Separation for the stationary Prandtl equation

Dalibard, Anne Laure; Masmoudi, Nader

doi:10.1007/s10240-019-00110-z

Cited by 38 publications

(34 citation statements)

References 39 publications

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“…e (θ, r) + p (1) p (θ, Y ) + ε 2 p (2) e (θ, r) + p (2) p (θ, Y ) + ε 3 p (3) p (θ, Y ) + • • • with the boundary conditions u (2) e (θ, 1) + u (2) p (θ, 0) = 0, v (3) e (θ, 1) + v (3) p (θ, 0) = 0, lim…”

Section: Linearized Prandtl Equations For (Umentioning

confidence: 99%

“…e ) which satisfies |∂ θ u (3) e + v (3) e |(θ, r) ≤Cηr, |∂ θ v (3) e − u (3) e |(θ, r) ≤ Cηr, ∀(θ, r) ∈ Ω,…”

Section: 66)mentioning

confidence: 99%

“…Due to the mismatch on the boundary, Prandtl in 1904 formally introduced the boundary layer theory to correct this mismatch, but the justification of this formal boundary expansion is a challenging problem. A first step for the justification of this formal boundary expansion is to understand the approximate terms in the formal expansion, that is the steady Prandtl equations and linearized steady Prandtl equations, see [3,13,32,35,40,41]. For the validity of Prandtl boundary layer expansion, there are some important results in recent years.…”

Section: Introductionmentioning

confidence: 99%

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Prandtl-Batchelor flows on a disk

Fei¹,

Gao²,

Lin³

et al. 2021

Preprint

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For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. In this paper, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying Batchelor-Wood formula.

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Section: Linearized Prandtl Equations For (Umentioning

confidence: 99%

“…e ) which satisfies |∂ θ u (3) e + v (3) e |(θ, r) ≤Cηr, |∂ θ v (3) e − u (3) e |(θ, r) ≤ Cηr, ∀(θ, r) ∈ Ω,…”

Section: 66)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Prandtl-Batchelor flows on a disk

Fei¹,

Gao²,

Lin³

et al. 2021

Preprint

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show abstract

“…The vorticity generated at the boundary is ejected into the bulk of the flow where it rolls up and is considered as one of the factors responsible for the anomalous dissipation of energy. In the steady case the detachment of the boundary layer from the flat plate was predicted by Goldstein [26] and has been proven recently in [12]. This breakdown of the assumptions on which Prandtl equations are derived signals the limitations of the classical Prandtl boundary layer theory, and new, higher order, theories are required in order to model the inviscid-boundary layer coupling near points of separation [7,10].…”

Section: B)mentioning

confidence: 99%

Real Analytic Local Well‐Posedness for the Triple Deck

Iyer

Vicol

2020

Comm Pure Appl Math

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The Triple Deck model is a classical high order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure-displacement relation which links pressure to the tangential slip. In order to overcome this, we split the Triple Deck system into two coupled equations: a Prandtl type system on H and a Benjamin-Ono type equation on R. This splitting enables us to extract a crucial leading order cancellation at the top of the lower deck. We develop a functional framework to subsequently extend this cancellation into the interior of the lower deck, which enables us to prove the local well-posedness of the model in tangentially real analytic spaces.

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“…However, the justification of this formal boundary expansion is a challenging problem. The studies on steady Prandtl equations and linearized steady Prandtl equations can be found in [3,14,33,41,42] and the validity of Prandtl boundary layer expansion has also some important progresses, see [11,15,16,17,12,8,19,9]. Moreover, the stability in Sobolev space for some class shear flow of Prandtl type has been studied in [3,2].…”

Section: Introductionmentioning

confidence: 99%

Prandtl-Batchelor flows on an annulus

Fei¹,

Gao²,

Lin³

et al. 2021

Preprint

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For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines) in a simply connected domain, Prandtl (1905) and Batchelor (1956) found that in the limit of vanishing viscosity, the vorticity is constant in an inner region separated from the boundary layer. In this paper, we consider the generalized Prandtl-Batchelor theory on an annulus that is non-simplyconnected. First, we observe that in the vanishing viscosity if steady Navier-Stokes solutions with nested closed streamlines on an annulus converge to steady Euler flows which are rotating shear flows, then the vorticity of Euler flows must be a ln r + b and the associated velocity must be (ar + b r + cr ln r, 0) in polar coordinates. We call solutions of steady Navier-Stokes equations with the above property Prandtl-Batchelor flows. Then, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on an annulus with the wall velocity slightly different from the rigid-rotation. In particular, for the above boundary conditions, we prove that there is a continuous curve (i.e. infinitely many) of solutions to the steady Navier-Stokes equations when the viscosity is sufficiently small.

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Separation for the stationary Prandtl equation

Cited by 38 publications

References 39 publications

Prandtl-Batchelor flows on a disk

Prandtl-Batchelor flows on a disk

Real Analytic Local Well‐Posedness for the Triple Deck

Prandtl-Batchelor flows on an annulus

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