Simulations of 2D and 3D thermocapillary flows by a least-squares finite element method

Numerical results for time-dependent 2D and 3D thermocapillary flows are presented in this work. The numerical algorithm is based on the Crank-Nicolson scheme for time integration, Newton's method for linearization, and a least-squares finite element method, together with a matrix-free Jacobi conjugate gradient technique. The main objective in this work is to demonstrate how the least-squares finite element method, together with an iterative procedure, deals with the capillary-traction boundary conditions at the free surface, which involves the coupling of velocity and teperature gradients. Mesh refinement studies were also carried out to validate the numerical results.