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 wave number 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 wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales 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 an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented.
Highlights
Waves at fluid interfaces are ubiquitous in two-phase flows across a wide range of scales, from the tidal wave with a wavelength of λ ∼ 107 m and tsunamis (λ > 105 m) down to wavelengths of the order of the size of individual m√olecules
To characterize the frequency dispersion of capillary waves in viscous fluids, a rational parametrization based on a harmonic-oscillator model has been proposed, from which a formulation for the critical wave number has been derived
This critical wave number has been shown to be a characteristic value of the frequency dispersion of capillary waves, as demonstrated by the consistent scaling of the undamped frequency ω0 as well as the damping rate computed with analytical initial-value solution (AIVS) and direct numerical simulation (DNS) for representative two-phase systems
Summary
Waves at fluid interfaces are ubiquitous in two-phase flows across a wide range of scales, from the tidal wave with a wavelength of λ ∼ 107 m and tsunamis (λ > 105 m) down to wavelengths of the order of the size of individual m√olecules. Longuet-Higgins [3] elegantly summarized the governing mechanisms for capillary waves: “At small scales, the role(s) of surface tension and viscosity are all-important.”. Capillary waves are observed at the front of short gravity waves [9,10,11,12], for instance in the ocean, where they enhance the heat and mass transfer between water and atmosphere [13,14]. Capillary waves have been identified as the key mechanism governing the formation of bound states of solitary waves in falling liquid films [15], and they have been observed to enhance film thinning between two approaching interfaces, for instance in foams
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