Analysis of the Fully Discrete Linearly Extrapolated Curvature Stabilize Method for the Thermal Micropolar Navier–Stokes Equations

This paper presents a new, efficient, accurate, and unconditionally stable second-order time-stepping method for the incompressible thermal micropolar Navier–Stokes equations (TMNSE) using mixed finite elements. The method linearizes the nonlinear convective terms in the momentum equation, microrotation equation, and temperature equation, requiring the solution of a linear problem at each time step. The discrete curvature of the solution is added as a stabilizing term for linear velocity [Formula: see text], microrotation velocity [Formula: see text], pressure [Formula: see text], and temperature [Formula: see text] in the equations, respectively. Curvature stabilization ([Formula: see text]) is a new concept in computational fluid dynamics (CFD) aimed at improving the commonly used velocity stabilization ([Formula: see text]), which only has first-order time accuracy and has adverse effects on important flow quantities such as drag coefficients. We derive a priori error estimates for the fully discrete linear extrapolation curvature stabilization method. The theoretical results and effectiveness of the new method are verified through a series of numerical experiments for [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] in 2D and 3D, respectively. In particular, this work considers the thermal cavity-driven flow experiment to validate the numerical scheme and obtains good results.