Viscous Bending Mitigates the Spontaneous Meandering of Rivulets in Hele-Shaw Cells

TL;DR

This study reveals that viscous bending is the key factor in selecting the most unstable wavelength in rivulet meandering within Hele-Shaw cells, resolving a 15-year-old open question.

Contribution

It demonstrates that viscous bending, incorporated into the Navier-Stokes equations, explains wavelength selection and the nature of the instability, providing a new physical interpretation.

Findings

Viscous bending determines the fastest-growing mode in rivulet meandering.

The analysis clarifies whether the instability is absolute or convective.

A simplified derivation of the instability criterion is provided.

Abstract

We investigate the spontaneous meandering of slender rivulets in Hele-Shaw cells and identify the physical mechanism that selects the most unstable wavenumber, a quantity that has remained elusive even since the identification of the instability threshold [Daerr et al., Phys. Rev. Lett. 106, 184501 (2011)]. Earlier criteria did not distinguish between wavelengths and thus predicted an undiscriminated amplification of arbitrarily short perturbations. By incorporating viscous bending into the depth-averaged Navier-Stokes equations, we show that this effect is responsible for the selection of a fastest-growing mode, answering a question that has remained open for 15 years. We answer the open question of whether the meandering instability is absolute or convective. Our analysis also provides a simpler alternative derivation of the instability criterion, based on a low-viscosity assumption, and finally it yields a new physical interpretation of the mechanism: the destabilization arises directly from friction effects, instead of being caused by inertial forces. Together, these results complete the linear-stability picture of rivulet meandering in confined geometries, and establish viscous bending as a key parameter governing wavelength selection. They lay the groundwork for future exploration of the nonlinear features of the spontaneous meandering instability.