Galerkin Method for the Solvability of a Micropolar Fluid Flow Model with Novel Frictional Boundary Conditions

Abstract

We investigate a mathematical model describing the flow of an incompressible micropolar fluid within a bounded domain of $\mathbb{R}^3$. The fluid's behavior is governed by a non-symmetric constitutive law, coupled with a couple stress tensor. Frictional boundary conditions are imposed through homogeneous Neumann conditions for the angular velocity field, along with a friction coefficient $h \in L^\infty(\partial\mathcal{O})$, which depends on the tangential component of the velocity field. To address the problem, we derive a variational formulation leading to a coupled system consisting of a variational equation with nonlinear terms governing the velocity field and a linear one describing the microrotational velocity. By applying the Galerkin method, the Cauchy-Lipschitz theorem, and compactness results, we obtain an approximate weak solution to this system.