Effect of velocity, fluid properties and drop shape on coalescence and neck oscillation

TL;DR

This work advances the understanding of droplet coalescence on a deep liquid pool by performing axisymmetric simulations across a broad parameter space spanning the Weber number $We$, Ohnesorge number $Oh$, Bond number $Bo$, and drop aspect ratio $AR$. A volume-of-fluid framework implemented in Gerris resolves the interfacial dynamics and captures neck formation, oscillations, and drainage with adaptive mesh refinement validated against experimental benchmarks. The study reveals four neck-oscillation–based regimes and shows that inertia and gravity promote complete coalescence while viscosity damps capillary waves; drop shape further modulates secondary-droplet formation, with prolate drops more prone to fragmentation. A three-dimensional regime map clarifies the conditions under which partial or complete coalescence occur, offering mechanistic insight into the competition between vertical and horizontal collapse and highlighting neck oscillations as a practical diagnostic for regime identification. Overall, the findings provide a comprehensive framework for predicting coalescence outcomes in air–liquid systems across diverse drop shapes and impact conditions, with implications for inkjet, spraying, and coating processes.

Abstract

We perform axisymmetric numerical simulations to investigate the coalescence dynamics of a liquid drop in a deep liquid pool. This study aims to generalize the mechanisms of partial coalescence across a range of drop shapes, elucidate the underlying mechanism of neck oscillations, and examine the roles of inertial, viscous and gravitational forces, quantified by the Weber, Ohnesorge, and Bond numbers, in governing the coalescence behavior. A phase diagram is constructed to delineate the boundaries between partial and complete coalescence regimes based on these dimensionless parameters. Our analysis of the height-to-neck ratio shows that, upon contact with the pool, the primary drop forms an upward liquid column that ultimately pinches off due to inwardly directed horizontal momentum. Additionally, the study suggests that as the dimensionless numbers increase, the effect of the vertical collapse rate plays a significant role in the outcome of the coalescence process. Notably, the Rayleigh-Plateau instability is found to be insignificant in driving partial coalescence within the explored parameter space. We identified a transition regime between partial and complete coalescence, characterized by multiple neck oscillations that delay the pinch-off of secondary droplets. The formation of secondary droplets is most prominent for prolate drops, followed by spherical and oblate drops of comparable volume. Furthermore, we observe that the tendency to form multiple droplets from elongated liquid columns diminishes with an increase in the impact velocity of the primary drop.

Figures (13)

• Figure 1: (a) Schematic diagram illustrating the initial configuration of a drop impacting a liquid pool. (b) Depiction of various shapes of drops. The aspect ratios, ${\it AR}=b/a<1$, ${\it AR}=1$, and ${\it AR}>1$, correspond to oblate, spherical, and prolate shapes, respectively.
• Figure 2: (a) Variation of the normalized neck radius $(r_D/D_{eq})$ with normalized time $(\tau = t/\sqrt{\rho_l D_{eq}^3/8\sigma})$ for different mesh sizes $\Delta x$, with ${\it AR} = 1.4$, ${\it We} = 40$, ${\it Bo} = 0.5$, and ${\it Oh} = 0.003$. (b) Variation of normalized pinch-off time $(\tau_{p} = t_{p}/\sqrt{\rho_l D_{eq}^3/8\sigma})$ with mesh size $\Delta x$ for ${\it AR} = 1$, ${\it We} = 0$, ${\it Bo} = 0.5$, and ${\it Oh} = 0.014$.
• Figure 3: The comparison between the results obtained using the current numerical solver and those of blanchette2006partial. Experimental results from blanchette2006partial are depicted in panel (a), while the results obtained in this study are displayed in panel (b). These panels illustrate the partial coalescence phenomenon of an ethanol drop of radius 0.535 mm in an ethanol pool with air as the surrounding medium. The times (in ms) are indicated on each image for reference.
• Figure 4: Figure adapted from hendrix2016universal. Log–log plot of the normalized bubble volume $(V_b / V_{sphere/drop})$ as a function of the Stokes number $(St)$. Here, $V_b$ and $V_{sphere/drop}$ are the volumes of the bubble and the impacting drop/sphere, respectively. The figure shows boundary-integral simulation results from hendrix2016universal, together with previously published experimental data from tran2013air, Marston_2011, and bouwhuis2012maximal. Results from the present study are superimposed as open squares. The solid line denotes the $-4/3$ scaling law originally reported by tran2013air.
• Figure 5: (a) Regime map showing the variation of the critical Weber number (${\it We}_c$) across the ${\it We}–{\it Oh}–{\it Bo}$ parameter space, with the color bar representing the magnitude of ${\it We}_c$. Cross-sectional phase diagrams highlight the boundaries between partial and complete coalescence regimes in (b) ${\it Oh}–{\it Bo}$ space at ${\it We} = 0$, (c) ${\it We}–{\it Bo}$ space for ${\it Oh} = 0.01$, and (d) ${\it Oh}–{\it We}$ space at ${\it Bo} = 0.1$. Panel (b) also includes a comparison of our findings with the results reported by blanchette2006partial.