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.

翻译：我们提出了一种高效且无条件能量稳定的方法，用于模拟液滴形成动力学。该方法基于标量辅助变量技术，开发了一种新型时间推进格式，专门用于求解具有可变密度与粘度的Cahn-Hilliard-Navier-Stokes相场模型。我们成功将该方法应用于牛顿流体通过垂直管注入另一种不相溶液体形成液滴的场景。为应对管道入口非齐次狄利克雷边界条件带来的挑战，我们引入了额外的非局部辅助变量及其关联的常微分方程，有效消除了边界项的影响。此外，我们在格式中加入了稳定性项以增强数值效果。值得注意的是，最终格式是完全解耦的，每个时间步仅需求解线性系统。通过适当修正，我们还证明了该格式的能量耗散性质。为评估算法的精度与稳定性，我们开展了大量数值模拟。同时，我们分析了液滴形成的动力学特性，并探讨了无量纲参数对过程的影响。总体而言，本工作提出了一种改进的液滴形成动力学模拟方法，在效率、能量稳定性和精度方面均有所提升。