Onset of oscillatory interfacial instability and wave motions in Bénard layers

Abstract

This chapter describes the onset of oscillatory interfacial instability and wave motions in Bénard layers. An account of the basic equations and approximations needed to study Bénard convection with heat or mass transfer and Marangoni stresses is presented. According to Pearson's theory, a liquid layer open to passive air is unstable to a well-defined short-wave planform of steady cellular convection for a critical value of the Marangoni number when the heating is from the liquid side. It is shown that that oscillatory instability is possible for gradients of the opposite sense if due account is taken of the dynamics of both the upper and lower phases as for the case of an interface between two liquids with transport from either side. It is found that when the Rayleigh numbers of two layers are very different from one another, the onset of instability in one of the layers drives the other, and hence the appearance of two counter-rotating cells. It is observed that when both Rayleigh numbers approach a common value, for about the same critical wavenumbers, the situation is more complex. The nonlinear waves and dissipative solitons are also elaborated.