Hele-Shaw flows with anisotropic surface tension

AbstractIn this note, we study Hele-Shaw flows in the presence of anisotropic surface tension when the fluid domain is bounded. The flows are driven by a sink, by a multipole, or solely by anisotropic surface tension. For a sink flow, we show that if the center of mass of the initial domain is not located at a certain point which is determined by the anisotropic surface tension and intensity of the sink, then either the solution will break down before all the fluid is sucked out or the fluid domain will eventually become unbounded in diameter. For a multipole driven flow, we prove that if the anisotropic surface tension, the order, and intensity of the multipole do not satisfy a certain equality, either the flow will develop finite-time singularities or the fluid domain will become unbounded in diameter as time goes to infinity. For a flow driven purely by anisotropic surface tension, we show that the center of mass of the fluid domain moves in a constant velocity, which is determined explicitly