We consider a diusive interface surface tension model under compressible ow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity eld. We prove that both model problems are well-posed. We are especially interested in the behavior of solutions with respect to droplet breakup phenomena. Numerical simulations of 1,2 and 3D all illustrate that the Cahn-Hilliard model is much more eective for droplet breakup. Using asymptotic methods we correctly predict the breakup condition for the Cahn-Hilliard model. Moreover, we prove that the Allen-Cahn model will not break up under certain circumstances due to a maximum principle.
BackgroundThere is a need to develop simple computational models for surface tension in the droplet breakup phenomena. As an example, consider a piece of material that expands under a sudden pulse of energy that comes from laser fusion [21] or heavy ion fusion [12]. The material will breakup, and surface tension plays an important role in the ensuing dynamics. There are many numerical methods that deal with surface tension in twophase uids. This problem is known for its computational stiness. It contains two dierent time scales, the small surface tension time scale and the convection time scale. Three main algorithms exist for two-phase uids. The sharp interface method tracks the interface explicitly, yet it requires extensive processing when the interface splits and merges. Since droplet breakup involves mainly merging and splitting of the interface, we * W. Liu and A. Bertozzi are with the