Numerical methods for phase separation of surface tension dominated immiscible fluid mixtures

Understanding the phase separation dynamics of immiscible fluid mixtures is of great interest for a variety of applications. Diffuse interface models based on minimizing a free energy functional are well established for describing the thermodynamics of such mixtures, however, they feature solutions with large gradients that introduce correspondingly large stresses when coupled with the Navier-Stokes equations. For a low Reynolds number, surface tension dominated flow, the coupled system requires integrating a convection dominated, nonlinear 4th order diffusion equation. Using spectral methods in space, we discuss strategies for accurate and efficient time integration of the stiff equations of motion and observe rich fluid dynamical structure when applied to droplet formation from two immiscible polymer solutions.