Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

Iyer, Sameer

doi:10.1137/18m1207351

Cited by 30 publications

(27 citation statements)

References 20 publications

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“…In 2016, the authors in [16] improved the results of Sammartino & Caflisch [53,54] in Gevery class. For the steady case, Guo & Nguyen in [26] justified the Prandtl boundary layer expansions for the steady Navier-Stokes flows over a moving plate, and similar results for steady flows were obtained in [29][30][31]. Very recently, Gérard-Varet & Maekawa in [15] shows the H 1 stability of shear flows of Prandtl type and verifies the Prandtl expansions for the steady Navier-Stokes equations with no-slip boundary condition.…”

Section: Introduction and Main Resultssupporting

confidence: 59%

Justification of Prandtl Ansatz for MHD Boundary Layer

Liu¹,

Xie²,

Yang³

2019

SIAM J. Math. Anal.

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As a continuation of [43], the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on velocity field and perfect conducting boundary condition on magnetic field. Under the assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a L 8 estimate on the error by multi-scale analysis.(1. 4)2000 Mathematics Subject Classification. 76N20, 35A07, 35G31, 35M33.

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Section: Introduction and Main Resultssupporting

confidence: 59%

Justification of Prandtl Ansatz for MHD Boundary Layer

Liu¹,

Xie²,

Yang³

2019

SIAM J. Math. Anal.

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“…For the incompressible steady Navier-Stokes equations, Guo and Nguyen [20] justified the boundary layer expansion for the flow with a non-slip boundary condition on a moving plate. This result has been extended to the case of a rotating disk and to the case of nonshear Euler flows( [23,24]). Recently, Guo and Iyer [17] studied the boundary layer expansion for the small viscous flows with the classical no slip boundary conditions or on the static plate.…”

Section: Introductionmentioning

confidence: 97%

Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime

Gao¹,

Li²,

Yao³

2021

Preprint

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In this paper, we investigate the higher regularity and asymptotic behavior for the 2-D magnetic Prandtl model in the Prandtl-Hartmann regime. Due to the degeneracy of horizontal velocity near boundary, the higher regularity of solution is a tricky problem. By constructing suitable approximated system and establishing closed energy estimate for a good quantity (called "quotient" in [18]), our first result is to solve this higher regularity problem. Furthermore, we show the global well-posedness and global-in-x asymptotic behavior when the initial data are small perturbation of the classical Hartmann layer in Sobolev space. By using the energy method to establish closed estimate for the quotient, we overcome the difficulty arising from the degeneracy of horizontal velocity near boundary. Due to the damping effect, we also point out that this global solution will converge to the equilibrium state(called Hartmann layer) with exponent decay rate.

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“…The leading order of Euler flows in [10] and [14] are both shear flows, and the width of the region in [10] and the angle of sector in [14] are small. Later, Iyer in [15] justified the global steady Prandtl expansions over a moving plane under the assumption of the smallness of the mismatch, and considered the situation that Euler flow is a perturbation of shear flow in [16]. For the no-slip boundary, there are also some important works.…”

Section: Introductionmentioning

confidence: 99%

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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Steady Prandtl Layers Over a Moving Boundary: Nonshear Euler Flows

Cited by 30 publications

References 20 publications

Justification of Prandtl Ansatz for MHD Boundary Layer

Justification of Prandtl Ansatz for MHD Boundary Layer

Higher regularity and asymptotic behavior of 2D magnetic Prandtl model in the Prandtl-Hartmann regime

Prandtl-Batchelor flows on a disk

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