Interfacial dynamics induced by impacts across rigid and soft substrates

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.

💡 Research Summary

This paper investigates the dynamics of a gas‑liquid interface that is driven by the impact of a liquid‑filled test tube on substrates spanning a wide range of elastic moduli (from 0.8 MPa to 2 × 10⁵ MPa). The authors systematically vary three experimental parameters: the impact velocity (V_i) (0.63–1.1 m s⁻¹) set by the drop height, the liquid filling height (H) (20–60 mm) which changes the mass of the impactor, and the substrate stiffness (E) using nine different materials (metals, epoxy, ABS, PE, rubber, and three PDMS formulations). High‑speed imaging (10 000 fps) captures the contact duration (\tau_{\text{impact}}), the focusing interval (\tau_{\text{focusing}}) required for jet formation, and the jet tip velocity (V_j).

When the jet velocity is normalized by the impact velocity ((V_j/V_i)) and plotted against the Cauchy number (Ca = \rho_e V_i^2 /E), all data collapse onto a single master curve. For (Ca < 10^{-4}) the normalized jet speed remains essentially constant (≈0.5), indicating that the substrate behaves as a rigid body and the classical pressure‑impulse theory applies. For (Ca > 10^{-4}) the normalized jet speed drops sharply, revealing that substrate compliance reduces the effective impulse transmitted to the interface.

To explain this transition, the authors introduce the concept of “partial impulse”. They argue that only the impulse delivered within the effective time window defined by the focusing interval contributes to the acceleration of the concave interface. When the contact interval exceeds the focusing interval, the later portion of the contact force does not affect jet formation. The contact force is modeled using an elastic foundation (spring) model: (F(t)=k,\delta(t)), where (k) is an equivalent stiffness derived from the measured modulus (E). The effective impulse is then (I_{\text{eff}} = \int_{0}^{\tau_{\text{focusing}}} F(t),dt). Substituting this into the pressure‑impulse framework yields a predictive expression for the jet speed, \