Waves beneath a drop levitating over a moving wall

TL;DR

This paper investigates the origin of traveling waves in the air film beneath a levitating droplet in a rotating drum, using a simplified lubrication model and stability analysis to match experimental observations.

Contribution

It introduces a linear stability analysis based on the Orr-Sommerfeld equation to explain wave formation beneath a levitating droplet, linking theory with experiments.

Findings

Traveling waves occur at higher rotation rates.

The most unstable wavelength and phase velocity are predicted theoretically.

The model's predictions agree with experimental data.

Abstract

In recent experiments, Sawaguchi et al. directly probed the lubrication layer of air beneath a droplet levitating inside a rotating cylindrical drum. For small rotation rates of the drum, the lubrication film beneath the drop adopted a steady shape, while at higher rotation rates, travelling waves propagated along the drop's lower surface with roughly half the wall velocity. We here rationalize the physical origin of these waves. We begin with a simplified model of the lubrication flow beneath the droplet, and examine the linear stability of this base state to perturbations of the Tollmien--Schlichting type. Our developments lead to the Orr-Sommerfeld equation (OSE), whose eigenvalues give the growth rates and phase speeds of the perturbations. By considering wavelengths long relative to the lubrication film thickness, we solve the OSE perturbatively and so deduce the wavelength and phase velocity of the most unstable mode. We find satisfactory agreement between experiment and theory over the parameter regime considered in the laboratory.