A note on gravity waves in a viscous liquid with surface tension
The author makes an interesting study of gravity waves in a viscous liquid with surface tension. Asymptotic formulas for the dispersion relation are derived. Introducing the surface tension parameter, \(S=M^{-1/3}\), where M is the Morton number, \(M=g\rho^ 3\nu^ 4/T^ 3=g\mu^ 4/\rho T^ 3, \mu =\rho\nu \) is the dynamic viscosity, T is the surface tension, the author discusses the nature of the phase and group velocities for different values of S, and determines the critical value of \(M_ c\) and \(S_ c\), where \(M_ c=S_ c^{-3}\). It is shown that, when \(\nu =0\), sufficiently short waves will always have group velocity larger than the phase velocity for non zero values of T. However in a viscous fluid, the group velocity may exceed the phase velocity only if \(M<M_ c\) (or \(S>S_ c)\). The critical value of S for static stability is found to be \(S_ c=2.8159\) which leads to the critical Morton number \(M_ c=0.044789\). For each value of S in case of static instability, an eventual spectrum of travelling waves includes a region where the group velocity exceeds the phase velocity. The author finds different regions in the S, \(\hat k\) plane and examines the equation of the bounding curve: \((\pm\hat k^{-3}+S\) \(\hat k{}^{-1})^{- 1/3}=k^*_ b=1.1981\), where the plus sign corresponds to static stability and the minus sign to static instability. An interesting discussion concerning stable or unstable region is included in the paper. Finally, it is shown that only in fluids with smaller Morton numbers, travelling ripples may exist with group velocity exceeding phase velocity. This study seems to be useful for instability of water waves.
scientific article; zbMATH DE number 3874810