3.3.1. Relative magnitude between the impact interval and the focusing interval

Figure 4 shows the evolution from the onset of impact to the stage where the jet elongates to approximately the test tube diameter for the conditions $H = 20$  mm and $V_{{i}} = 0.63$  m s⁻¹, using seven images at equal time intervals. Figures 4(a–c) correspond to $ E = 2.0 \times 10^{5}$ , $1.7 \times 10^{0}$ and $8.1 \times 10^{-1}$ MPa, respectively. The time $t = 0$  ms denotes when the container begins to contact the substrate. The time when the container separates from the substrate is defined as $t = \tau _{\textit{impact}}$ . In this subsubsection, $t=\tau _{\textit{impact}}$ is interpreted as the time when the impact interval ends, and $t=\tau _{\textit{focusing}}$ as the time when the focusing interval ends, namely when the centre of the interface rises and a protrusion becomes visible from the side (see ). The values of $t = \tau _{\textit{impact}}$ and $t=\tau _{\textit{focusing}}$ were determined by analysing the complete image sequences recorded at intervals of 0.1 ms for all experiments. Among the seven equally time-spaced images displayed in figure 4, the images whose time stamps are closest to the evaluated values of $t = \tau _{\textit{impact}}$ and $t=\tau _{\textit{focusing}}$ are indicated by ( $\color {red}{\blacktriangle }$ ) and (), respectively. Here, we focus on the relative magnitudes of $\tau _{\textit{impact}}$ and $\tau _{\textit{focusing}}$ . In the rigid-impact regime, for $E = 2.0 \times 10^{5}$ MPa, $\tau _{\textit{impact}} = 0.2$ ms and $\tau _{\textit{focusing}} = 1.9$ ms, giving $\tau _{\textit{impact}} \ll \tau _{\textit{focusing}}$ (see figure 4 a). This means that the impact interval is shorter than the focusing interval, and the jet forms after contact has ended. In contrast, in the soft-impact regime, the order of $\tau _{\textit{impact}}$ and $\tau _{\textit{focusing}}$ is reversed. For $E = 1.7 \times 10^{0}$ MPa, $\tau _{\textit{impact}} = 6.5$ ms and $\tau _{\textit{focusing}} = 5.0$ ms. For $E = 8.1 \times 10^{-1}$ MPa, $\tau _{\textit{impact}} = 9.2$ ms and $\tau _{\textit{focusing}} = 5.7$ ms, and in both cases, $\tau _{\textit{impact}} \gt \tau _{\textit{focusing}}$ . This means that the impact interval is longer than the focusing interval, and the jet is formed during the contact (see figures 4 b,c). The key point is that as $E$ decreases, the relationship between $\tau _{\textit{impact}}$ and $\tau _{\textit{focusing}}$ is reversed, and the impact interval becomes longer than the focusing interval. This suggests that in the soft-impact regime, the total impulse from the substrate to the container is not fully transmitted to the concave gas–liquid interface before the jet is formed.

Figure 4.

The moments when the container separates from the substrate (see $\color {red}{\blacktriangle }$ ) and when the centre of the interface rises and the protrusion becomes visible from the side (see ) 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_{{i}} = 0.63$  m s⁻¹). Here, $t = 0$  ms denotes the time when the container begins to make contact with the substrate. For $E = 2.0 \times 10^{5}$ MPa, $\tau _{\textit{impact}} = 0.2$  ms and $\tau _{\textit{focusing}} = 1.9$  ms, giving $\tau _{\textit{impact}} \ll \tau _{\textit{focusing}}$ . In other words, the impact interval is short compared to the focusing interval. In contrast, for $E = 1.7 \times 10^{0}$ and $8.1 \times 10^{-1}$ 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. Here, $H_{{m}}$ represents the meniscus thickness just before the container and the substrate come into contact.

