Sloshing in a Hele-Shaw cell: experiments and theory

Abstract

The response of the free liquid surface in a Hele-Shaw cell subjected to a horizontal oscillation is investigated. We study the low-oscillation-amplitude regime and we show, by varying the fluid viscosity, $\unicode[STIX]{x1D708}$, and the forcing frequency, $\unicode[STIX]{x1D714}$, that the ratio between the Stokes viscous length, $\sqrt{2\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}}$, and the cell thickness greatly affects the amplitude and phase lag of the gravity waves. In particular, the sloshing system undergoes an underdamped/overdamped transition for sufficiently large viscosities. A consistent theoretical model, based on a modification of Darcy’s law to include unsteadiness, is then introduced to rationalize the experimental observations. Contrary to traditional sloshing wave theory, the viscous flow dissipation comes at leading order in the analysis, rather than as a higher-order asymptotic correction to the inviscid sloshing dynamics. The analytical expression for the resonance curves agrees well with experimental results without tunable parameters.