Overstability in an Infinite Liquid Layer under Simultaneous Surface Tension, Buoyancy and Shear Effects

Abstract

There has been a long tradition concerning the theoretical study of Rayleigh-Bénard and Bénard-Marangoni instabilities, with papers by Rayleigh (1916), Pearson (1958), Scriven and Sternling (1964) and Nield (1964) among others. The principle of exchange of stability holds for Rayleigh-Bénard problems. Conversely, for Bénard-Marangoni problems, the liquid layer may exhibit overstability. Linear analysis of overstability is much more difficult to perform than in the case of exchange of stability because eigenvalues at marginality are no more equal to 0. This is likely to be the reason why the overstability problem has been solved in the case of a pure Marangoni mechanism only about ten years ago by Takashima (1981). Thereafter, the question naturally arose to know how overstability would set in when both buoyancy and surface tension mechanisms were simultaneously acting. This problem was solved recently by Gouesbet and Maquet (1989), Gouesbet et al (1990) and, independently, by Perez-Garcia and Carneiro (1991). Some specific calculations were devoted to the case of silicon oils in Gouesbet and Maquet (1989). For this liquid, it appears that the onset of oscillatory behaviour required very high driving temperature differences of about typically 500 to 5000 K, conflicting even with the very existence of the liquid state. We have then been interested in knowing whether supplementary effects like extra destabilization by basic state convection would not decrease these values to lower more reasonable ones.