Asymptotic Behavior of the Navier--Stokes System in a Thin Domain with Navier Condition on a Slightly Rough Boundary
Juan Casado‐Díaz, Manuel Luna‐Laynez, Francisco J. Suárez‐Grau · SIAM Journal on Mathematical Analysis · 2013
We study the asymptotic behavior of the solutions of the Navier--Stokes system in a thin domain $\Omega_\varepsilon$ of thickness $\varepsilon$ satisfying the Navier boundary condition on a periodic rough set $\Gamma_\varepsilon\subset \partial\Omega_\varepsilon$ of period $r_\varepsilon$ and amplitude $\delta_\varepsilon$, with $\delta_\varepsilon \ll r_\varepsilon \ll \varepsilon$. We prove that the limit behavior as $\varepsilon$ goes to zero depends on the limit $\lambda$ of $\delta_\varepsilon \varepsilon^{1\over 2}/r_\varepsilon^{3\over 2}$. Namely, if $\lambda=+\infty$, the roughness is so strong that the fluid behaves as if we had imposed the adherence condition on $\Gamma_\varepsilon$. If $\lambda=0$, the roughness is too weak and the fluid behaves as if $\Gamma_\varepsilon$ were a plane. Finally, if $\lambda\in(0,+\infty)$, the roughness is strong enough to make a new friction term appear in the limit.