Shape of pendant droplets under a tilted surface

Abstract

For a pendant drop whose contact line is a circle of radius $r_0$, we derive the relation $mg\sin\alpha={\pi\over2}\gamma r_0\,(\cos\theta^{\rm min}-\cos\theta^{\rm max})$ at first order in the Bond number, where $\theta^{\rm min}$ and $\theta^{\rm max}$ are the contact angles at the back (uphill) and at the front (downhill), $m$ is the mass of the drop and $\gamma$ the surface tension of the liquid. The Bond (or E\"otv\"os) number is taken as $Bo=mg/(2r_0\gamma)$. The tilt angle $\alpha$ may increase from $\alpha=0$ (sessile drop) to $\alpha=\pi/2$ (drop pinned on vertical wall) to $\alpha=\pi$ (drop pendant from ceiling). The focus will be on pendant drops with $\alpha=\pi/2$ and $\alpha=3\pi/4$. The drop profile is computed exactly, in the same approximation. Results are compared with surface evolver simulations, showing good agreement up to about $Bo=1.2$, corresponding for example to hemispherical water droplets of volume up to about $50\,\mu$L. An explicit formula for each contact angle $\theta^{\rm min}$ and $\theta^{\rm max}$ is also given and compared with the almost exact surface evolver values.