The onset of free convection near the liquid–vapour critical point.: Part II: Unsteady heating

The analysis of the Rayleigh–Bénard stability in supercritical fluids, conducted in the case of an initial state of steady conduction in Part I of this paper [Physica D 126 (1999) 69] is now extended to an unsteady bottom-heating. In contrast to the steady case, an unsteady heating results in the appearance of a regime of “Piston effect” in the fluid cell, which constitutes the basic flow. This flow is calculated by means of matched asymptotic expansions, and its stability is then analysed using standard linear stability techniques applied to different typical sizes of perturbations. The dispersion relation is obtained, showing that the only influence of the Piston effect on the stability is indirect, through the particular shape of the temperature profile it creates. A numerical method is then proposed to solve the dispersion relation and predict the rate of growth of perturbations of different wave numbers at different moments of the evolution of the basic flow. This method should permit, given a heating law and an experimental configuration, to predict if and when convection appears.