Upper Bounds for the Complex Growth Rate of Thermohaline Convection of Veronis and Stern Types with Viscosity Variations

Jyoti Prakash, Rajeev Kumar

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

*Corresponding Author E-mail: jpsmaths67@gmail.com., rajeevkumar2012math@gmail.com

ABSTRACT:

Upper bounds for the complex growth rate of an arbitrary oscillatory perturbation which may be neutral or unstable of thermohaline convection of Veronis (G.Veronis,J.MarineRes.,23,(1965) 1-17) type with the viscosity variation effects included heated from below are obtained which in particular yield sufficient condition for the validity the “principle of the exchange of stabilities” for this configuration. Similar results are also obtained for thermohaline convection of Stern (ME Stern, Tellus, 12,(1960), 171-175) type with the viscosity variation effect included. The results obtained herein are uniformly valid for all combinations of dynamically free and rigid boundaries.

KEYWORDS: Thermohaline instability, Veronis type, Stern type, Oscillatory motions, Variable viscosity.

1. INTRODUCTION:

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. (Turner (1974)). Two fundamental configurations have been studied in the context of thermohaline convection problem, one by Veronis (Veronis(1965)), wherein the temperature gradient is destabilizing and the concentration gradient is stabilizing and another by Stern (Stern (1960)) wherein the temperature gradient is stabilizing and the concentration gradient is destabilizing. The main findings of Veronis and Stern for their respective configuration are that both allow the occurrence of steady motion or oscillatory motion of growing amplitude, provided the destabilizing temperature gradient or the concentration gradient is sufficiently large. In the case of Veronis’ configuration, oscillatory motion of growing amplitude are preferred mode of onset of instability whereas in case of Sterns’ configuration, stationary convection is the preferred mode of onset of instability and these results are independent of the initially gravitational stable or unstable character of these two configurations.

The problem of obtaining the bounds for the complex growth rate of an arbitrary oscillatory motion of growing amplitude in thermohaline convection heated from below is an important problem especially when both the boundaries are not dynamically free so that the exact solution in closed form are not obtainable and one has to depend on numerical solutions which are rather laborious. Banerjee et al.(1981) formulated a noble way of combining the governing equations and boundary conditions of thermohaline instability problem and obtained the desired bounds for the complex growth rates.

For the field of applications of the Banerjeeet al.’s results 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. 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 thermohaline 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 Banerjee et al.’s results and extend the domain of validity of the earlier results in the literature.

4. REFERENCES:

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

2. M. E. Stern,The Salt-fountain and thermohaline convection,Tellus 12, 172, (1960).

3. J. S. Turner, Double-diffusive phenomena, Annual Review of Fluid Mechanics, 6, 37-54 (1974).

4. M. B. Banerjee, D. C.Katoch, G. S. Dube and K. Banerjee, Bounds for growth rate of a perturbation in thermohaline convection, Proceeding Royal Society of London A, 378,301-304(1981).

5. M. B. Banerjee, J. R. Gupta and Jyoti Prakash,Onthermohaline convection of the Veronis type, J. Math. Anal. Appl. 179, 327(1993).

6. M.H Schultz, Spline Analysis, Prentice Hall, Englewood Cliffs, New Jersy (1973).

7. Jyoti Prakash,A mathematical theorem for thermohaline convection of Veronis type with viscosity variation,Indian J. pure appl. Math, 26(8), 813-821(1995).

Received on 14.01.2014 Accepted on 31.01.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 117-120