Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions

María Anguiano; Francisco J. Suárez-Grau

Paper

Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions

Abstract

This theoretical study deals with the Navier-Stokes equations posed in a 3D thin domain with thickness

, assuming power law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced in (Djoko {\it et al.} {\it Comput. Math. Appl.} 128 (2022) 198--213), represents a generalization of the Navier slip boundary condition. The goal is to study the influence of the power law slip boundary conditions with an anisotropic tensor of order

, with

and flow index

, on the behavior of the fluid with thickness

by using asymptotic analysis when

, depending on the values of

. As a result, we deduce the existence of a critical value of

given by

and so, three different limit boundary conditions are derived. The critical case

corresponds to a limit condition of type power law slip. The supercritical case

corresponds to a limit boundary condition of type perfect slip. The subcritical case

corresponds to a limit boundary condition of type no-slip.