Efficient, second-order in time, and energy stable scheme for a new hydrodynamically coupled three components volume-conserved Allen–Cahn phase-field model

Abstract

In this paper, we establish a new hydrodynamically coupled phase-field model for three immiscible fluid components system. The model consists of the Navier–Stokes equations and three coupled nonlinear Allen–Cahn type equations, to which we add nonlocal type Lagrange multipliers to conserve the volume of each phase accurately. To solve the model, a linear and energy stable time-marching method is constructed by combining the stabilized-Invariant Energy Quadratization (S-IEQ) approach and the projection method. The well-posedness of the scheme and its unconditional energy stability are rigorously proved. Several numerical simulations in 2D and 3D are carried out, including spinodal decomposition, dynamical deformations of a liquid lens and rising liquid drops, to validate the model and demonstrate the efficiency and energy stability of the proposed scheme, numerically.