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Three‐layer flows in the shallow water limit
Author(s) -
Viríssimo Francisco,
Milewski Paul A.
Publication year - 2019
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12266
Subject(s) - limit (mathematics) , bounded function , waves and shallow water , invariant (physics) , shallow water equations , boussinesq approximation (buoyancy) , mathematical analysis , mathematics , flow (mathematics) , type (biology) , mechanics , physics , geology , geometry , mathematical physics , thermodynamics , paleontology , convection , natural convection , rayleigh number
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 nonlocal 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 nonsymmetric cases. Reduced models consisting of systems of two equations are also proposed and compared to the full system.