An efficient unconditional energy stable scheme for the simulation of droplet formation
Jinpeng Zhang, Changjuan Zhang, Xiaoping Wang
TL;DR
This work develops an efficient, unconditionally energy-stable, fully decoupled scheme for simulating droplet formation in a CH-NS framework with variable density and viscosity. By extending the DSAV approach with nonlocal auxiliary variables ($\boldsymbol{K}, \boldsymbol{Q}, \boldsymbol{R}, \boldsymbol{T}, \boldsymbol{U}$) and a modified penalty mechanism, the authors achieve linear, decoupled solves while maintaining energy dissipation under inflow boundary conditions. The scheme is proven to be unconditionally energy stable at the discrete level and is validated through accuracy tests, grid/time refinement studies, and comparisons to experimental data, including gravity effects. A parametric study shows how key dimensionless groups influence the maximum droplet radius, demonstrating the method’s practical utility for predicting droplet formation across operating regimes.
Abstract
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.