Three-layer flows in the shallow water limit

Francisco de Melo Viríssimo, Paul Milewski

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We formulate and discuss the shallow water limit dynamics of the layered flow with three layers of immiscible fluids of different densities bounded above and below by horizontal walls. We obtain a resulting system of four equations, which may be non-local in the non-Boussinesq case. We provide a systematic way to pass to the Boussinesq limit, and then study those equations, which are first order PDEs of mixed type, more carefully. We show that in a symmetric case the solutions remain on an invariant surface and using simple waves we illustrate that this is not the case for non-symmetric cases. Reduced models consisting of systems of 2 equations are also proposed and compared to the full system.
Original languageEnglish
Pages (from-to)487-512
Number of pages26
JournalStudies in Applied Mathematics
Issue number4
Early online date2 Apr 2019
Publication statusPublished - 1 May 2019


  • hyperbolic systems
  • interfacial waves
  • nonlinear waves
  • stratified flows

ASJC Scopus subject areas

  • Applied Mathematics


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