Phase-Locking Parametric Instability Coupling Longitudinal and Transverse Waves on Rivulets in a Hele-Shaw Cell

We report an instability exhibited by a fluid system when coupling two distinct types of waves, both linearly damped. While none of them is unstable on its own, they amplify one another, resulting in a previously unreported convective instability. An external excitation is used to induce a parametric cross-coupling between longitudinal and transverse deformations of a liquid bridge between two vertical glass plates. Coherent amplification results in waves satisfying a synchronization condition, which selects a precise wavelength. We derive a model for this instability using depth-averaged Navier-Stokes equations, showing the physical origin of the coamplification, and confirm its relevance experimentally. Our findings open new perspectives in the study of parametrically controlled pattern formation, and invites the search for analogous parametric cross-coupling instabilities in other systems exhibiting distinct wave types, from plasma to elastic media.