Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities

Abstract

We develop spatial-temporally second-order, energy stable numerical schemes for two classes of hydrodynamic phase field models of binary viscous fluid mixtures of different densities. One is quasi-incompressible while the other is incompressible. We introduce a novel energy qudratization technique to arrive at fully discrete linear schemes, where in each time step only a linear system needs to be solved. These schemes are then proved to be unconditionally energy stable rigorously subject to periodic boundary conditions so that a large time step is plausible. Both spatial and temporal mesh refinements are conducted to illustrate the second order accuracy of the schemes. The linearization technique developed in this paper is so general that it can be applied to any thermodynamically consistent hydrodynamic theories so long as their energies are bounded below. Numerical examples on coarsening dynamics of two immiscible fluids and a heavy fluid drop settling in a lighter fluid matrix are presented to show the effectiveness of our proposed linear schemes. Predictions by the two fluid mixture models are compared and discussed. In the end, we believe the quasi-incompressible model is more reliable than the incompressible one.