Gravity-driven motion of a highly viscous non-wetting drop on an inclined wall

• Published in: Journal of fluid mechanics 2024-11, Vol.1000, Article A16
• Main Author: Zinchenko, Alexander Z.
• Format: Article
• Language: English
• Subjects: Acceleration; Contact angle; Dimpling; Formability; Glycerol; Gravity; Green's function; Green's functions; Half spaces; JFM Papers; Kinematics; Lubrication; Movement; Multipoles; Regularization; Simulation; Steady state; Thin films; Viscosity; Viscosity ratio; Wetting
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2024.998

Extensive three-dimensional boundary-integral simulations are presented for the steady-state, low-Reynolds-number motion of a non-wetting deformable drop in another liquid on an inclined solid wall. The drop remains separated from the wall by a lubricating film. The boundary-integral formulation is based on the half-space Green function. The focus is on the challenging case of small tilt angles $\theta$ combined with high drop-to-medium viscosity ratios $\lambda$, when the drop travels with strong hydrodynamical interaction very close to the wall. Simulations to steady state have required ultrahigh drop surface resolutions (to $3\times 10^5$ boundary elements) achieved through multipole acceleration and combined with novel regularization to fully eliminate near-singular behaviour of the double-layer integrals due to small clearance. Non-dimensional drop speed $U$ is presented for $\theta \geq 7.5^\circ$, $\lambda \leq 300$ and in a broad range of Bond numbers $B$, covering from nearly spherical to strongly pancaked drops. The results are consistent with published experiments on liquid–liquid systems. At small $\theta$ and $\lambda \gg 1$, $U$ is a strong, decreasing function of $B$; the asymptotic regime $U\to 0$ at $B\to 0$ is not observed in the simulated range. For small $B$, the dimpled thin-film geometry is insensitive to $\lambda =1\unicode{x2013}300$. For pancaked drops, the lubrication film is much thicker for $\lambda =1$ than for $\lambda \gg 1$ drops. Approximate thin-film uniformity in the drop motion direction is confirmed for pancaked, but not for $B\ll 1$, drops. Kinematics of drop motion shows that neither perfect tank treading, nor perfect rolling can be approached for liquid–liquid systems in the purely hydrodynamical formulation. The methodology is applicable to other problems and can allow for direct inclusion of short-range colloidal forces in three-dimensional boundary-integral simulations.