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The Two-Component Bénard Problem with Flow. II. Further Numerical Results

A numerical solution for the two-component Benard problem with flow is presented, taking into account the contribution of thermal diffusion to the total density gradient. The results are compared with the approximate solution obtained by the variational technique of the local potential introduced some years ago by Glansdorff and Prigogine. With a transverse rolls pattern, the destabilizing effect of a flow in a fluid mixture stabilized by the thermal diffusion effect is confirmed. A further checking of these results is presented. Using again the local potential technique, calculations are conducted with a Chebyshev polynomial expansion as trial functions. The two methods produce critical Rayleigh numbers which are in agreement with each other. Thus it is conjectured that a decrease of the critical Rayleigh number with the Reynolds number, is a real phenomenon. Introduction In a previous paper [1 ] we presented a study of the linear hydrodynamic stability of a binary liquid mixture heated from below and flowing with a parabolic velocity distribution. We investigated the whole range of Soret coefficients, positive and negative, and flows with Reynolds numbers up to 125. It was demonstrated that rolls perpendicular to the velocity flow are the more unstable mode when the Soret number is negative. In this case, the system is stabilized by the concentration gradient. The thermal diffusion effect is swept away rather rapidly by the flow effect. Let us recall that without flow, the system is stabilized by the thermal diffusion effect; when the velocity of the flow is increased the only possibility for the critical Rayleigh number to reach rapidly the values of the pure liquids, is to decrease down to this value. J. Non-Equilib. Thermodyn., Vol. 3, 1978, No. 1 2 J. C. Legros, J. K. Platten In order to verify that the critical Rayleigh number actually begins to decrease for small flow velocities and negative Soret coefficient and that this behaviour is not linked to a particular technique of calculation, namely the local potential theory with only one variational parameter in each unknown, we have made a completely different approach to the problem. It is based on a direct numerical integration of the starting system of equations. This method has already been used a few times in the theory of linear hydrodynamic stability [2, 3]. In section 3, we used again the local potential technique, but this time with more sophisticated trial functions, namely Chebyshev polynomials. 1. The numerical integration The starting equations for the two-dimensional problem with transverse rolls, after normal modes analysis may be written as (for more details and notations, see [1 ]): (D-a + aPr-iaRePrUx) θ =-w , (D a) (D a + σ) w ic*Re[Ux(D a) D x] w = = Ra α θ — Rth α χ , (D a + aSc sD iaDxReSc) ÷ + [N, (D á) + sD] θ Sc (1)