of the Veronis type with viscosity variations, Ind. J. Pure Appl.Math., 26(8)(1995)813.

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9. Turner J. S., Double diffusive phenomena, Ann. Rev. Fluid mech. 6(1974)37.

10. Veronis G., On finite amplitude instability in thermohaline convection, J. Mar. Res., 23(1965)1.

Received on 15.01.2013 Accepted on 13.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 140-143