Numerical simulations of simultaneous pair-drop impacts and their energetics

TL;DR

This work tackles the dynamics of simultaneous pair-drop impacts on a hydrophobic substrate using high-resolution 3D direct numerical simulations to resolve the central rising sheet formed by colliding lamellae. By validating against experiments and developing an energetic framework, it connects the maximum central-sheet height $H_{s,max}$ to the initial inertial–capillary balance through two geometric approximations—the cylindrical disk and the lollipop—while incorporating viscous dissipation. The study finds robust scaling trends, notably $H_{s,max} \sim We^{0.68} Re^{0.2}$ in the large-$Re$ regime (with geometry-dependent variations) and demonstrates near collapse of height dynamics under appropriate $We$ and $Re$ rescalings, providing a practical route to predict central-sheet evolution in multi-drop impacts. These results enhance understanding of inter-drop interactions in sprays and coatings and offer a framework applicable to more complex fluids and surface conditions.

Abstract

We present three-dimensional direct numerical simulations of the simultaneous impact of two identical drops on an hydrophobic substrate, varying the relative strength of capillary and viscous effects respectively through Weber and Reynolds numbers of impact. The interaction between the two drops is characterized by the appearance of a lamella arising from the collision of the two droplets' spreading rims. We examine the width, the height, and the general morphological evolution of the central sheet; the numerical data is validated against prior experiments and used to guide the development of an energetic model for the maximum elevation of the central sheet. In particular, the rise of the central sheet resembles the spreading behaviour single-drop impacts, especially at high Weber and Reynolds numbers. This fact can be used to estimate scalings in the capillary- and viscous-dominated regimes, which can be used to collapse the trajectories. These insights provide a route for a more complete understanding of the dynamics for the central rising sheet, and anticipate the detailed study of its fragmentation characteristics

Figures (27)

• Figure 1: Problem formulation illustrated using a two-dimensional schematic. Carets denote dimensional variables throughout the figure
• Figure 2: Definition of the measured variables of the central rising sheet formed by two drops after the second impact during the rising stage. The two blue squares indicate the assumed centers of the drops. $D_{rim}$, $W$, and $H_s$ are defined as the corresponding maximum values
• Figure 3: Comparison of the evolution of the central sheet with non-dimensional time $\tau$ between numerical simulations and experiments for $We=130$ at $\phi=40\%$ solution. Note: the experimental images are adapted from GoswamiPhDthesis, CC BY 4.0
• Figure 4: Comparison of the maximum heights between numerical results and experiments: (a) $\phi=0\%$ solutions and (b) $\phi=40\%$ solutions for three experimental cases with different $We$ (the corresponding $Re$ is obtained accordingly). Note: the experimental results are adapted from Goswami2023JFM. For clarity, the definitions of variables are omitted from the labels in the following figures
• Figure 5: Evolution of the maximum spreading ratio $\beta \equiv D/D_0$, shown by solid lines, and the dimensionless width of the central interaction surface $W$, shown by dotted lines of the same color, for various $\mathrm{We}$ and $\mathrm{Re}$. Reference experimental data adapted from Goswami2023JFM, are plotted as filled circles and show good agreement with the corresponding numerical results.