Long-scale nonlinear evolution of parametrically excited Marangoni convection

Abstract

We consider a horizontal liquid layer with a deformable free upper surface heated from below so that on the bottom the heat flux is periodically changing and the averaged temperature of the layer equals zero. A set of nonlinear evolution equations is derived for the description of the spatiotemporal dynamics of the long-wave Marangoni instability. A bifurcation analysis near the threshold of the convection onset shows the existence of the subcritical as well as supercritical type of bifurcations. The region of supercritical bifurcation regime is found. The evolution equation for the surface deviation in the form of the well-known Cahn-Hilliard equation is obtained in the vicinity of the critical value of the Marangoni number.