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Two-fluid Marangoni--Bénard convection with a deformable interface

Two immiscible fluid layers that are subjected to a temperature gradient perpendicular to their interface exhibit a range of behaviors that is considerably richer than for the single-fluid case. We describe a numerical technique for calculating thermally driven flows in two fluid layers which uses a simple technique based on a Landau transformation to map the physical domain into a reference domain, enabling the unknown location of the deformable interface to be determined. The coupled system of nonlinear partial differential equations, comprising mapping, continuity, momentum, and energy equations and the appropriate boundary conditions, is solved using the finite-element method in two-dimensional domains. Numerical bifurcation techniques are used to investigate the multiplicity of the solution set. The case of heating from above is considered in some detail and the results of finite-element computations are compared with linear stability calculations performed on unbounded domains. The principal advantages of the finite-element approach are the ability to determine the effect of non-90° contact angles (when the conducting solution no longer exists and traditional linear stability approaches fail), the ability to determine the role of finite aspect ratio domains and the relative volume fractions of the two fluids, and the capability of calculating the nonlinear development of flows beyond the critical temperature gradient.