The micropolar Navier-Stokes equations: A priori error analysis

TL;DR

This paper develops and analyzes stable, convergent numerical schemes for the unsteady Micropolar Navier-Stokes Equations, which model fluids with rotational effects, and explores their qualitative properties.

Contribution

It introduces a first order semi-implicit scheme for MNSE that is unconditionally stable and nearly optimal, along with a second order scheme with similar stability properties.

Findings

The first order scheme is unconditionally stable and optimally convergent.

The second order scheme is almost unconditionally stable.

Qualitative properties related to ferrofluid manipulation are explored.

Abstract

The unsteady Micropolar Navier-Stokes Equations (MNSE) are a system of parabolic partial differential equations coupling linear velocity and pressure with angular velocity: material particles have both translational and rotational degrees of freedom. We propose and analyze a first order semi-implicit fully-discrete scheme for the MNSE, which decouples the computation of the linear and angular velocities, is unconditionally stable and delivers optimal convergence rates under assumptions analogous to those used for the Navier-Stokes equations. With the help of our scheme we explore some qualitative properties of the MNSE related to ferrofluid manipulation and pumping. Finally, we propose a second order scheme and show that it is almost unconditionally stable.