Motion of drops on inclined surfaces in the inertial regime

We present experimental results on high-Reynolds-number motion of partially non-wetting liquid drops on inclined plane surfaces using: (i) water on fluoro-alkyl silane (FAS)-coated glass; and (ii) mercury on glass. The former is a high-hysteresis ( 35∘ ) surface while the latter is a low-hysteresis one ( 6∘ ). The water drop experiments have been conducted for capillary numbers 0. 0003< Ca< 0. 0075 and for Reynolds numbers based on drop diameter 137< Re< 3142 . The ranges for mercury on glass experiments are 0. 0002< Ca< 0. 0023 and 3037< Re< 200.167em 069 . It is shown that when Re 10³ for water and Re 10 for mercury, a boundary layer flow model accounts for the observed velocities. A general expression for the dimensionless velocity of the drop, covering the whole $Re$ range, is derived, which scales with the modified Bond number ( Boₘ ). This expression shows that at low $Re$ , Ca~ Boₘ and at large $Re$ , Ca √Re ~ Boₘ . The dynamic contact angle ( θ d ) variation scales, at least to first-order, with $Ca$ ; the contact angle variation in water, corrected for the hysteresis, collapses onto the low- $Re$ data of LeGrand, Daerr & Limat ( J. Fluid Mech. , vol. 541, 2005, pp. 293–315). The receding contact angle variation of mercury has a slope very different from that in water, but the variation is practically linear with $Ca$ . We compare our dynamic contact angle data to several models available in the literature. Most models can describe the data of LeGrand et al. (2005) for high-viscosity silicon oil, but often need unexpected values of parameters to describe our water and mercury data. In particular, a purely hydrodynamic description requires unphysically small values of slip length, while the molecular-kinetic model shows asymmetry between the wetting and dewetting, which is quite strong for mercury. The model by Shikhmurzaev ( Intl J. Multiphase Flow , vol. 19, 1993, pp. 589–610) is able to group the data for the three fluids around a single curve, thereby restoring a certain symmetry, by using two adjustable parameters that have reasonable values. At larger velocities, the mercury drops undergo a change at the rear from an oval to a corner shape when viewed from above; the corner transition occurs at a finite receding contact angle. Water drops do not show such a clear transition from oval to corner shape. Instead, a direct transition from an oval shape to a rivulet appears to occur.