Figure 4. Long description

Panel A: A series of photos showing the impact and focusing intervals of a liquid in a container with an elastic modulus of 2.0 times 10 to the power of 5 megapascals. The images capture the liquid’s behavior at different time intervals, from 0 to 4.2 milliseconds. The impact interval is marked in red and lasts for 0.2 milliseconds, while the focusing interval is marked in blue and lasts for 1.9 milliseconds. A close-up image shows the frame closest to the moment when the container separates from the substrate. Panel B: A series of photos showing the impact and focusing intervals of a liquid in a container with an elastic modulus of 1.7 times 10 to the power of 0 megapascals. The images capture the liquid’s behavior at different time intervals, from 0 to 10.2 milliseconds. The impact interval is marked in red and lasts for 6.5 milliseconds, while the focusing interval is marked in blue and lasts for 5.0 milliseconds. A close-up image shows the frame closest to the moment when the container separates from the substrate. Panel C: A series of photos showing the impact and focusing intervals of a liquid in a container with an elastic modulus of 8.1 times 10 to the power of -1 megapascals. The images capture the liquid’s behavior at different time intervals, from 0 to 12 milliseconds. The impact interval is marked in red and lasts for 9.2 milliseconds, while the focusing interval is marked in blue and lasts for 5.7 milliseconds. A close-up image shows the frame closest to the moment when the container separates from the substrate.

3.3.2. Development of a new model equation for the jet velocity

To discuss the mechanism behind the decrease in $V_{\!{j}}/V_{{i}}$ on soft substrates, we model the jet velocity in terms of the impulse transmitted from the substrate during the time that is dynamically relevant for formation. In Pokrovski’s experiment, the impact generates a force on the container over a finite duration, which initiates the motion of the concave gas–liquid interface and is subsequently amplified by flow focusing (Antkowiak et al. Reference Antkowiak, Bremond, Dizès and Villermaux2007; Kiyama et al. Reference Kiyama, Noguchi and Tagawa2014). In the rigid-impact regime, the contact ends well before the jet forms, and it is therefore natural to relate the jet to the total impulse delivered during the contact. In the soft-impact regime, however, we observe $\tau _{\textit{impact}} \gt \tau _{\textit{focusing}}$ , meaning that the jet forms while contact is still ongoing. In this case, the interface cannot be driven by the total impulse accumulated over the entire contact duration; instead, jet formation can only depend on the impulse delivered up to the end of the focusing interval.

Figure 5.

(a) A schematic of the impactor and substrate at $t = 0$ and $t = \tau _{\textit{focusing}}$ . Here, $V'$ denotes the velocity of the impactor at $t = \tau _{\textit{focusing}}$ . 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 $V_{{i}} = 0.63$  m s⁻¹. The shaded area represents $I$ calculated from (3.11), i.e. the effective impulse time window. For $E \ge 7.0 \times 10^{2}$ 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.

This idea can be expressed without introducing additional variables by integrating the contact force over a finite time window. Importantly, this formulation automatically recovers the rigid-impact limit. When $\tau _{\textit{impact}} \lt \tau _{\textit{focusing}}$ , the contact force vanishes for $t \gt \tau _{\textit{impact}}$ , thus integrating up to $t=\tau _{\textit{focusing}}$ is equivalent to integrating over the full contact. In contrast, when $\tau _{\textit{impact}} \gt \tau _{\textit{focusing}}$ , the upper limit $t=\tau _{\textit{focusing}}$ cuts off the force history, so that only a portion of the total impulse contributes to jet formation (see figures 5(b) and 6). In the remainder of this section, we translate this concept into a predictive model by combining momentum conservation with a contact-force model based on the elastic foundation model.

We consider the motion of the impactor of mass $m$ over the interval from first contact, $t = 0$ , to the end of the focusing interval, $t = \tau _{\textit{focusing}}$ . Figure 5(a) shows a schematic of the impactor and the substrate over the time interval $[0,\tau _{\textit{focusing}}]$ . Let $F(t)$ denote the magnitude of the normal contact force exerted by the substrate on the container. Over $[0,\tau _{\textit{focusing}}]$ , momentum conservation gives

(3.4) \begin{equation} m(V_{{i}} + V') = \int _{0}^{\tau _{\textit{focusing}}} F(t)\, {\rm d}t. \end{equation}

Here, $V'$ is the velocity of the impactor at $t = \tau _{\textit{focusing}}$ . Equation (3.4) therefore relates the impulse delivered up to $t=\tau _{\textit{focusing}}$ to the momentum change of the impactor over $[0,\tau _{\textit{focusing}}]$ .

