Frequency dispersion of small-amplitude capillary waves in viscous fluids

TL;DR

This paper provides an analytical and numerical study of small-amplitude capillary wave dispersion in viscous fluids, introducing a self-similar solution and a new damping rate definition that improve prediction accuracy.

Contribution

It introduces a rational parametrization for capillary wave dispersion in viscous fluids and proposes a new damping rate for better accuracy in the underdamped regime.

Findings

Self-similar solution for frequency dispersion across the underdamped regime

Critical damping occurs when dispersive and dissipative timescales are balanced

Hydrodynamic theory may not accurately predict dispersion with significant viscous damping

Abstract

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wavenumber of critical damping based on a harmonic oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wavenumber, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic timescales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant and a new definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.