  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.