Next, we relate the left-hand side of (3.4) to the jet velocity $V_{\!{j}}$ . Based on pressure-impulse theory, Kiyama et al. (Reference Kiyama, Noguchi and Tagawa2014) related the change in momentum of the impactor to $V_{\!{j}}$ . That is, $V_{\!{j}}$ can be described by considering the change in momentum of the impactor over $[0, \tau _{\textit{focusing}}]$ as

(3.5) \begin{equation} V_{\!{j}} = \alpha (V_{{i}} + V'), \end{equation}

where $\alpha$ is the strength of the flow-focusing effect after the interface acquires velocity due to the impact (Kiyama et al. Reference Kiyama, Tagawa, Ando and Kameda2016). The value of $\alpha$ depends on the interface geometry (e.g. the contact angle at the wall) and on viscous effects (Kiyama et al. Reference Kiyama, Noguchi and Tagawa2014; Cheng et al. Reference Cheng, Chen, Ding, Zhang, Hu and Jia2024). In this study, the strength of the flow-focusing effect is calculated using the semi-empirical formula derived by Cheng et al. (Reference Cheng, Chen, Ding, Zhang, Hu and Jia2024) from experiments and simulations on rigid substrates. This semi-empirical formula accounts for the viscous boundary layer and is expressed as

(3.6) \begin{equation} \alpha = 2.05(1-{\rm e}^{- \zeta\, \textit{Re}_{{p}}}), \end{equation}

where $\textit{Re}_{{p}}$ denotes the Reynolds number, which can be written as

(3.7) \begin{equation} \textit{Re}_{{p}} = \sqrt {\frac {R U_0}{\nu }}, \end{equation}

with $U_0$ the velocity of the concave gas–liquid interface just after the impact, $\nu$ the kinematic viscosity, and $\zeta = 0.076$ a fitting parameter (Cheng et al. Reference Cheng, Chen, Ding, Zhang, Hu and Jia2024). For metal substrates, $ U_0$ can be defined as $U_0 = V_{{i}} + V_{{r}}$ , where $V_{{r}}$ is the rebound velocity of the impactor (Kiyama et al. 2014, Reference Kiyama, Tagawa, Ando and Kameda2016; Cheng et al. Reference Cheng, Chen, Ding, Zhang, Hu and Jia2024). We obtained $U_0$ from the metal substrate cases following the same procedure as in Kiyama et al. (Reference Kiyama, Noguchi and Tagawa2014, Reference Kiyama, Tagawa, Ando and Kameda2016) and Cheng et al. (Reference Cheng, Chen, Ding, Zhang, Hu and Jia2024), which yielded $\alpha = 1.91$ . We also confirmed that $\alpha$ can reasonably be treated as approximately constant within the range of experimental conditions considered in the present study (see Appendix C). Equations (3.4)–(3.7) therefore relate the jet velocity to the time integral of the contact force over $[0,\tau _{\textit{focusing}}]$ .

We now model the right-hand side of (3.4). To describe the time history of the contact force magnitude $F(t)$ , we consider the one-dimensional contact dynamics of an impactor with a hemispherical bottom impacting a compliant substrate. Let $z(t)$ denote the indentation of the substrate surface at the contact point (so that $z = 0$ at first contact, and $z \lt 0$ during compression). Since the gravitational term is negligibly small compared with the other terms, the equation of motion is

(3.8) \begin{equation} m \frac {{\rm d}^2 z}{{\rm d}t^2} - F(z) = 0. \end{equation}

