Control of long-wavelength Marangoni-Benard convection

A nonlinear feedback control strategy for delaying the onset and eliminatin g the subcritical nature of long-wavelength Marangoni-Benard convection is investigated based on an evolution equation. A control temperature is appli ed 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 contr ol temperature with a cubic-order polynomial dependence on the deflection i s capable of delaying the onset as well as eliminating the subcritical inst ability altogether, at least on the basis of a weakly nonlinear analysis. T he analytical results are supported by direct numerical simulations. The co ntrol coefficients required for stabilization are O(1) for both delaying on set indefinitely and eliminating subcritical instability. In order to discu ss the effects of control, a review is made of the dependence of the weakly nonlinear subcritical solutions without control upon the various governing parameters.