Parametric instability of the interface between two fluids

TLDR

A vertically vibrating vessel can parametrically excite the flat interface between two fluids, producing standing waves, but viscosity prevents reduction to a simple Mathieu equation with linear damping. The study proposes a method to measure interfacial tension and the combined density–viscosity of two fluid phases near the liquid‑vapour critical point. The authors derive the governing equations for the viscous interface under sinusoidal forcing, perform a Floquet analysis, and use the results to develop a measurement method for interfacial tension and combined density–viscosity near the critical point. The analysis reveals that instability occurs only for specific wavelength–amplitude combinations, producing tongue‑shaped stability zones, that viscosity qualitatively shifts the onset wavelength—decreasing it at low viscosities and increasing it at higher viscosities—and that the predicted stability thresholds agree well with experiments.

Abstract

The flat interface between two fluids in a vertically vibrating vessel may be parametrically excited, leading to the generation of standing waves. The equations constituting the stability problem for the interface of two viscous fluids subjected to sinusoidal forcing are derived and a Floquet analysis is presented. The hydrodynamic system in the presence of viscosity cannot be reduced to a system of Mathieu equations with linear damping. For a given driving frequency, the instability occurs only for certain combinations of the wavelength and driving amplitude, leading to tongue-like stability zones. The viscosity has a qualitative effect on the wavelength at onset: at small viscosities, the wavelength decreases with increasing viscosity, while it increases for higher viscosities. The stability threshold is in good agreement with experimental results. Based on the analysis, a method for the measurement of the interfacial tension, and the sum of densities and dynamic viscosities of two phases of a fluid near the liquid-vapour critical point is proposed.