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 et al. 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 ${\epsilon }^{\frac{\gamma }{s}}$, with $\gamma \in R$ and flow index , on the behavior of the fluid with thickness $\epsilon$ by using asymptotic analysis when $\epsilon \to 0$, depending on the values of $\gamma$. As a result, we deduce the existence of a critical value of $\gamma$ given by ${\gamma }_s^{\ast }=3-2s$ and so, three different limit boundary conditions are derived. The critical case $\gamma ={\gamma }_s^{\ast }$ corresponds to a limit condition of type power-law slip. The supercritical case $\gamma >{\gamma }_s^{\ast }$ corresponds to a limit boundary condition of type perfect slip. The subcritical case corresponds to a limit boundary condition of type no-slip.