Three-dimensional simulations of Marangoni-Benard convection in small containers by the least-squares finite element method Original Research Article

This paper reports a numerical study of the Marangoni-Benard (MB) convection in a planar fluid layer. The least-squares finite element method (LSFEM) is employed to solve the three-dimensional Stokes equations and the energy equation. First, the governing equations are reduced to be first-order by introducing variables such as vorticity and heat fluxes. The resultant first-order system is then cast into a div-curl-grad formulation, and its ellipticity and boundary conditions are illustrated. This numerical approach provides an equal-order discretization for velocity, pressure, vorticity, temperature and heat conduction fluxes. Numerical results reported include the critical Marangoni numbers (Mac) for the onset of the convection in containers with various aspect ratios, and the planforms of supercritical MB flows. The numerical solutions compared favorably with the experimental results reported by Koschmieder and Prahl.