On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow

Jäger, Willi; Mikelić, Andro

doi:10.1006/jdeq.2000.3814

Cited by 274 publications

(268 citation statements)

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“…There are numerous publications devoted to the study of various boundary-value problems in domains with locally periodic boundaries (see, e.g., [11, §7.5] and [12]- [15] for scalar equations, [16]- [18] for the Stokes system, and [19]- [21] for the elasticity theory system); however, the results pertain largely to the case of γ = 1, where the period, the depth of the hollows, and the height of the ridges are of one and the same order. The papers that address the case of γ = 0 (we do not consider this case in the present paper; cf.…”

Section: Plan Of the Paper And A Brief Review Of Publicationsmentioning

confidence: 99%

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary

Kozlov

Назаров

2011

St. Petersburg Math. J.

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Abstract. Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity γ that describes the depth O(ε γ ) of irregularity (ε is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula. §1. Introduction Setting of problems.Let a domain Ω on the plane R 2 be bounded by a simple, closed, and smooth (of class C ∞ ) contour Γ = ∂Ω. By scaling, we make the length of Γ equal to 1. In a neighborhood V of Γ we introduce a system of natural curvilinear coordinates (n, s), where n is the distance to Γ, taken with the minus sign inside of Ω, and s is the arclength on Γ. The rapidly oscillating boundary Γ ε of the perturbed domain Ω ε (see Figure 1) is defined by the formulawhere ε = 1/N is a small parameter, N ∈ N is a large integer, γ is a quantity measuring the "irregularity" of the boundary (the greater is γ, the smaller is the perturbation irregularity), and H is a profile function that is smooth relative to both variables, the "slow" variable s and the "fast" variable η = ε −1 s, and 1-periodic relative to η. Note that, somewhat loosely, in our notation we do not distinguish between a point s ∈ Γ and its coordinate. For the role of V it is convenient to take the -neighborhood V with an appropriate > 0. Concerning the nonregular perturbation of the boundary (see Figure 2), we assume thatIn all other respects, the perturbation of Ω is arbitrary. To avoid numerous duplication of formulas, we keep the notation Ω and Ω ε also in §4. It should be emphasized that The paper was written during S. A. Nazarov's visit to the University of Linköping, whose financial support is acknowledged gratefully. Also, V. A. Kozlov thanks the Swedish Research Council (VR), and S. A. Nazarov thanks RFBR (project no. 09-01-00759).

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Section: Plan Of the Paper And A Brief Review Of Publicationsmentioning

confidence: 99%

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary

Kozlov

Назаров

2011

St. Petersburg Math. J.

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“…These coercive Ventcel boundary conditions also appear in the context of wall laws for rough boundaries (see [1,24,2]). Due to the physical range of the parameters, problem (1.4) is not coercive.…”

Section: Introductionmentioning

confidence: 85%

Artificial conditions for the linear elasticity equations

Bonnaillie‐Noël

Dambrine

Hérau³

et al. 2014

Math. Comp.

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In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the wellposedness except for a countable set of parameters. A perturbation argument allows to consider near-circular domains. We complete the analysis by some numerical simulations.

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“…This situation is more likely the rougher the boundary is. In fact, it can be shown rigorously in certain situations that homogenization of the no-slip boundary condition on a highly oscillating boundary will give rise in the limit to slip boundary conditions (see [73,18,119] and references therein). With Navier friction boundary conditions, the vanishing viscosity limit holds in two and three space dimensions under additional regularity conditions on the initial data, even if vorticity is produced at the boundary and the boundary is characteristic.…”

Section: Case Of Slip-type Boundary Conditionmentioning

confidence: 99%

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

Maekawa¹,

Mazzucato²

2016

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.

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On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow

Cited by 274 publications

References 8 publications

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary

Artificial conditions for the linear elasticity equations

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

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