An Efficient Unconditional Energy Stable Scheme for the Simulation of Droplet Formation

We have developed an efficient and unconditionally energy-stable method forsimulating droplet formation dynamics. Our approach involves a noveltime-marching scheme based on the scalar auxiliary variable technique,specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase fieldmodel with variable density and viscosity. We have successfully applied thismethod to simulate droplet formation in scenarios where a Newtonian fluid isinjected through a vertical tube into another immiscible Newtonian fluid. Totackle the challenges posed by nonhomogeneous Dirichlet boundary conditions atthe tube entrance, we have introduced additional nonlocal auxiliary variablesand associated ordinary differential equations. These additions effectivelyeliminate the influence of boundary terms. Moreover, we have incorporatedstabilization terms into the scheme to enhance its numerical effectiveness.Notably, our resulting scheme is fully decoupled, requiring the solution ofonly linear systems at each time step. We have also demonstrated the energydecaying property of the scheme, with suitable modifications. To assess theaccuracy and stability of our algorithm, we have conducted extensive numericalsimulations. Additionally, we have examined the dynamics of droplet formationand 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.