Numerical simulation of the behavior of a rising bubble by an energy-stable Lagrange-Galerkin Scheme

We consider two-fluid flow problems,where each fluid is governed by the Navier-Stokes equations and the surface tension proportional to the curvature acts on the interface. The domain which each fluid occupies is unknown,and the interface moves with the velocity of the particle on it. We have developed an energy-stable Lagrange-Galerkin finite element scheme for the two-fluid flow problems. It maintains not only the advantages of Lagrange-Galerkin method of the robustness to high-Reynolds numbers and of the symmetry of the resultant matrix but also the property of energy-stability under the condition of the smoothness of the interface. Here we perform numerical simulation of the behavior of a rising bubble by the scheme.