Diffuse interface surface tension models in an expanding flow

We consider a diffusive interface surface tension model under compressible flow.The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity field.These are two reduced models which show important properties of the full-scale surface tension model.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 effective 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.