On the Stability and Bifurcation of Marangoni Convection of Two Immiscible Liquids with a Nondeformable Interface\ast

This article investigates the nonlinear stability and dynamic bifurcation for the equations governing the Marangoni convection of two superimposed immiscible liquids subject to temperature gradient perpendicular to the plate. First, we obtain the critical value of the Marangoni number and verify the stability exchange principle by adopting a hybrid method that combines theoretical analysis and numerical calculations. Second, we employ the energy method, probing the nonlinear stability and establishing the nonlinear thresholds of the Marangoni number. Third, we apply the technique of center manifold reduction to reduce the corresponding infinite-dimensional model to finite-dimensional ODEs. According to the ODEs, we establish a dynamic bifurcation theorem with the transition number that determines the bifurcation type of the model. Finally, we determine the nondimensional transition number and present related temporal and flow patterns by performing careful numerical computation.