Fully-decoupled, energy stable second-order time-accurate and finite element numerical scheme of the binary immiscible Nematic-Newtonian model

Abstract

We consider the numerical approximation of a two-phase complex fluid flow model that couples the immiscible liquid crystal fluid component immersed in the ambient free flow field, where the free interface motion between the two fluids is simulated by using the phase-field approach via the energy variational method. The model is a highly nonlinear coupling system, which is composed of the Navier–Stokes equations for the flow field, the Cahn–Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. This paper proposes an effective full-discrete finite element numerical scheme to solve the model and makes the scheme not only have the characteristics of linearity, second-order time accuracy, unconditional energy stability, decoupling structure but also only need to solve a few constant-coefficient elliptic equations at each time step. The unconditional energy stability of the scheme and the detailed implementation process are further given. Various numerical experiments including the well-known “beads-on-string” phenomena are also simulated to illustrate the effectiveness of the scheme.