Inertial Bénard–Marangoni convection

Abstract

Two-dimensional surface-tension-driven Bénard convection in a layer with a free-slip bottom is investigated in the limit of small Prandtl number using accurate numerical simulations with a pseudospectral method complemented by linear stability analysis and a perturbation method. It is found that the system attains a steady state consisting of counter-rotating convection rolls. Upon increasing the Marangoni number Ma the system experiences a transition between two typical convective regimes. The first one is the regime of weak convection characterized by only slight deviations of the isotherms from the linear conductive temperature profile. In contrast, the second regime, called inertial convection, shows significantly deformed isotherms. The transition between the two regimes becomes increasingly sharp as the Prandtl number is reduced. For sufficiently small Prandtl number the transition from weak to inertial convection proceeds via a subcritical bifurcation involving weak hysteresis. In the viscous zero-Prandtl-number limit the transition manifests itself in an unbounded growth of the flow amplitude for Marangoni numbers beyond a critical value Mai. For Ma<Mai the zero-Prandtl-number equations provide a reasonable approximation for weak convection at small but finite Prandtl number. The possibility of experimental verification of inertial Bénard–Marangoni convection is briefly discussed.