Interfacial dynamics induced by impacts across rigid and soft substrates

TL;DR

This work reveals that impact-driven jetting from a concave gas–liquid interface, created by a liquid-filled container impacting substrates, can be unified across rigid and soft substrates using a Cauchy-number-based framework. By introducing the partial-impulse concept and coupling an elastic-foundation contact model with momentum balance, the authors show that only the impulse accumulated within the jet-formation window drives acceleration in the soft regime, while the full impulse governs jetting in the rigid regime. The resulting model quantitatively matches experimental jet velocities over six orders of magnitude in substrate stiffness and provides a clear criterion Ca ≈ 10^{-4} separating rigid-like from soft-like impacts. This framework extends classical impulse-based theories to compliant substrates and offers practical insights for controlling impact-induced jetting in engineering and biological contexts.

Abstract

We investigate impact-induced gas-liquid interfacial dynamics through experiments in which a liquid-filled container impacts substrates with elastic moduli from $O(10^{-1})$ MPa to $O(10^{5})$ MPa. Upon impact, the concave gas-liquid interface inside the container deforms and emits a focused jet. When the jet velocity is normalized by the container impact velocity, all data collapse onto a single curve when plotted against the Cauchy number, $Ca = ρ_{\rm e} V_{\rm i}^2 / E$, which represents the ratio of the inertial force of the container-liquid system to the elastic restoring force of the substrate. The dimensionless jet velocity remains nearly constant for $Ca< 10^{-4}$, but decreases significantly for $Ca > 10^{-4}$. Based on this observation, we define the boundary between the rigid-impact and soft-impact regimes using the Cauchy number, providing a quantitative criterion for what constitutes ``softness'' in impact-driven interfacial flows. To explain the reduction in jet velocity observed in the soft-impact regime, we introduce a framework in which only the impulse transferred within the effective time window for jet formation contributes to interface acceleration. This concept, referred to as the partial impulse, captures the situation where the impact interval (the duration of contact between the container and the substrate) exceeds the focusing interval (the time required for jet formation). By modelling the contact force using an elastic foundation model and solving the resulting momentum equation over the finite impulse window, we quantitatively reproduce the experimental results. This partial impulse framework unifies the dynamics of impact-driven jetting across both rigid and soft substrate regimes, extending the applicability of classical impulse-based models.

Figures (11)

• Figure 1: A schematic of the experimental setup. A test tube filled with silicone oil of 10 mm$^2$/s was released in free fall using an electromagnet to impact the substrate. The experimental parameters were the elastic moduli of the substrates $E$, the impact velocity of the impactor $V_{\rm{i}}$, and the liquid filling height $H$. Nine substrates with different $E$ made of metals, resins, rubber and elastomers were used.
• Figure 2: A comparison of the interface motion and jet behaviour for $H = 20$ mm and $V_{\rm i} = 0.63$ m/s: (a) $E = 2.0 \times 10^{5}$ MPa, (b) $E = 8.1 \times 10^{-1}$ MPa. $t = 0$ ms represents the time at which the container first makes contact with the substrate. For $E = 8.1 \times 10^{-1}$ MPa, the time required for the concave interface to deform is longer, and the jet tip position is lower, compared with $E = 2.0 \times 10^{5}$ MPa. (c) The time evolution of the jet velocity for different values of $E$. Here, $t = 0$ ms represents the time at which the container first makes contact with the substrate; the plot starts at $t=\tau_{\rm focusing}$ and ends at the time of pinch-off. The results indicate that as $E$ decreases, the jet velocity becomes slower, while $\tau_{\rm focusing}$ increases.
• Figure 3: (a) $V_{\rm{j}}$ vs. $E$. $V_{\rm j}$ increases with increasing $E$ at fixed $V_{\rm i}$ and also increases with $V_{\rm i}$ at fixed $E$, indicating that $V_{\rm j}$ strongly depends on both $E$ and $V_{\rm i}$. However, the data points do not collapse onto a single curve. (b) $V_{\rm{j}}/V_{\rm{i}}$ vs. $\rho_{\rm{e}}V_{\rm{i}}^2/E$. The black dashed line shows the $V_{\rm j}/V_{\rm i} = 3.66$ estimated from the rigid substrate experiments by kiyamaEffectsWaterHammer2016, and the red line shows equation (\ref{['eq:Vj model']}). All data collapse onto a single curve. When $\rho_{\rm e} V_{\rm i}^2 / E< 10^{-4}$, the variation in $V_{\rm j}/V_{\rm i}$ is small. Since this regime can be explained by the pressure--impulse theory established in previous studies using rigid substrates, we refer to it as the rigid-impact regime in this paper. In contrast, when $\rho_{\rm e} V_{\rm i}^2 / E > 10^{-4}$, $V_{\rm j}/V_{\rm i}$ clearly decreases. This regime cannot be explained by conventional pressure--impulse theory. Because the influence of the soft substrate on the interfacial motion is pronounced, we refer to this region as the soft-impact regime.
• Figure 4: The moment when the container separates from the substrate (see $\blacktriangle$) and when the centre of the interface rises and the protrusion becomes visible from the side (see $\blacktriangledown$) are shown for (a) $E = 2.0 \times 10^{5}$ MPa, (b) $E = 1.7 \times 10^{0}$ MPa and (c) $E = 8.1 \times 10^{-1}$ MPa ($H = 20$ mm and $V_{\rm i} = 0.63$ m/s). $t = 0$ ms denotes the time when the container begins to make contact with the substrate. For $E = 2.0 \times 10^{5}$ MPa, $\tau_{\rm impact} = 0.2$ ms and $\tau_{\rm focusing} = 1.9$ ms, giving $\tau_{\rm impact} \ll \tau_{\rm focusing}$. In other words, the impact interval is short compared to the focusing interval. In contrast, for $E = 1.7 \times 10^{0}$ MPa and $E = 8.1 \times 10^{-1}$ MPa, the centre of the interface rises, and the protrusion becomes visible from the side while the container and the substrate are in contact. In other words, the impact interval becomes longer than the focusing interval. $H_{\rm m}$ represents the meniscus thickness just before the container and the substrate come into contact.
• Figure 5: (a) A schematic of the impactor and substrate at $t = 0$ and $t = \tau_{\rm focusing}$. $V'$ denotes the velocity of the impactor at $t = \tau_{\rm focusing}$. In this study, $F$ is evaluated using an elastic foundation model, in which the elastic response of the substrate is modelled as a spring. (b) The time evolution of $F$ for each $E$, obtained from the elastic foundation model for $H = 20$ mm and $V_{\rm i} = 0.63$ m/s. The shaded area represents $I$ calculated from equation (\ref{['eq:Vj model']}), i.e., the effective impulse time window. For $E \ge 5.7 \times 10^{0}$ MPa, which falls within the rigid-impact regime, the jet is driven by $I$ obtained by integrating $F$ over its entire duration, i.e. the total impulse. In contrast, for $E \le 1.9 \times 10^{1}$ MPa, which belongs to the soft-impact regime, the jet is driven by the impulse $I$ obtained from a partial integration of $F$, corresponding to a partial impulse.