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Some explicit solutions of the three-dimensional Euler equations with a free surface.
1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Mathematische Annalen
|October 24, 2022
Summary
We found new radial solutions for the 3D Euler equations in finite-depth fluids with free surfaces. These solutions show complex vertical structure and depth-dependent density, analyzed for stability using a Wentzel-Kramers-Brillouin ansatz.
Area of Science:
- Fluid dynamics
- Mathematical physics
Background:
- The three-dimensional Euler equations govern fluid motion.
- Understanding free surface flows in finite-depth domains is crucial.
- Radial solutions offer simplified yet insightful models.
Purpose of the Study:
- To present novel radial solutions to the 3D Euler equations.
- To analyze fluid flows with free surfaces and finite depth.
- To investigate solutions exhibiting vertical structure and depth-dependent density.
Main Methods:
- Utilizing Eulerian coordinates for solution representation.
- Employing functional analytic methods to solve for the free surface.
- Performing stability analysis via a Wentzel-Kramers-Brillouin (WKB) ansatz.
Main Results:
- A family of radial solutions with vertical structure and non-constant vorticity was derived.
- Density was found to be dependent on fluid depth.
- Explicit formulas for velocity and pressure were obtained, with the free surface implicitly defined by a functional equation.
Conclusions:
- The derived solutions provide new insights into complex fluid dynamics.
- The stability analysis offers a pathway to understanding the behavior of these flows.
- Simplification of the free surface equation is possible for specific density functions.
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