On the convective instability of a horizontal binary mixture layer with Soret effect under transversal high frequency vibration

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Abstract

The stability of the mechanical equilibrium of a plane horizontal binary mixture layer with Soret effect in the presence of high frequency transversal vibration is studied.

The asymptotic analysis for long-wave disturbances and the numerical solution of the spectral amplitude problem for cellular disturbance has shown that independently of properties of the binary mixture the effect of transversal vibration is always stabilizing. The critical values of Rayleigh number and the characteristics of critical disturbances are determined.

Introduction

It is well known that vibration of a cavity filled with fluid has a strong effect on the convective flows in the presence of non homogeneous temperature distribution. In some cases the vibration can provoke a mean flow with the structure and intensity depending on the direction and characteristics of the vibration. Thus, in the general case it is possible to distinguish two mechanisms of thermal convection excitation—thermogravitational and thermovibrational. The problem of thermovibrational convective stability of mechanical equilibrium in the state of weightlessness has been investigated 1, 2, 3, 4.

The problem of vibrational convective instability for the case of a binary mixture with Soret effect has been investigated recently [5]. The effect of longitudinal high frequency vibration on a plane horizontal layer of the mixture, with rigid and isothermal boundaries, was studied. It was shown that convective instability is caused by both mechanisms of excitation—thermogravitational and thermovibrational, which are superimposed.

In the present paper we consider the case where the axis of vibration is vertical, i.e. transversal with respect to the layer. Physically the situation is quite different from the one considered in [5]. For a one component medium it has been proven [2]that the equilibrium is absolutely stable if the axis of vibration is parallel to the temperature gradient. Thus one expects that for a binary mixture the mechanical equilibrium will be stable if the axis of vibration and the density gradient are mutually parallel. The analysis performed confirms this expectation.

In Section 2the problem is described and the main equations are written down. In Section 3the state of mechanical equilibrium is considered and the problem of its linear convective stability is formulated. In Section 4the asymptotic analysis is developed for the limiting case of long-wave normal disturbances. The wave number of the normal mode is used as a small parameter for regular perturbation theory. In Section 5the numerical results for arbitrary wave numbers (cellular modes) are presented and discussed.

Section snippets

The problem description and the basic equations

We consider an infinite plane horizontal binary mixture layer with Soret effect. The horizontal boundaries of the layer are assumed to be rigid, isothermal and impermeable to the mixture components. The Cartesian coordinate system is chosen with the origin on the lower plate z = 0 and with the z-axis directed vertically upward. The temperature of the lower plane z = 0 is maintained constant and equal to Θ, the temperature of the upper plane z = h is chosen as a reference point 0, so both cases

Mechanical equilibrium and stability problem formulation

An important question is whether the state of mechanical quasiequilibrium (i.e. the state at which the mean velocity is zero, but the pulsational component is not in general ) exists or not in our situation.

To find the quasiequilibrium conditions it is necessary to set up v = 0, ∂⧹∂t = 0, P = P0, T = T0, C = C0 and w = w0, where P0, T0, C0 and w0 are the distributions in the state of mechanical equilibrium. The general system leads to the following necessary conditions for quasiequilibrium

The limiting case of long-wave disturbances

In the general case the solution of the eigenvalue problem (3.6) , (3.7) must be found numerically. But the condition of impermeability enables us to expect that for some ranges of parameter values, long-wave disturbances (with the wave number k = 0) are responsible for instability excitation. To study the behaviour of long-wave disturbances one may develop the regular perturbation method with the wave number k as a small parameter.

So, let us try to construct the asymptotic solution of the

Numerical results

The numerical solution of the complete spectral eigenvalue problem has been obtained by straight forward step-by-step numerical integration of the system of amplitude equations by the Runge–Kutta–Merson method in combination with shooting procedure. Some numerically determined instability boundary characteristics are presented: the critical values of the Rayleigh number Ram (obtained as a result of minimization of critical Ra with respect to the wave number k) , the critical wave number km

Conclusions

For a binary mixture in an infinite horizontal layer in the presence of vertical high frequency vibration, contrary to the case of longitudinal h.f. vibration, the specific vibrational mechanism of instability by excitation is not operative. The effect of vibration is purely stabilizing: at arbitrary values of binary mixture parameters the critical Rayleigh number increases monotonously with increasing vibrational parameter.

The critical values of Rayleigh number and the characteristics of the

Acknowledgements

The research presented was partially performed under the contracts NASA No. 920⧹18-5208⧹96, INTAS No. 94-0529 and the Belgian Program on Interuniversity Pole of Attraction (PAI-IVAP Nr 21) initiated by the Belgian state, Prime Ministers Office, Science Policy Programming.

References (10)

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  • G.Z. Gershuni, E.M. Zhukhovitsky, Convective instability of a fluid in a vibrational field under conditions of...
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