The Uniform Rugosity Effect

Abstract

Relying on the effect of microscopic asperities, one can mathematically justify that viscous fluids adhere completely on the boundary of an impermeable domain. The rugosity effect accounts asymptotically for the transformation of complete slip boundary conditions on a rough surface in total adherence boundary conditions, as the amplitude of the rugosities vanishes. The decreasing rate (average velocity divided by the amplitude of the rugosities) computed on close flat layers is definitely influenced by the geometry. Recent results prove that this ratio has a uniform upper bound for certain geometries, like periodical and "almost Lipschitz" boundaries. The purpose of this paper is to prove that such a result holds for arbitrary (non-periodical) crystalline boundaries and general (non-smooth) periodical boundaries.

Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary

“…The existence of solution for system (5) with (8)-(9), the existence and uniqueness of solution for system (6) with (8)-(9), and a priori estimates are given by the following result.…”

“…In the present section we describe the asymptotic behavior of the solutions (u ε , π ε ) of the non-Newtonian Navier-Stokes system (5) (9/5 ≤ p < +∞) and the unique solution of the non-Newtonian Stokes system (6) (1 < p < 9/5) posed in Ω ε , assuming Navier condition (8) on the rough part of the boundary Γ ε and no-slip (9) on the rest of the boundary, where Ω ε and Γ ε are defined by (2) and (4), respectively.…”

“…Along this proof, we will work with Navier-Stokes system (5) (9/5 ≤ p < +∞), giving the corresponding computations for the case of Stokes system (6) (1 < p < 9/5) when necessary.…”

“…Let u ε be a solution of (5) (9/5 ≤ p < +∞) or the unique solution of (6) (1 < p < 9/5), and  u ε given by (20) the corresponding rescaled solutions of (38) or (39). Then we have…”

“…More precisely, these results account asymptotically for the transformation of pure slip boundary conditions on a rough surface in no-slip boundary conditions, as the amplitude of the roughness vanishes, provided that the energy of the solutions is uniformly bounded and there is enough roughness of the oscillating boundaries. We refer to the pioneering paper [14] for the case of periodic and very smooth boundaries, and to [6,[10][11][12] (see also [16,19,23,28]) for having a quite complete understanding for arbitrary boundaries.…”