Velocity of focused liquid jets induced by impact across rigid and soft substrates

When a liquid-filled test tube impacts a substrate, the concave gas–liquid interface inside the tube deforms and emits a focused liquid jet. This configuration, known as Pokrovski’s experiment, is investigated here with a focus on how substrate elasticity affects the jet velocity. By varying the substrate elastic modulus from upper O left parenthesis 10 Superscript negative 1 Baseline right parenthesis O ( 10 − 1 ) O(10⁻¹) to upper O left parenthesis 10 Superscript 5 Baseline right parenthesis O ( 10 5 ) O(10⁵) MPa, we show that the jet velocity decreases as the substrate becomes softer. The results are organised using the Cauchy number italic Ca Ca Ca , which represents the ratio of the inertia of the container–liquid system to the elastic restoring force of the substrate. The dimensionless jet velocity remains nearly constant for italic Ca less than 10 Superscript negative 4 Ca < 10 − 4 Ca < 10⁻⁴ , whereas it decreases for italic Ca greater than 10 Superscript negative 4 Ca > 10 − 4 Ca > 10⁻⁴ . To explain this reduction, we introduce the concept of partial impulse. When the impact interval, during which the container and substrate remain in contact, exceeds the focusing interval, during which the interface forms a jet, the jet is formed before the total impulse has been delivered. Thus only the impulse accumulated within the effective time window for jet formation contributes to the interface motion. By modelling the contact force and integrating it over this effective impulse window, we quantitatively reproduce the experimental trend. This partial-impulse framework provides a unified description of impact-induced focused jetting across both rigid and soft substrates.