Marangoni–Bénard Convection with a Deformable Free Surface

Computations of Marangoni convection are usually performed in two- or three-dimensional domains with rigid boundaries. In two dimensions, allowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure. We describe a finite-element technique for calculating bifurcations that arise due to thermal gradients in a two-dimensional domain with a deformable free surface. The fluid is assumed to be Newtonian, to conform to the Boussinesq approximation, and to have a surface tension that varies linearly with temperature. An orthogonal mapping from the physical domain to a reference domain is employed, which is determined as the solution to a pair of elliptic partial differential equations. The mapping equations and the equilibrium equations for the velocity, pressure, and temperature fields and their appropriate nonlinear boundary conditions are discretized using the finite-element method and solved simultaneously by Newton iteration. Contact angles other than 90 degrees are shown to disconnect the transcritical bifurcations to flows with an even number of cells in the expected manner. The loss of stability to single cell flows is associated with the breaking of a reflectional symmetry about the middle of the domain and therefore occurs at a pitchfork bifurcation point for contact angles both equal to, and less than, 90 degrees.