Control of long-wavelength Marangoni–Bénard convection

Abstract

A nonlinear feedback control strategy for delaying the onset and eliminating the subcritical nature of long-wavelength Marangoni–Bénard convection is investigated based on an evolution equation. A control temperature is applied to the lower wall in a gas–liquid layer otherwise heated uniformly from below. It is shown that, if the interface deflection is assumed to be known via sensing as a function of both horizontal coordinates and time, a control temperature with a cubic-order polynomial dependence on the deflection is capable of delaying the onset as well as eliminating the subcritical instability altogether, at least on the basis of a weakly nonlinear analysis. The analytical results are supported by direct numerical simulations. The control coefficients required for stabilization are O(1) for both delaying onset indefinitely and eliminating subcritical instability. In order to discuss the effects of control, a review is made of the dependence of the weakly nonlinear subcritical solutions without control upon the various governing parameters.