Consistent energy-stable method for the hydrodynamics coupled PFC model

In this study, we propose an efficient mathematical equation to describe incompressible fluid flow-coupled crystallization. By using the L²-gradient flow, the evolutional equation is derived from an energy functional and a nonlocal Lagrange multiplier is introduced to preserve the total mass. Various physical phenomena of colloidal crystals can be modeled using a generalized nonlinearity and the fluid dynamics are described by incompressible Navier–Stokes (NS) equations. To efficiently manage the nonlinear and coupled terms in this complex fluid system and maintain the discrete version of the energy stability, we develop a linear, second-order time-accurate, and energy-stable numerical method for the fluid flow-coupled phase-field crystal model. The energy stability of the discrete method is estimated. In each time step, all variables are completely decoupled, and we only need to solve several linear elliptic-type equations. We adopt a relaxation method to correct the energy. The main merits of this study are as follows: (i) a simple fluid flow-coupled crystal equation with generalized nonlinearity is derived; (ii) the proposed scheme not only satisfies the energy stability but also achieves improved consistency; and (iii) the calculation in each time step is highly efficient in implementing the proposed scheme. Numerical experiments, including flow-coupled phase transition, crystallization in shear flow, and sedimentation of crystals, are performed to validate the accuracy, stability, and superior performance of the proposed method.