A Second Order Fully-discrete Linear Energy Stable Scheme for a Binary Compressible Viscous Fluid Model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flow mixtures derived from the generalized Onsager Principle. The hydrodynamic model not only possesses the variational structure, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model in an equivalent form using the energy quadratization method and then discretize the reformulated model to obtain a semi-discrete partial differential equation system using the Crank-Nicolson method in time. The numerical scheme so derived preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using the 2nd order finite difference method, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and two numerical examples on phase separation due to the spinodal decomposition in two polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications

An efficient unconditional energy stable scheme for the simulation of droplet formation

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy