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 $0<\varepsilon\ll 1$, 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 $\varepsilon^{\gamma\over s}$, with $\gamma\in \mathbb{R}$ and flow index $1<s<2$, on the behavior of the fluid with thickness $\varepsilon$ by using asymptotic analysis when $\varepsilon\to 0$, depending on the values of $\gamma$. As a result, we deduce the existence of a critical value of $\gamma$ given by $\gamma_s^*=3-2s$ and so, three different limit boundary conditions are derived. The critical case $\gamma=\gamma_s^*$ corresponds to a limit condition of type power law slip. The supercritical case $\gamma>\gamma_s^*$ corresponds to a limit boundary condition of type perfect slip. The subcritical case $\gamma<\gamma_s^*$ corresponds to a limit boundary condition of type no-slip.