Well-Posedness of a Nonlinear Shallow Water Model for an Oscillating Water Column with Time-Dependent Air Pressure

We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device are reformulated as a transmission problem related to the interaction between waves and a fixed partially immersed structure. B...

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Bibliographic Details
Published in:Journal of nonlinear science 2023-12, Vol.33 (6), Article 103
Main Authors: Bocchi, Edoardo, He, Jiao, Vergara-Hermosilla, Gastón
Format: Article
Language:English
Subjects:
Analysis
Boundary conditions
Boundary value problems
Classical Mechanics
Economic Theory/Quantitative Economics/Mathematical Methods
Estimates
Fluid dynamics
Iterative methods
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Physical simulation
Pressure dependence
Shallow water equations
Theoretical
Time dependence
Water circulation
Well posed problems
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Summary:We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow water equations in the presence of this device are reformulated as a transmission problem related to the interaction between waves and a fixed partially immersed structure. By imposing the conservation of the total fluid-OWC energy in the non-damped scenario, we are able to derive a transmission condition that involves a time-dependent air pressure inside the chamber of the device, instead of a constant atmospheric pressure as in Bocchi et al. (ESAIM Proc Surv 70:68–83, 2021). We then show that the transmission problem can be reduced to a quasilinear hyperbolic initial boundary value problem with a semi-linear boundary condition determined by an ODE depending on the trace of the solution to the PDE at the boundary. Local well-posedness for general problems of this type is established via an iterative scheme by using linear estimates for the PDE and nonlinear estimates for the ODE.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-023-09964-4