Bouncing to coalescence transition for droplet impact onto moving liquid pools

TL;DR

This work investigates how translational motion of a liquid bath affects the bouncing-to-coalescence transition for a droplet impact. It combines controlled experiments with moving baths and three-dimensional direct numerical simulations to uncover the mechanism by which bath motion promotes air-layer drainage, lowering the critical normal Weber number $We=\frac{\rho V^2 R}{\sigma}$. A simple geometric collapse using a film tilt angle $\phi \approx 25.2^{\circ}$ explains the data across viscosities and droplet sizes, linking upstream air-film drainage to the transition. The findings advance understanding of gas-film dynamics in oblique 3D impacts and have implications for droplet deposition and coating processes on moving substrates.

Abstract

A droplet impacting a deep fluid bath is as common as rain over the ocean. If the impact is sufficiently gentle, the mediating air layer remains intact, and the droplet may rebound completely from the interface. In this work, we experimentally investigate the role of translational bath motion on the bouncing to coalescence transition. Over a range of parameters, we find that the relative bath motion systematically decreases the normal Weber number required to transition from bouncing to merging. Direct numerical simulations demonstrate that the depression created during impact combined with the translational motion of the bath enhances the air layer drainage on the upstream side of the droplet, ultimately favoring coalescence. A simple geometric argument is presented that rationalizes the collapse of the experimental threshold data, extending what is known for the case of axisymmetric normal impacts to the more general 3D scenario of interest herein.

Figures (5)

• Figure 1: (a) Schematic of a droplet impacting a moving bath of the same fluid. (b) Rendering of experimental setup. A droplet is generated by a piezoelectric droplet generator and impacts a moving fluid layer atop a rotating table. The dynamics are filmed from the side with a high-speed video camera. (c,d) A 2 cSt silicone oil droplet of radius $R=0.230 \pm 0.006$ mm impacts a fluid bath moving with horizontal speed $U=35$ cm/s. Images spaced by 1/540 seconds are directly superimposed. (c) With an impact velocity of $V=59.3 \pm 0.9$ cm/s, the droplet rebounds from the fluid bath while obtaining a horizontal velocity from the bath during contact. (d) At $V=60.7\pm0.8$ cm/s, the droplet merges with the bath and the residual interfacial disturbance is transported downstream following coalescence. Corresponding Videos available in Supplementary Movie 1.
• Figure 2: (a) Three-dimensional computational domain, with adaptive mesh highlighted in the inset. Supplementary videos of representative test cases (impact onto both static and moving pools) are also made available as Supplementary Movies 2 and 3. (b) Trajectory of highest point of droplet in the symmetry plane for a case of a 2 cSt silicone oil droplet with $R=0.23$ mm, $V=60.3$ cm/s, and $U=15$ cm/s. Yellow markers are predictions without inlet airflow, green markers are predictions with uniform inlet airflow, and solid lines are experimental measurements. The shaded region represents two standard deviations across experimental trials.
• Figure 3: Droplets (2 cSt silicone oil) of radius $R=0.230\pm 0.006$ mm with incident vertical speed $V$ impact a fluid bath moving with horizontal speed $U$. (a-c) Fixed impact velocity $V=73.1 \pm 1.0$ cm/s with increasing bath speed. Successive images in each sequence are spaced 1/750 seconds apart. Bouncing is observed for (a) $U=0$ cm/s and (b) $U=10$ cm/s, with merging at (c) $U=20$ cm/s. Corresponding videos available in Supplementary Movie 4. (d) Bouncing to coalescence transition as a function of bath speed. Gray triangles are trials where bouncing was observed, gray $\times$s are trials where coalescence was observed, and black markers are the mean transition values at each bath speed. (e) Critical $We$ as a function of normalized bath speed $U/V$. In all cases error bars represent propagated error, including one standard deviation across trials.
• Figure 4: Simulation results of droplets (2 cSt silicone oil) of radius $R=0.23$ mm with incident vertical speed $V=60.3$ cm/s impacting a fluid bath moving with horizontal speed $U$. (a) Evolution of air layer thickness profile along symmetry plane for the case of $U=0$ (left) and $U=15$ cm/s (right). Circular markers indicate the point of minimum thickness, which systematically occurs on the upstream side of the contact region for $U>0$. (b) Minimum air layer thickness as a function of bath speed $U$. Inset shows a typical slice from the simulation, with the circular marker indicating the position of minimum thickness. (c) Vertical coefficient of restitution ($\alpha$), horizontal coefficient of restitution ($\epsilon$), and non-dimensional contact time ($t_c/t_\sigma$) as a function of $U$ for simulations ($\square$) and experiments ($\bullet$). $t_\sigma$ is the inertio-capillary timescale defined as $\sqrt{\rho R^3/\sigma}$. For experimental data, error bars represent propagated error, including one standard deviation across trials.
• Figure 5: (a) Bouncing to coalescence transition for silicone oil droplets of different viscosities (marker color) and radii (marker size) as a function of the normalized bath speed. For 2 cSt oil (blue): small, medium, and large markers correspond to $R=0.169 \pm 0.008$, $0.230 \pm 0.006$, and, $0.403 \pm 0.008$ mm, respectively. For 20 cSt oil (purple): medium and large markers correspond to $R=0.208 \pm 0.009$ and $0.398 \pm 0.008$ mm, respectively. For 50 cSt oil (red): large markers correspond to $R=0.443 \pm 0.016$ mm. (b) Critical vertical velocity $V$ normalized by the critical velocity for a still bath $V_0$ (i.e. with $U=0$) under otherwise equivalent conditions. Dashed line (and shaded region) shows equation \ref{['eqn:collapse']} with film angle parameter $\phi=25.2\pm5.2^{\circ}$. In all cases error bars represent propagated error, including one standard deviation across trials.