A Semi-circle Theorem in Triply Diffusive Convection

Jyoti Prakash*, Sanjay Kumar Gupta, Renu Bala and Kanu Vaid

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

*Corresponding Author E-mail: jpsmaths67@gmail.com; sanjay6571@gmail.com

ABSTRACT:

The paper mathematically establishes that the complex growth rate of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude, in a triply diffusive fluid layer with one of the components as heat with diffusivity , must lie inside a semicircle in the right- half of the -plane whose centre is origin and radius equals where and are the Rayleigh numbers for the two concentration components with diffusivities and (with no loss of generality, ) and σ is the Prandtl number. The bounds obtained herein, in particular, yield a sufficient condition for the validity of ‘the principle of the exchange of stability’. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces.

KEYWORDS: Triply Diffusive convection; Oscillatory motions; complex growth rate; Principle of the exchange of Stability.

INTRODUCTION:

Convective motions can occur in a stably stratified fluid when there are two components contributing to the density which diffuse at different rates. This phenomenon is called double-diffusive convection. To determine the conditions under which these convective motions will occur, the linear stability of two superposed concentration (or one of them may be temperature gradient) gradients has been studied by Stern (1960), Veronis (1965), Nield (1967), Baines and Gill (1969) and Turner (1968) etc.

All these researchers have considered the case of two component systems. However, it has been recognized later on (Griffiths (1979), 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, geothermally heated lakes, magmas and their laboratory models and sea water. Griffiths (1979), Pearlstein et al (1989) and Lopez et al. (1990) have theoretically studied the onset of convection in a horizontal layer, of infinite extension, of a triply diffusive fluid (where the density depends on three independently diffusing agencies with different diffusivities). These researchers found 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.

Thus oscillatory motions of growing amplitude can occur in Triply diffusive convection. The problem of obtaining bounds for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude in triply diffusive convection problem is an important feature of fluid dynamics, especially when both the boundaries are not dynamically free so that exact solutions in the closed form are not obtainable, the bounds for the complex growth rate of an arbitrary oscillatory perturbation of growing amplitude in triply diffusive case must be found. Banerjee et al. (1981) formulated a noble way of combining the governing equations and boundary conditions to obtain such bounds. We used their technique to prove the following theorem:

The complex growth rate of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude, in a triply diffusive fluid layer with one of the components as heat with diffusivity , must lie inside a semicircle in the right- half of the -plane whose centre is origin and radius equals where and are the Rayleigh numbers for the two concentration components with diffusivities and respectively (with no loss of generality, ) and σ is the Prandtl number. The bounds obtained herein, in particular, yield a sufficient condition for the validity of ‘the principle of the exchange of stability’. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces and the results of Banerjee et al. (1981) for double diffusive convection follow as a consequence.

CONCLUSIONS:

A linear stability analysis is used to derive the upper bounds for complex growth rates in triply diffusive convection problem. These bounds are important especially when both the boundaries are not dynamically free so that exact solutions in the closed form are not obtainable. Further, the results so obtained are uniformly valid for all the combinations of rigid and free boundaries.

REFERENCES:

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Banerjee M B, Katoch D C, Dube G S, Banerjee K (1981) Bounds for growth rate of perturbation in thermohaline convection, Proc Roy Soc London, Ser A, 378, 301 – 304.

Griffiths R W (1979) The influence of a third diffusing component upon the onset of convection, J Fluid Mech, 92, 659 – 670.

Lopez A R, Romero L A, Pearlstein A J (1990) Effect of rigid boundaries on the onset of convective instability in a triply diffusive fluid layer, Physics of fluids A, 2(6), 897 – 902.

Nield D A (1967) The thermohaline Rayleigh-Jeffreys problem, J Fluid Mech, 29, 545 – 558.

Pearlstein A J, Harris R M, Terrones G (1989) The onset of convective instability in a triply diffusive fluid layer, J Fluid Mech, 202, 443 – 465.

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Received on 08.01.2014 Accepted on 01.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 210-213