On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

Duchêne, Vincent

doi:10.1007/s00332-014-9200-2

Cited by 11 publications

(9 citation statements)

References 37 publications

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“…Theorem 1.2 is restricted to d = 2 because we use dispersive decay estimates on rapidly propagating acoustic waves in order to control nonlinear coupling effects between the fast and slow modes. In the case of dimension d = 1, and provided that the initial data is sufficiently localized in space, we justified in [16] a similar mode decomposition of the flow, by making use of the different spatial support of each mode after small time. Proposition 4.4 therein, together with Proposition 3.8 in the present work, offer a convergence between the exact and the approximate solution with rate O( ).…”

Section: Resultsmentioning

confidence: 99%

“…In [16], the author studied the so-called inviscid bilayer Saint-Venant (or shallow water) system in the limit of small density contrast. The change of variables allowing for uniform energy estimates was exhibited therein, and convergence towards a solution of the rigid-lid limit, as well as a secondorder approximation, was deduced in the case of well-prepared initial data.…”

Section: Related Earlier Resultsmentioning

confidence: 99%

“…The ability to construct symmetrizers depending strongly on the shear velocities but only weakly on a background velocity (or on the total volume flux) is also essential for us to obtain results outside of the scope of well-prepared initial data; see [16] for an analysis where this property is not used.…”

Section: Standard Well-posedness Theorymentioning

confidence: 99%

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The multilayer shallow water system in the limit of small density contrast

Duchêne

2016

ASY

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We study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow and a smallness assumption on the initial surface deformation, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that there exists a small initial layer in time for ill-prepared initial data, accounting for rapidly propagating "acoustic" waves (here, the so-called barotropic mode) which interact only weakly with the "incompressible" component (here, baroclinic).1 Because we assume that the layers are all of comparable depth and the vertical stratification is balanced, in the sense that we fix m > 0 such that sup i∈{1,...,N } {δ i , δ −1 i , r i , r −1 i } ≤ m, then c 0 measures the typical velocity of propagation of the baroclinic modes; see Appendix B.

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Section: Resultsmentioning

confidence: 99%

Section: Related Earlier Resultsmentioning

confidence: 99%

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The multilayer shallow water system in the limit of small density contrast

Duchêne

2016

ASY

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“…In the bilayer setting, the use of the rigid-lid assumption is well-grounded only when the density contrast, 1 − γ, is small. In this situation, one may use the Boussinesq approximation, that is set γ = 1; see [22] in the dispersionless setting. Notice however that system (1.2) exhibits unstable modes that are reminiscent of Kelvin-Helmholtz instabilities when the Fourier multipliers F i satisfy Definition 1.1(iv) with θ ∈ [0, 1); see [23].…”

Section: Definition 12 (Strongly Admissible Class Of Fourier Multipmentioning

confidence: 99%

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Duchêne

Nilsson

Wahlén

2017

J. Math. Fluid Mech.

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Abstract. We provide the existence and asymptotic description of solitary wave solutions to a class of modified Green-Naghdi systems, modeling the propagation of long surface or internal waves. This class was recently proposed by Duchêne et al. (Stud Appl Math 137:356-415, 2016) in order to improve the frequency dispersion of the original Green-Naghdi system while maintaining the same precision. The solitary waves are constructed from the solutions of a constrained minimization problem. The main difficulties stem from the fact that the functional at stake involves low order non-local operators, intertwining multiplications and convolutions through Fourier multipliers.

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“…In general, the rigid-lid approximation is used by many researchers (see, for example, [2,6,12]) to simplify their models. In a rigid-lid approximation, the free-surface displacements compared to the interface displacements are neglected.…”

Section: Introductionmentioning

confidence: 99%

Three-Layer Fluid Flow Over a Small Obstruction on the Bottom of A channel

2015

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Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

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On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

Cited by 11 publications

References 37 publications

The multilayer shallow water system in the limit of small density contrast

The multilayer shallow water system in the limit of small density contrast

Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

Three-Layer Fluid Flow Over a Small Obstruction on the Bottom of A channel

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