On a non-homogeneous bi-layer shallow-water problem: smoothness and uniqueness results

Muñoz-Ruiz, Marı́a Luz

doi:10.1016/j.na.2004.01.009

Cited by 4 publications

(4 citation statements)

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“…Concerning the well-possessedness of the proposed system, if we restrict just to the hyperbolic problem (20), without non-hydrostatic pressure, some studies can be found although there is no final answer. For classical two-layer shallow water systems some results can be found in [54][55][56][57]. In the case of multi-layer shallow water systems, we may cite the work [7], where the authors prove that the multilayer model is hyperbolic for the case of two layers, but the question remains open for the general case.…”

Section: Remarkmentioning

confidence: 99%

An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves

et al. 2018

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In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH ≈ 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method.

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Section: Remarkmentioning

confidence: 99%

An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves

et al. 2018

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“…There are various examples of such viscous layers over fluid substrates as for instance oil slicks over water [19] float glasses [35], and atlantic waters over the deeper denser Mediterranean sea in the Strait of Gibraltar [31].…”

Section: A Thin Viscous Sheet Modelmentioning

confidence: 99%

“…On the other hand, in the Strait of Gibtraltar, the denser Mediterranean sea flows below Atlandic waters penetrating in the Alboran sea. These phenomena may be modeled by using bi-layer Saint-Venant shallow water equations [31]. More recently, multi-layer Saint-Venant equations have also been investigated [26,31,1,2].…”

Section: A Thin Viscous Sheet Modelmentioning

confidence: 99%

Mathematical Analysis of a Saint-Venant Model With Variable Temperature

Giovangigli

Tran

2010

Math. Models Methods Appl. Sci.

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We investigate the derivation and the mathematical properties of a Saint-Venant model with an energy equation and with temperature-dependent transport coefficients. These equations model shallow water flows as well as thin viscous sheets over fluid substrates like oil slicks, atlantic waters in the Strait of Gilbraltar or float glasses. We exhibit an entropy function for the system of partial differential equations and by using the corresponding entropic variable, we derive a symmetric conservative formulation of the system. The symmetrized Saint-Venant quasilinear system of partial differential equations is then shown to satisfy the nullspace invariance property and is recast into a normal form. Upon establishing the local dissipative structure of the linearized normal form, global existence results and asymptotic stability of equilibrium states are obtained. We finally derive the Saint-Venant equations with an energy equation taking into account the temperature-dependence of transport coefficients from an asymptotic limit of a three-dimensional model.

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“…This analysis uses the method developed by Orenga in [24] and the system is energetically consistent only for small enough initial data. Others works concerning the existence of global weak solution of a bi-layer Shallow-Water using the preceding method can also find in [11] and [23]. In this work we consider in a periodic domain Ω, a system composed by two layers of immiscible fluids with different and constant densities (ρ 1 and ρ 2 , resp.)…”

Section: Introductionmentioning

confidence: 99%