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 90° (Gau et al 1999 Science 283 46–9). 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 all cylindrical segments are linearly unstable if (and only if) their apparent contact angle is larger than 90°. 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 90°, 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.
Highlights
Is in the tens of nanometers, which is much smaller than the typical cross-sectional size in the experiments of Gau et al [5]. Within these assumptions, the most unstable wavelength for the capillary instability of the cylindrical segment tends toward infinity as the contact angle approaches 90◦, thereby allowing us to re-interpret the creation of bulges in the experiment as a zero-wavenumber capillary instability [5]
It was shown that a circular segment of fluid located on a hydrophilic stripe on an otherwise hydrophobic substrate becomes unstable when its volume reaches that at which its apparent contact angle on the surface is 90◦
Instead of breaking up into droplets, the instability leads to the excess fluid collecting into a single bulge along each stripe
Summary
The geometrical set-up for our linear stability calculation is illustrated in figure 3. The basic state is a cylindrical segment of fluid of radius R, whose two-dimensional contact line is pinned along a stripe. The apparent contact angle of the fluid at the contact line is denoted θc. We assume θc to be sufficiently larger (smaller) than the receding (advancing) angle on the hydrophilic (hydrophobic) substrate so that we can safely assume that during the initial stages of the instability the contact line remains pinned along the same location. Since we know from earlier work that the case where θc < 90◦ is stable [6]–[11], we will focus here on determining the dynamics of the instability in the case where θc 90◦
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
