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

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 , 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.