Upper limits to the Linear Growth Rate in Triply Diffusive Convection

Jyoti Prakash*, Renu Bala, Kultaran Kumari

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 mathematically established that the linear growth rate of an arbitrary neutral or unstable oscillatory perturbation of growing amplitude in a triply diffusive fluid layer (with one of the component as heat with diffusivity κ) must lie inside a semicircle in the right half of the - plane whose centre is at the origin and radius equals where R and R₁ are the thermal Rayleigh number and concentration Rayleigh number with diffusivities κ and κ_1. Further, it is proved that this result is uniformly valid for quite general nature of the bounding surfaces.

KEY WORDS: Triply diffusive convection; Oscillatory motions; complex growth rate; Concentration Rayleigh number.

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 is the thermal Rayleigh number, and are the concentration Rayleigh numbers for the two concentration components, and σ is the Prandtl number. Further, it is proved that above result is uniformly valid for quite general nature of the bounding surfaces

Mathematical Formulation and Analysis

A viscous heat conducting Boussinesq fluid of infinite horizontal extension is statistically confined between two horizontal boundaries and which are respectively maintained at uniform temperatures and uniform concentrations (as shown in Fig.1). In other words the fluid layer is heated from above, salted from below by concentration component and salted from above by concentration component

Fig.1 Physical Configuration

The above theorem may be stated in an equivalent form as: the complex growth rate of an arbitrary, neutral or unstable oscillatory perturbation of growing amplitude in a triply diffusive fluid layer must lie inside a semicircle in the right half of the - plane whose centre is at the origin and radius equals . Further, it is proved that this result is uniformly valid for quite general nature of the bounding surfaces.

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:

1. Baines P G, Gill A E (1969) On thermohaline convection with linear gradient J Fluid Mech, 37, 289.

2. 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.

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

4. 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.

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

6. 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.

7. Prakash J, Vaid K, Bala R (2014) Upper Limits to the Complex Growth Rates in Triply Diffusive Convection, Proc. Indian Nat. Sci. Acad. Vol. 80(1), 2014, pp. 115-122.

8. Stern M E (1960) The Salt Fountain and thermohaline convection, Tellus, 12, 172.

9. Turner J S (1968) The behaviour of a stable salinity gradient heated from below, J Fluid Mech, 33, 183.

10. Turner J S (1985) Multicomponent convection, Ann Rev Fluid Mech, 17, 11.

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

Received on 05.01.2015 Accepted on 20.01.2015

Research J. Engineering and Tech. 6(1): Jan.-Mar. 2015 page 47-49

DOI: 10.5958/2321-581X.2015.00008.2