MARANGONI-BENARD 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, al lowing the free surface to deform can result in a solution set with a qualitatively different bifurcation structure. We describe a finite-el ement 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 ap proximation, and to have a surface tension that varies linearly with t emperature. An orthogonal mapping from the physical domain to a refere nce 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 discret ized using the finite-element method and solved simultaneously by Newt on iteration. Contact angles other than 90 degrees are shown to discon nect the transcritical bifurcations to flows with an even number of ce lls in the expected manner. The loss of stability to single cell flow is associated with the breaking of a reflectional symmetry about the m iddle of the domain and therefore occurs at a pitchfork bifurcation po int for contact angles both equal to, and less than, 90 degrees. (C) 1 998 Academic Press.