Bénard–Marangoni instability in rigid rectangular containers

Abstract

Thermocapillary convection in three-dimensional rectangular finite containers with rigid lateral walls is studied. The upper surface of the fluid layer is assumed to be flat and non-deformable but is submitted to a temperature-dependent surface tension. The realistic ‘no-slip’ condition at the sidewalls makes the method of separation of variables inapplicable for the linear problem. A spectral Tau method is used to determine the critical Marangoni number and the convective pattern at the threshold as functions of the aspect ratios of the container. The influence on the critical parameters of a non-vanishing gravity and a non-zero Biot number at the upper surface is also examined. The nonlinear regime for pure Marangoni convection (Ra = 0) and for Pr = 104, Bi = 0 is studied by reducing the dynamics of the system to the dynamics of the most unstable modes of convection. Owing to the presence of rigid walls, it is shown that the convective pattern above the threshold may be quite different from that predicted by the linear approach. The theoretical predictions of the present study are in very good agreement with the experiments of Koschmieder & Prahl (1990) and agree also with most of Dijkstra's (1995a, b) numerical results. Important differences with the analysis of Rosenblat, Homsy & Davis (1982b) on slippery walls containers are emphasized.