A reduction-consistent phase field model for non-isothermal multiphase flows of N immiscible incompressible fluids

Abstract

In this study, we developed a reduction-consistent phase field model for non-isothermal incompressible N-phase flows. The model is based on the high-order spectral element method. To account for thermocapillary effects, we reformulated the continuum surface force model specifically for N-phase flows. In order to enhance computational efficiency, time-independent coefficient matrices for all variables involved were reconstructed, and an unsymmetrized multifrontal LU factorization was employed to solve the linear algebraic equations, which is derived by discretization. To verify the model effectiveness in calculating surface tension and describing three-phase interfacial dynamics, we conducted several experiments, including the stationary two-droplet example, the three-phase droplet spreading, and the equilibrium morphology of double emulsion droplet. Through these experiments, both the reduction-consistency and robustness of our model were demonstrated. Moreover, we validated the proposed model's applicability to non-isothermal three-phase flows by investigating the thermocapillary migration of single droplet and two droplets of different phases. Notably, we explored the thermocapillary migration of three-phase double droplets, focusing particularly on how the encapsulation process affects overall thermocapillary motion. Our findings indicate that the interaction between the two vortices near the interfaces of the two droplets strongly influences their migration behavior.