Unveiling crown-finger instability of a non-spherical drop impacting a liquid surface

TL;DR

This work addresses how non-spherical droplet shape influences crown formation and splashing when impacting a liquid film. It combines three-dimensional DNS with a linear stability analysis that couples Rayleigh–Plateau and Rayleigh–Taylor instabilities to predict crown-finger formation, validated by regime maps across $A_r$ and $We$. The results show that oblate drops promote earlier finger development due to stronger rim deceleration, while prolate drops favor canopy formation, with hole ruptures beneath the rim driving jetting and detachment. The integrated numerical and theoretical approach yields a predictive framework for regime transitions and finger counts, offering practical insights for processes where droplet shape impacts are critical.

Abstract

We present a three-dimensional numerical study of the splashing dynamics of non-spherical droplets impacting a quiescent liquid film, covering a wide range of aspect ratios (Ar) and Weber numbers (We). The simulations reveal distinct impact dynamics, such as spreading, splashing type-1, splashing type-2, and canopy formation, which are delineated in a regime map constructed in the Ar-We parameter space. Our results demonstrate that droplet morphology during the impact significantly influences crown evolution and splash initiation, with oblate drops promoting finger growth and fragmentation due to enhanced rim deceleration, while prolate drops tend to form canopies. We observe that the hole instability, which becomes more prominent at higher Weber numbers, arises from lamella rupture in the thinnest region of the film, located just beneath the crown rim. A linear stability analysis, supplemented by the temporal evolution of the crown obtained from the numerical simulations, adequately predicts the number of fingers formed along the crown rim by accounting for both Rayleigh-Plateau (RP) and Rayleigh-Taylor (RT) instabilities. The theoretical analysis demonstrates the dominant role of the Rayleigh-Plateau instability in determining the number and wavelength of early undulations, with the Rayleigh-Taylor instability serving to amplify the growth rate of the disturbances. Our findings highlight the critical role of the droplet shape in splash dynamics, which is relevant to a range of applications involving droplet impact.

Figures (24)

• Figure 1: (a) Schematic representation of the three-dimensional computational domain ($H \times H \times H$) illustrating the initial configuration of a non-spherical drop impacting a liquid pool with velocity $U_0$. (b) Illustration of the various initial shapes of the non-spherical drop, classified as oblate ($A_r = a/b > 1$), spherical ($A_r = 1$), and prolate ($A_r < 1$). Here, $R_{eq}$ denotes the equivalent spherical radius of the drop, $h$ represents the depth of the liquid film, and $q$ is the separation distance between the bottom of the drop and the liquid film.
• Figure 2: Grid-independence test showing the temporal evolution of the crown diameter $D_{{crown}}$ for three minimum grid sizes at ${\it We} = 729$ and $A_r = 1.5$.
• Figure 3: Grid-independence test: Panel ($a$) shows the crown morphology for three minimum grid sizes, $\Delta x_{\min} = 0.035R_{eq}$, $0.03R_{eq}$, and $0.025R_{eq}$, at ${\it We} = 729$ and $A_r = 1.5$ at $\tau = 42.86$. Panel ($b$) presents a magnified view of a representative finger for all three grids.
• Figure 4: Effect of the adaptive mesh refinement (AMR) parameter on the crown morphology for ${\it We} = 729$ and $A_r = 1.5$ at $\tau = 42.86$. The bottom panels display the cross-sectional front view of the crown in the $x$–$z$ plane along with the corresponding computational grids for each refinement level.
• Figure 5: Effect of the adaptive mesh refinement (AMR) parameter on the temporal evolution of the crown diameter $D_{{crown}}$ for ${\it We} = 729$ and $A_r = 1.5$.