Impacting spheres: from liquid drops to elastic beads

TL;DR

This work unifies liquid-drop and elastic-bead impact physics by modeling a viscoelastic sphere impacting a non-contacting rigid surface with the Oldroyd-B constitutive framework. Two key dimensionless controls, the elasticity number $El = \frac{G}{\rho_l V_0^2}$ and the Weissenberg number $Wi$, govern the transition from Wagner-like inertial-capillary behavior to Hertz-like elastic contact, including a capillary-dominated regime. The authors derive asymptotic scalings for the peak impact force: a Wagner plateau $F_{\max}/(\rho_l V_0^2 R_0^2) \approx 3.24$ and a Hertz-like scaling $\approx 5.3\,El^{2/5}$, and propose a unified interpolating model $\frac{F_{\max}}{\rho_l V_0^2 R_0^2} = 5.3\,f(El)\,El^{2/5} + (1-f(El))(\frac{3.2}{We} + 3.24)$ with $f(El)=\tanh(\log_{10} El)$. Elastic memory (large $Wi$) accelerates the onset of Hertz-like behavior, while memoryless liquids (small $Wi$) align with Wagner predictions; together these results provide a continuous, predictive framework for the liquid-to-elastic transition across systems and applications.

Abstract

A liquid drop impacting a non-wetting rigid substrate laterally spreads, then retracts, and finally jumps off again. An elastic solid, by contrast, undergoes a slight deformation, contacts briefly, and bounces. The impact force on the substrate - crucial for engineering and natural processes - is classically described by Wagner's (liquids) and Hertz's (solids) theories. This work bridges these limits by considering a generic viscoelastic medium. Using direct numerical simulations, we study a viscoelastic sphere impacting a rigid, non-contacting surface and quantify how the elasticity number ($El$, dimensionless elastic modulus) and the Weissenberg number ($Wi$, dimensionless relaxation time) dictate the impact force. We recover the Newtonian liquid response as either $El \to 0$ or $Wi \to 0$, and obtain elastic-solid behavior in the limit $Wi \to \infty$ and $El \ne 0$. In this elastic-memory limit, three regimes emerge - capillary-dominated, Wagner scaling, and Hertz scaling - with a smooth transition from the Wagner to the Hertz regime. Sweeping $Wi$ from 0 to $\infty$ reveals a continuous shift from materials with no memory to materials with permanent memory of deformation, providing an alternate, controlled route from liquid drops to elastic beads. The study unifies liquid and solid impact processes and offers a general framework for the liquid-to-elastic transition relevant across systems and applications.

Figures (6)

• Figure 1: Schematic: a viscoelastic sphere (radius $R_0$) impacts a non‑contacting rigid surface with velocity $V_0$. Material properties are $\rho_l$, $\eta_l$, $G$, $\lambda$, and $\gamma$ for the sphere; $\rho_g$ and $\eta_g$ for the gas.
• Figure 2: (a) Phase space in the $Ec$-$We$ plane illustrating the range of simulations conducted in this work colored according to the elasticity number $El = Ec/We$. The four highlighted symbols locate typical cases representing the range of parameters explored. We chose ($We, Ec$) $=$ (b) ($5,1$), (c) ($500,2$), (d) ($5, 1000$), (e) ($500, 1000$). For each case, the color scheme of each snapshot represents the magnitude of the velocity normalized by the impact velocity, alongside the corresponding force history $F(t)/(\rho_l V_0^2 R_0^2)$ plotted versus $t/t_{\max}$ (right). The force traces are plotted up to $t/t_{\max}=2$: for the liquid-drop reference, the second peak associated with the Worthington jet zhangImpactForcesWater2022sanjayRoleViscosityDrop2025 occurs at later times $t_2 \gg t_{\max}$ (with $t_2/t_{\max} \sim \sqrt{We}$) and is therefore outside the plotted window.
• Figure 3: Peak force in the elastic-memory limit ($De\to\infty$): (a) Variation of the normalized peak force, $F_{\max}/(\rho_l V_0^2 R_0^2)$, with the Weber number $We=\rho_l V_0^2R_0/\gamma$ for different elastocapillary numbers $Ec=GR_0/\gamma$. For $Ec\lesssim\mathcal{O}(1)$ the data follow the liquid-impact result: a high-$We$ Wagner plateau $\simeq 3.24$, with the low-$We$ correction $F_{\max}/(\rho_l V_0^2 R_0^2)\approx 3.2/We+3.24$ (dashed line, eq. \ref{['eq:fmax_liq']}). As $Ec$ increases, $F_{\max}$ rises, most clearly at low $We$, and for sufficiently large $Ec$ the curves acquire a log–log slope $-2/5$, i.e. $F_{\max}/(\rho_l V_0^2 R_0^2)\sim We^{-2/5}$ at fixed $Ec$, consistent with the approach to Hertzian elastic contact. (b) Dependence on $Ec$ at fixed $We$ (curves labelled by $We$). At small $Ec$ all series collapse to the liquid-like level ($\approx 3.24$); above a $We$-dependent crossover, $F_{\max}$ increases monotonically with $Ec$, following $F_{\max}/(\rho_l V_0^2 R_0^2)\sim Ec^{2/5}$, again, consistent with the approach to Hertzian elastic contact. Together, (a,b) show a continuous evolution from Wagner (liquid) to Hertz (elastic) behavior as $Ec$ increases.
• Figure 4: Unified scaling and regime map: (a) Collapse of the normalized peak force versus the elasticity number $El=Ec/We=G/(\rho_l V_0^2)$. For $El\lesssim 1$ and sufficiently large $We$ the data sit on the Wagner plateau $F_{\max}/(\rho_l V_0^2 R_0^2)\approx 3.24$; deviations at very small $We$ reflect capillary corrections in eq. \ref{['eq:fmax_liq']}. For $El\gtrsim 1$ all cases collapse onto a single power law with slope $2/5$, $F_{\max}/(\rho_l V_0^2 R_0^2)\sim El^{2/5}$ (eq. \ref{['eq:FmaxElscale']}), the hallmark of Hertz scaling. (b) Contours of $F_{\max}/(\rho_l V_0^2 R_0^2)$ in the $(We,Ec)$ plane (symbols: simulation points). The dashed guide $El=1$ marks the smooth crossover from the Wagner region (lower right) to the Hertz region (upper left); the low‑$We$ corner is capillary‑dominated and requires the correction in eq. \ref{['eq:fmax_liq']}. The map visualizes the continuous transition from liquid‑like to solid‑like impact forces as $We$ and $Ec$ are varied.
• Figure 5: Relaxing material memory at fixed $We$ and $El$. Evolution of shape (left) and force (right) when $Wi$ decreases from $\infty$ to $0$ at $We=100$ and $El=40$: (a) $Wi\to\infty$ (elastic‑memory limit); (b) $Wi=10^{-1}$; (c) $Wi=10^{-2}$; (d) $Wi=0$ (Newtonian). For each case, the color scheme of each snapshot represents the magnitude of the velocity normalized by the impact velocity, alongside the corresponding force history $F(t)/(\rho_l V_0^2 R_0^2)$ plotted versus $t/t_{\max}$ (right). As $Wi$ decreases, contact time increases and $F(t)$ becomes increasingly liquid‑like with a reduced $F_{\max}$.