Asymptotic behavior of 2D incompressible ideal flow around small disks

Lacave, Christophe; Filho, Milton C. Lopes; Lopes, Helena J. Nussenzveig

doi:10.3233/asy-171456

Cited by 3 publications

(4 citation statements)

References 20 publications

(61 reference statements)

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“…Therefore, if we consider an inclusion which shrinks to a point with zero initial circulation, at the limit the circulation around the point is zero (see e.g. [16,21,26]).…”

Section: 4mentioning

confidence: 99%

“…The assumption concerning the zero circulation of u ε 0 is crucial to obtain a uniform L 2 estimate. For non-zero circulations γ i , the authors in [21] have had to develop an L p theory for p < 2: when an inclusion shrinks to a point x i,j , a reminiscent term appears of the form γ i…”

Section: 1mentioning

confidence: 99%

“…The asymptotic behavior of the Euler equations around inclusions which shrink to points was the subject of several recent works: the case of one shrinking obstacle in the whole plane [16], of one shrinking obstacle in a bounded domain [26], of an infinite number of shrinking disks [21]. In all these works, the authors have also treated the case where the initial circulations are non zero, which adds some difficulties and provides interesting asymptotics.…”

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confidence: 99%

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Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

Lacave

Masmoudi

2016

Arch Rational Mech Anal

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International audienceWe study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size $\varepsilon$ separated by distances $d_\varepsilon$ and the fluid fills the exterior.If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of $\frac{d_{\varepsilon}}\varepsilon$ when $\varepsilon$ goes to zero. If $\frac{d_{\varepsilon}}\varepsilon\to \infty$, then the limit motion is not perturbed by the porous medium, namely we recover the Euler solution in the whole space. On the contrary, if $\frac{d_{\varepsilon}}\varepsilon\to 0$, then the fluid cannot penetrate the porous region, namely the limit velocity verifies the Euler equations in the exterior of an impermeable square.If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}}$ where $\gamma\in (0,\infty]$ is related to the geometry of the lateral boundaries of the obstacles. If $\frac{d_{\varepsilon}}{\varepsilon^{2+\frac1\gamma}} \to \infty$, then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions. In particular it is equal to $\varepsilon^3$ for balls

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“…Therefore, if we consider an inclusion which shrinks to a point with zero initial circulation, at the limit the circulation around the point is zero (see e.g. [16,21,26]).…”

Section: 4mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

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Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

Lacave

Masmoudi

2016

Arch Rational Mech Anal

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“…Various models for the fluids reaching from incompressible inviscid flows (see e.g. [HLW22,LLN18,LM16a,MP99]) to compressible viscous flows (see e.g. [BO23,HKS21,Mas02,Osc22]) and even non-Newtonian fluids (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit

Höfer

2023

Nonlinearity

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We study the solution u ɛ to the Navier–Stokes equations in R 3 perforated by small particles centered at ( ε Z ) 3 with no-slip boundary conditions at the particles. We study the behavior of u ɛ for small ɛ, depending on the diameter ɛ α , α > 1, of the particles and the viscosity ɛ γ , γ > 0, of the fluid. We prove quantitative convergence results for u ɛ in all regimes when the local Reynolds number at the particles is negligible. Then, the particles approximately exert a linear friction force on the fluid. The obtained effective macroscopic equations depend on the order of magnitude of the collective friction. We obtain (a) the Euler–Brinkman equations in the critical regime, (b) the Euler equations in the subcritical regime and (c) Darcy’s law in the supercritical regime.

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Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

Iftimie

Filho

Lopes

2019

Communications in Partial Differential Equations

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In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data. We formulate our results for flows outside a finite number of smooth obstacles. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulations around connected components of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Our first result is existence of a solution for the weak velocity formulation with vortex sheet initial data for flow outside a finite number of smooth obstacles. The proof is a straightforward adaptation of Delort's original existence result and requires the usual sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Next, we focus on weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove that the circulations around each boundary component cannot be smaller than the initial data circulation, so that nonnegative vorticity may be absorbed by the boundary, but not produced by the boundary. In addition, we prove that if the weak solution conserves circulation at the boundary components it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation at the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. In addition, we discuss the definition of the mechanical force which the flow exerts on material boundary components and its relation with conservation of circulation. Finally, we describe the corresponding results for a bounded domain with holes, and the adaptations required in the proofs.

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Asymptotic behavior of 2D incompressible ideal flow around small disks

Cited by 3 publications

References 20 publications

Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit

Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

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