Kelvin-Helmholtz instability and formation of viscous solitons on highly viscous liquids

Abstract

Viscous solitons are strongly non-linear surface deformations generated by blowing wind over a liquid beyond a critical viscosity. Their shape and dynamics result from a balance between wind drag, surface tension and viscous dissipation in the liquid. We investigate here the influence of the liquid viscosity in their generation and propagation. Experiments are carried out using silicon oils, covering a wide range of kinematic viscosities $\nu_\ell$ between 20 and 5000~mm$^2$~s$^{-1}$. \modif{We show that, for $\nu_\ell> 200$~mm$^2$~s$^{-1}$, viscous solitons are sub-critically generated from an unstable initial wave train at small fetch, where the wind shear stress is larger. The properties of this initial wave train are those expected from Miles's theory of the Kelvin-Helmholtz instability of a highly viscous fluid sheared by a turbulent wind: the critical friction velocity and critical wavelength are independent of $\nu_\ell$, and the phase velocity decreases as $\nu_\ell^{-1}$.} We demonstrate the subcritical nature of the transition to viscous solitons by triggering them using a wavemaker for a wind velocity below the natural threshold. Finally, we analyze the flow field induced by a viscous soliton, and show that it is well described by a two-dimensional Stokeslet singularity in the far field. The resulting viscous drag implies a propagation velocity with a logarithmic correction in liquid depth, in good agreement with our measurements.