Rayleigh–Taylor instability for nonhomogeneous incompressible fluids with Navier‐slip boundary conditions

Ding, Shijin; Ji, Zhijun; Li, Quanrong

doi:10.1002/mma.6376

Cited by 6 publications

(5 citation statements)

References 27 publications

(35 reference statements)

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“…In [2], the authors Ding, Zi and Li construct an approximate solution generated by the maximal growing mode, pσ a , u a , q a qpt, xq " δe λ1pkqt U 1 p xq with k being fixed such that 2Λ 3 ă λ 1 pkq ă Λ. Applying Proposition 5.1, the perturbed problem (2.1)-(2.2) with the initial data pσ δ , u δ , q δ qp0q " pσ d , u d , q d qp0q.…”

Section: It Yieldsmentioning

confidence: 99%

“…By using the inequality We are not clear about the way in [2] to remove all integral terms over 2πLT in the r.h.s of (D.9) to get (D.2) for all µ ą 0, especially the following term…”

Section: It Yieldsmentioning

confidence: 99%

“…The first important things are the local existence of strong solutions to the nonlinear perturbed equation and a prior nonlinear energy estimates to those solutions. We restate Proposition 4.1 of[2].…”

mentioning

confidence: 95%

“…2 dx 2 ´βpk, λ, µqpλB k,λ,µ pφ, φq ´1q. (5.16) Thanks to the Lagrange multiplier theorem, it suffices to find a stationary point pβ, φq of L B , that satisfies λB k,λ,µ p φ, φq " 1(5.17…”

mentioning

confidence: 99%

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Linear and nonlinear analysis of the Rayleigh-Taylor system with Navier-slip boundary conditions

Nguyen¹

2022

Preprint

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In this paper, we are interested in the linear and the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a laminar smooth density profile ρ 0 px 2 q being increasing in an infinite slab 2πLT ˆp´1, 1q (L ą 0, T is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ordinary differential equation of the vertical component of perturbed velocity on the finite interval p´1, 1q,with the boundary conditionswhere g ą 0 is the gravity constant, λ is the growth rate in time, k is the wave number transverse to the density profile and two Navier-slip coefficients ξ ˘are nonnegative constants. For each k P L ´1Zzt0u, we define a threshold of viscosity coefficient µcpk, Ξq for linear instability. So that, in the ksupercritical regime, i.e. µ ą µcpk, Ξq, we provide a spectral analysis adapting the operator method of Lafitte-Nguyễn in [12] and then prove that there are infinite solutions of (0.1)-(0.2). Based on infinitely unstable modes of the linearized problem, we consider a wide class of initial data to the nonlinear perturbation problem, extending Grenier's framework [6], to prove nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely µ ą 3 sup kPL ´1Zzt0u µcpk, Ξq. ContentsData availability statement 2

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Section: It Yieldsmentioning

confidence: 99%

“…By using the inequality We are not clear about the way in [2] to remove all integral terms over 2πLT in the r.h.s of (D.9) to get (D.2) for all µ ą 0, especially the following term…”

Section: It Yieldsmentioning

confidence: 99%

mentioning

confidence: 95%

mentioning

confidence: 99%

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Linear and nonlinear analysis of the Rayleigh-Taylor system with Navier-slip boundary conditions

Nguyen¹

2022

Preprint

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“…where n = ( n 1 , n 2 ) T denotes the outward normal unit vector on ∂Ω, Dv = (∇v + ∇v T )/2 the strain tensor, and the subscript "tan" the tangential component of a vector (for example [7,8,34,38]. Here and in what follows, we always use the superscript 0 to emphasize the initial data.…”

Section: Introductionmentioning

confidence: 99%

On Inhibition of Rayleigh--Taylor Instability by a Horizontal Magnetic Field in Non-resistive MHD Fluids: the Viscous Case

Jiang¹,

Jiang²,

Zhao³

2022

Preprint

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It is still open whether the phenomenon of inhibition of Rayleigh-Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive viscous magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly proved in the linearized case by Wang in [41]. In this paper, we prove such inhibition phenomenon by the (nonlinear) inhomogeneous, incompressible, viscous case with Navier (slip) boundary condition. More precisely, we show that there is a critical number of field strength m C , such that if the strength |m| of a horizontal magnetic field is bigger than m C , then the small perturbation solution around the magnetic RT equilibrium state is algebraically stable in time. In addition, we also provide a nonlinear instability result for the case |m| ∈ [0, m C ). The instability result presents that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.

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Linear and nonlinear analysis of the viscous Rayleigh–Taylor system with Navier-slip boundary conditions

Nguyễn

2024

Calc. Var.

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Rayleigh–Taylor instability for nonhomogeneous incompressible fluids with Navier‐slip boundary conditions

Cited by 6 publications

References 27 publications

Linear and nonlinear analysis of the Rayleigh-Taylor system with Navier-slip boundary conditions

Linear and nonlinear analysis of the Rayleigh-Taylor system with Navier-slip boundary conditions

On Inhibition of Rayleigh--Taylor Instability by a Horizontal Magnetic Field in Non-resistive MHD Fluids: the Viscous Case

Linear and nonlinear analysis of the viscous Rayleigh–Taylor system with Navier-slip boundary conditions

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