On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries

Abstract

We study the asymptotic behavior of solutions to the incompressible Navier-Stokes system considered on a sequence of spatial domains, whose boundaries exhibit fast oscillations with amplitude and characteristic wave length proportional to a small parameter. Imposing the complete slip boundary conditions we show that in the asymptotic limit the fluid sticks completely to the boundary provided the oscillations are nondegenerate, meaning not oriented in a single direction.

On the Navier boundary condition for viscous fluids in rough domains

“…Remark that in our case Φ ε = δ ε Ψ( x ′ ε ) converges strongly to zero in W 1,∞ (ω) and therefore the results in [3] do not apply.…”

“…We focus in the case λ ∈ (0, +∞) which, as we said in Remark 2, can be considered as the general one. Assuming δ ε = λε 3 2 , with λ ∈ (0, +∞), we prove the following theorem Theorem 3 If the function u defined by (14) and (18) belongs to H s (Ω) 3 , with s > 3/2, then we have…”

“…Due to the freedom of choice of boundary conditions, a natural question is if there is any relationship between conditions (1) and (3). In this sense, it was considered in [10] a three-dimensional domain Ω ε with a rough boundary described by the equation (see Figure 1)…”

Continuity of Drag and Domain Stability in the Low Mach Number Limits

“…More specifically, the scaled domains 1/ε α Ω ε are the uniform C 2 -domains discussed in [4]. In addition to the previous hypotheses, we assume, following [2], that the boundaries ∂B ε "oscillate" as ε → 0, mimicking the effect of "roughness", see the following section.…”

“…• Roughness (see [2]): The limit obstacle B is Lipschitz, in particular almost any point y ∈ ∂B (in the sense of the 2-D Hausdorff measure) admits an (outer) normal vector n y . We require the boundaries ∂B ε to oscillate in the following sense:…”