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Nonlinear stability of two-layer shallow water flows with a free surface
Francisco de Melo Viríssimo1, Paul A Milewski1
1Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|May 14, 2020
Summary
This study analyzes fluid dynamics in a two-layer system, revealing stability depends on baroclinic parameters. Numerical simulations show nonlinear instability, with hyperbolic data evolving into elliptic regions.
Area of Science:
- Fluid Dynamics
- Partial Differential Equations (PDEs)
- Nonlinear Stability Analysis
Background:
- Investigates a fluid dynamics problem involving two immiscible fluid layers bounded by a passive fluid and a flat bed.
- Formulates the system using a first-order, four-dimensional system of PDEs of mixed-type.
Purpose of the Study:
- To analyze the dynamics and nonlinear stability of this fluid system, particularly the free surface case.
- To establish criteria for well-posedness and understand solution behavior.
Main Methods:
- Non-dimensionalization of governing equations using physical parameters.
- Explicitly writing and discussing six conservation laws for both non-Boussinesq and Boussinesq cases.
- Analyzing the Cauchy problem for dynamics and nonlinear stability.
Main Results:
- Proves solution stability is determined by two baroclinic parameters: shear and layer thickness difference.
- Establishes a precise criterion for the system's well-posedness.
- Numerically demonstrates nonlinear instability: hyperbolic data transitions to elliptic regions before shock formation.
Conclusions:
- The shear parameter is identified as the most critical factor for stability.
- Simple waves are proposed as a tool to bound solutions and prevent hyperbolic-to-elliptic transitions.
- A mathematical proof for nonlinear instability is provided using simple wave analysis.
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