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Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. (English) Zbl 1217.76018
Summary: Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli’s equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.
Reviewer: Reviewer (Berlin)

MSC:
76B07 Free-surface potential flows for incompressible inviscid fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
35Q55 NLS equations (nonlinear Schrödinger equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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