Efficient fully discrete and decoupled scheme with unconditional energy stability and second-order accuracy for micropolar Navier–Stokes equations

Abstract

This article focuses on the numerical approximation of the micropolar Navier–Stokes (MNS) system for micropolar fluids, which consists of the Navier–Stokes equations and the angular momentum equations. A significant challenge in developing efficient numerical algorithms for this model is the complex coupling structure, involving both linear and nonlinear couplings. In particular, the linear coupling between flow velocity and angular velocity requires innovative methods for effective decoupling. Recognizing that the terms associated with this linear coupling constitute a diffusion term in the form of a complete square, we introduce a new nonlocal auxiliary variable and construct an ordinary differential equation with an ingenious structure. Reformulating the MNS system into an equivalent form allows us to decouple the linear coupling through explicit discretization. This novel method integrates the zero-energy-contribution decoupling method for handling nonlinear couplings, the second-order projection method for hydrodynamics, and the spatial finite element method, resulting in a fully discrete scheme that is unconditionally energy stable, fully decoupled, linear, and second-order accurate in time. Moreover, the proposed scheme is highly efficient, as only a few independent linear elliptic problems with constant coefficients need to be solved at each time step. The unconditional energy stability and well-posedness of the scheme are also established. Numerical simulations, including 2D/3D driven cavity flows and stirring of a passive scalar, are implemented to verify the stability and accuracy of the scheme, with the numerical results exhibiting interesting phenomena in micropolar fluids.