Shape of pendent droplets under a tilted surface

Abstract

For a pendent drop whose contact line is a circle of radius r0, we derive the Furmidge-like relation mgsinα=π2γr0(cosθmin−cosθmax) at first order in the Bond number, where θmin and θmax are the contact angles at the back (uphill) and at the front (downhill), m is the mass of the drop and γ the surface tension of the liquid. The Bond (or Eötvös) number is taken as Bo=mg∕(2r0γ). The tilt angle α may increase from α=0 (sessile drop) to α=π∕2 (drop pinned on vertical wall) to α=π (drop pendent from ceiling). The focus will be on pendent drops with α=π∕2 and α=3π∕4, while α=π∕4 is also included for comparison. 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 50μL. An explicit formula for each contact angle θmin and θmax is also given and compared with the almost exact Surface Evolver values.