Nonlinear Partial Differential Equations in Fluid Dynamics: Interfaces, Microstructure, and Stability

Abstract

We derive the equations of motion of Newtonian incompressible homogeneous micropolar fluids by following the path of rational continuum mechanics. By contrast with classical fluids, micropolar fluids allows for the non-trivial behaviour of a rigid microstructure at the microscopic scale. This introduces an additional kinematic quantity, an additional conserved quantity, and an additional stresstensor responsible for the mediation of couples at the microscopic scale, namely the angular velocity and the microinertia of the microstructure and the couple stress tensor, respectively. To be more precise, we derive the equations by postulating (1) the integral balance laws for conserved physical quantities such as mass, linear, and angular momen- tum, (2) the frame-invariance of the constitutive equations for the stress and couple-stress tensor, and (3) the satisfaction of the Onsager reciprocity relations.