Onset of Rayleigh–Bénard–Marangoni convection with time-dependent nonlinear concentration profiles

Abstract

Taking into account the effects of the time-dependent nonlinear base concentration profile on the concentration and velocity disturbances in the Rayleigh–Bénard, Bénard–Marangoni, and Rayleigh–Bénard–Marangoni problems, a spatial base-profile influenced frozen-time marginal state analysis has been developed to predict the onset of cellular convection caused by the Rayleigh effect (buoyancy) and Marangoni effect (surface tension gradient). The relations between the base velocity and concentration profiles and the velocity and concentration fluctuations have been established, analogous to the theory of statistical physics which describes the relation between the fluctuating forces and velocities for particles in non-equilibrium systems. Compared with the quasi-static analysis and the initial-value technique, the numerical calculations of this spatial base-profile influenced frozen-time marginal state analysis are straightforward and involve no subjective decisions on either the initial conditions of disturbances or the criterion for the onset time of convection. Using the spatial base-profile influenced frozen-time marginal state analysis, the calculated critical parameters for the onset of convection are in good agreement with the published experimental results.