Linear Stability Analysis of Multicomponent convection

Jyoti Prakash, Shweta Manan*, Virender Singh

Department of Mathematics and Statistics, Himachal Pradesh University, Shimla – 171005 (India)

*Corresponding Author Email: mananshweta882@gmail.com

ABSTRACT:

Condition for characterizing nonoscillatory motions, which may be neutral or unstable, for multicomponent convection is derived. It is analytically proved that the principle of the exchange of stabilities, in multicomponent convection, is valid in the regime , where ,,... are the Rayleigh numbers for the concentration components and is the Prandtl number and , are the Lewis numbers for the concentration components respectively. When the complement of this sufficient condition holds good, oscillatory motions of neutral or growing amplitude can exist, and thus it is important to derive upper bounds for the complex growth rate of such motions when atleast one of the bounding surfaces is rigid so that exact solutions of the problem in closed form are not obtainable. Thus as a second problem bounds for the growth rates are also obtained. It is further proved that these results are uniformly valid for all combinations of rigid and dynamically free boundaries.

KEYWORDS: Multicomponent convection,the principle of the exchange of stabilities,oscillatory motions, complex growth rate, concentration Rayleigh number.

1 INTRODUCTION:

If gradients of two stratifying agencies, such as heat and salt, with different diffusivities are simultaneously present in a fluid layer, the convective phenomena, which occurs, is generally known as double diffusive convection. This phenomenon is now well known and has been extensively studied. For review on this subject one may be referred to Turner [1-2], Baines and Gill [3] and Brandt and Fernando [4].

Although the subject of double diffusive convection is still an important area of research (Kellner and Tilgner [5] and Schmitt [6]), there are many fluid systems where more than two components are present (Turner [2] and Griffith [7]). Examples of such systems include the solidification of molten alloys, earth core, geothermally heated lakes, sea water, and magmas and their laboratory models. The presence of more than one salt in fluid mixtures is very often requested for describing natural phenomena such as contaminant transport, acid rain effects, underground water flow, and warming of the stratosphere. The subject of more than two stratifying agencies has attracted many researchers (Griffith [7-8], Pearlstein et al. [9], Rionero [10], Lopez et al. [11], Terrones [12] and Prakash et al. [13]). The essence of the work of these researchers is that small concentrations of a third diffusing component with a smaller diffusivity can have a significant effect upon the nature of diffusive instabilities and diffusive convection (oscillatory modes) and salt finger (steady modes) 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 even if the overall density stratification is hydrostatically stable. These researchers also notice some fundamental differences between double and triply diffusive convection. One is that if the gradients of two of the stratifying agencies are held fixed, then three critical values of the Rayleigh number of the third agency are sometimes required to specify the linear stability criteria (in double diffusive convection only one critical Rayleigh number is required). The other difference is that the onset of convection for the case of free boundaries may occur via a quassiperiodic bifurcation from the motionless basic state.

Now the triply diffusive convection despite its complexities has also been well studied. But, to the author knowledge, not many investigations have been conducted in stability theory when more than three components are present which may be, perhaps, due to the complexities involved in mathematical calculations and numerical computations. Some worth researches which may be referred here are due to Terrones and Pearlstein [14] who derived analytical results for components and numerical results for using dynamically free boundary conditions. Later Lopez et al. [11] predicted that the results of triply diffusive convection may be extended to multicomponent convection with n components for rigid surfaces also. Further significant contributions to multicomponent convection are due to Ryzhkov and Shevtsova [15-16].

In the present work, we analyse the onset of buoyancy driven convection in a multicomponent fluid layer. We generalize the existing results of triply diffusive convection problem concerning the validity of the principle of the exchange of stabilities (Prakash et al. [17]) and arresting the complex growth rate of an oscillatory motion (when it occurs) (Prakash et al. [18]) which are important especially when atleast one boundary is rigid so that exact solutions in the closed form are not obtainable. The results derived herein are uniformly valid for any combination of the rigid and free boundaries and the results of double diffusive (Banerjee et al. [19], Gupta et al. [20]) and triply diffusive convection (Prakash et al. [17-18]) follow as a consequence. Further the importance of the results obtained herein lies in that these results may be used for any hydrodynamic multicomponent system where no mathematical calculation or numerical computation is possible.

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Received on 21.08.2016 Accepted on 06.09.2016

Int. J. Tech. 2016; 6(2): 118-122.

DOI: 10.5958/2231-3915.2016.00019.5