Existence and Stability of Nonmonotone Hydraulic Shocks for the Saint Venant Equations of Inclined Thin-Film Flow

Abstract Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number $$F&lt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams, are all rigorously obtained by mathematical analysis and explicit calculation.

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Publication Details

Journal
Archive for Rational Mechanics and Analysis
Published
2024-09-13
DOI
https://doi.org/10.1007/s00205-024-02033-4
Citations
5
Primary Topic
Fluid Dynamics and Thin Films
Type
article
Field-Weighted Citation Impact
1.87

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article

Existence and Stability of Nonmonotone Hydraulic Shocks for the Saint Venant Equations of Inclined Thin-Film Flow

Grégory Faye, Zhao Yang, Kevin Zumbrun, L. Miguel Rodrigues
5 citations
Archive for Rational Mechanics and Analysis
Fluid Dynamics and Thin Films
1.87
article

Existence and Stability of Nonmonotone Hydraulic Shocks for the Saint Venant Equations of Inclined Thin-Film Flow

Grégory Faye, Zhao Yang, Kevin Zumbrun, L. Miguel Rodrigues
article en
5 citations

Abstract

Abstract Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number $$F&lt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams, are all rigorously obtained by mathematical analysis and explicit calculation.

Archive for Rational Mechanics and AnalysisVol. 248(5)
Centre National de la Recherche Scientifique (FR), Université Toulouse III - Paul Sabatier (FR), Institut National des Sciences Appliquées de Toulouse (FR), Chinese Academy of Sciences (CN), Toulouse Mathematics Institute (FR), Indiana University Bloomington (US), Academy of Mathematics and Systems Science (CN), Institut de recherche mathématique de Rennes (FR), Université de Rennes (FR), Institut de Mathématiques de Toulouse (FR)
National Science Foundation, Agence Nationale de la Recherche, Engineering and Physical Sciences Research Council, Centre International de Mathématiques et Informatique de Toulouse
Openalex Percentile: Top 19%
Fluid Dynamics and Thin Films
1.87
DOIOpen Access PDFOpenAlex
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