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Effect of viscosity of long gravity waves. (English) Zbl 1334.76018
Summary: The effect of viscosity is examined on long gravity waves of small but finite amplitude. The reductive perturbation method combined with the usual boundary layer theory reveals that the inviscid Korteweg-de Vries equation is not affected by the viscosity if \(O(\alpha^{-5})< R\), where \(R\) is the Reynolds number and \(\alpha(\gg 1)\) the wavenumber. For \(O(\alpha^{-1})< R \geq O(\alpha^{-5})\), the effect of viscosity modifies the Korteweg-de Vries equation and yields new types of equation. On the other hand, for \(R <O(\alpha^{-1})\), the complex phase velocity becomes purely imaginary and the free surface is found to be governed by a nonlinear diffusion equation which was first obtained by Nakaya.

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
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