Steady Stokes Flow in Exterior Domains

Galdi, Giovanni P.

doi:10.1007/978-1-4757-3866-7_5

Cited by 2 publications

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References 13 publications

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“…Similarly, the pressures (σ j ) 1≤j≤d are uniquely determined by the condition σ j ∈ L 2 (R d \T ) (see e.g. Lemma 1.1, article V. of [36]).…”

Section: 3mentioning

confidence: 99%

High Order Homogenization of the Stokes System in a Periodic Porous Medium

Feppon¹

2021

SIAM J. Math. Anal.

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We derive high order homogenized models for the incompressible Stokes system in a cubic domain filled with periodic obstacles. These models have the potential to unify the three classical limit problems (namely the "unchanged" Stokes system, the Brinkman model, and the Darcy's law) corresponding to various asymptotic regimes of the ratio η ≡ aε/ε between the radius aε of the holes and the size ε of the periodic cell. What is more, a novel, rather surprising feature of our higher order effective equations is the occurrence of odd order differential operators when the obstacles are not symmetric. Our derivation relies on the method of two-scale power series expansions and on the existence of a "criminal" ansatz, which allows to reconstruct the oscillating velocity and pressure (uε, pε) as a linear combination of the derivatives of their formal average (u * ε , p * ε ) weighted by suitable corrector tensors. The formal average (u * ε , p * ε ) is itself the solution to a formal, infinite order homogenized equation, whose truncation at any finite order is in general ill-posed. Inspired by the variational truncation method of [53,27], we derive, for any K ∈ N, a well-posed model of order 2K + 2 which yields approximations of the original solutions with an error of order O(ε K+3 ) in the L 2 norm. Furthermore, the error improves up to the order O(ε 2K+4 ) if a slight modification of this model remains well-posed. Finally, we find asymptotics of all homogenized tensors in the low volume fraction limit η → 0 and in dimension d ≥ 3. This allows us to obtain that our effective equations converge coefficient-wise to either of the Brinkman or Darcy regimes which arise when η is respectively equivalent, or greater than the critical scaling η crit ∼ ε 2/(d−2) .

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“…Similarly, the pressures (σ j ) 1≤j≤d are uniquely determined by the condition σ j ∈ L 2 (R d \T ) (see e.g. Lemma 1.1, article V. of [36]).…”

Section: 3mentioning

confidence: 99%

High Order Homogenization of the Stokes System in a Periodic Porous Medium

Feppon¹

2021

SIAM J. Math. Anal.

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“…The existence and uniqueness of a solution to (4.9) is standard by using layer potential theory [20,19] or variational arguments in homogeneous Sobolev spaces [14,25] (also called Deny-Lions or Beppo-Levi spaces). We denote by F := (F ij ) 1≤i,j≤d the matrix collecting the drag force components:…”

Section: High Order Homogenized Stokes Models Capture All Three Regim...mentioning

confidence: 99%

High Order Homogenized Stokes Models Capture all Three Regimes

Feppon¹,

Jing²

2022

SIAM J. Math. Anal.

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This article is a sequel to our previous work [13] concerned with the derivation of high-order homogenized models for the Stokes equation in a periodic porous medium. We provide an improved asymptotic analysis of the coefficients of the higher order models in the low-volume fraction regime whereby the periodic obstacles are rescaled by a factor η which converges to zero. By introducing a new family of order k corrector tensors with a controlled growth as η → 0 uniform in k ∈ N, we are able to show that both the infinite order and the finite order models converge in a coefficient-wise sense to the three classical asymptotic regimes. Namely, we retrieve the Darcy model, the Brinkman equation or the Stokes equation in the homogeneous cubic domain depending on whether η is respectively larger, proportional to, or smaller than the critical size η crit ∼ ε 2/(d−2) . For completeness, the paper first establishes the analogous results for the perforated Poisson equation, considered as a simplified scalar version of the Stokes system.

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Steady Stokes Flow in Exterior Domains

Cited by 2 publications

References 13 publications

High Order Homogenization of the Stokes System in a Periodic Porous Medium

High Order Homogenization of the Stokes System in a Periodic Porous Medium

High Order Homogenized Stokes Models Capture all Three Regimes

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