A second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-Nematic model for two-phase complex fluids confined in the Hele-Shaw cell

We consider the numerical approximation of the binary immiscible mixture of nematic liquid crystals and viscous Newtonian fluids confined in a Hele-Shaw cell, where the free interface motion is simulated by using the phase-field approach via the energy variational method. The governing system is highly complicated nonlinear and coupled, consisting of the Darcy equations for the flow field, the Cahn-Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. The numerical scheme developed in this paper is the first "ideal" scheme, namely, it not only has the following characteristics: linearity, second-order time accuracy, unconditional energy stability, and decoupling structure, but also at each time step, only needs to solve a few elliptic equations with constant coefficients. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical experiments are further carried out to prove the effectiveness of the scheme, in which the influence of the initial orientations and anchoring elastic energy of the liquid crystal on the Saffman-Taylor fingering instability are studied. (C) 2022 Elsevier Inc. All rights reserved.