Capillary instability on a hydrophilic stripe

TL;DR

This paper analyzes the capillary instability of water segments on hydrophilic stripes, revealing a long-wavelength instability when the apparent contact angle exceeds ninety degrees, with implications for understanding fluid behavior on patterned surfaces.

Contribution

It provides a linear stability analysis of the capillary flow on hydrophilic stripes, identifying a zero-wavenumber instability as the cause of bulge formation.

Findings

Cylindrical segments are unstable if contact angle > 90°

Most unstable wavenumber approaches zero near 90° contact angle

Bulge formation corresponds to zero-wavenumber capillary instability

Abstract

A recent experiment showed that cylindrical segments of water filling a hydrophilic stripe on an otherwise hydrophobic surface display a capillary instability when their volume is increased beyond the critical volume at which their apparent contact angle on the surface reaches ninety degrees (Gau et al., Science, 283, 1999). Surprisingly, the fluid segments did not break up into droplets -- as would be expected for a classical Rayleigh-Plateau instability -- but instead displayed a long-wavelength instability where all excess fluid gathered in a single bulge along each stripe. We consider here the dynamics of the flow instability associated with this setup. We perform a linear stability analysis of the capillary flow problem in the inviscid limit. We first confirm previous work showing that that all cylindrical segments are linearly unstable if (and only if) their apparent contact angle is larger than ninety degrees. We then demonstrate that the most unstable wavenumber for the surface perturbation decreases to zero as the apparent contact angle of the fluid on the surface approaches ninety degrees, allowing us to re-interpret the creation of bulges in the experiment as a zero-wavenumber capillary instability. A variation of the stability calculation is also considered for the case of a hydrophilic stripe located on a wedge-like geometry.