We apply the initial condition at $t = 0$ , ${\rm d}z/{\rm d}t = -V_{{i}}$ and $z = 0$ , to (3.8). The contact force $F(z)$ can be theoretically obtained from the elastic foundation model using the displacement of the substrate surface (Johnson Reference Johnson1987), which is appropriate for finite-thickness substrates. For a rigid sphere (radius $R$ ) indenting a substrate of thickness $h$ , the model gives

(3.9) \begin{equation} F(z) = \frac {\pi \textit{RE}}{h} z ^2. \end{equation}

The force model is validated through comparison with our measurements. The validity of the elastic foundation model for the present experiments is discussed in detail in Appendix D. As shown in figure 5(a), the elastic foundation model represents the substrate’s elastic force using springs, and describes the relationship between the contact force and the substrate surface displacement. The time variation of the contact force $F(t)$ from the elastic foundation model can be obtained numerically using (3.8) and (3.9). First, by substituting (3.9) into (3.8), we obtain a nonlinear second-order ordinary differential equation (ODE) in $z$ and $t$ . We solve this ODE numerically using an explicit Runge–Kutta method (the solve_ivp function, an ODE solver from the SciPy library in Python) to obtain the time evolution of the substrate surface displacement $z = z(t)$ . Finally, by substituting the numerically obtained $z = z(t)$ into (3.9), we can calculate $F(t)$ . As seen from (3.8) and (3.9), $F(t)$ is determined by $m$ , $h$ , $R$ , $V_{{i}}$ and $E$ , but in the present experiments, the most dominant parameter is $E$ . In general, the container radius $R$ and the bottom geometry are also important because they determine the contact-force history through the adopted contact model. Moreover, the container geometry can influence the pressure-impulse field and the subsequent focusing dynamics (Watanabe et al. Reference Watanabe, Kusuno and Tagawa2025); see Appendix D. However, since $R$ and the bottom geometry were fixed throughout the present experiments, their effects were not investigated systematically and are therefore not considered further in the present analysis.

Figure 5(b) shows the time evolution of $F$ for each $E$ as solid lines for $H = 20$  mm and $ V_{{i}} = 0.63$  m s⁻¹. The time at which $F$ first becomes positive, corresponding to the onset of contact between the container and the substrate, is defined as $t = 0$ ms. The end of the solid line indicates the time $t = \tau _{\textit{impact}}$ , when $F$ returns to zero as the substrate and the container separate. As $E$ decreases, the peak of $F$ decreases, while $\tau _{\textit{impact}}$ increases, consistent with softer contact.

Finally, we estimate $\tau _{\textit{focusing}}$ by assuming that a concave gas–liquid interface with meniscus thickness $H_{{m}}$ (see figure 4) deforms at velocity $ V_{\!{j}}$ , as

(3.10) \begin{equation} \tau _{\textit{focusing}} \approx \frac {H_{{m}}}{V_{\!{j}}}. \end{equation}

As shown in figure 4, $H_{{m}}$ is defined as the distance between the upper and lower edges in the central region of the concave gas–liquid interface. By measuring the meniscus thickness just before impact for all experimental conditions, we obtained $H_{{m}} = 5.41$  mm. The estimate based on (3.10) was compared with the experimental values and was confirmed to show overall good agreement (see Appendix E). Here, $H_{{m}}$ can also be expressed using $R$ . Using the contact angle $\theta$ formed between the concave gas–liquid interface and the container wall, and neglecting the wall thickness, $H_{{m}}$ can be written geometrically as $H_{{m}} = (R / {\cos \theta} ) (1 - \sin \theta )$ .

By using (3.4)–(3.10), the impulse used to drive the concave gas–liquid interface $I$ can be modelled. Therefore, a new closed model equation for the jet velocity can be written as

(3.11) \begin{equation} \frac {V_{\!{j}}}{V_{{i}}} = \frac {\alpha }{m V_{{i}}} \int _{0}^{{{H_{{m}}}/{V_{\!{j}}}}} F(t) \, {\rm d}t. \end{equation}

Equation (3.11) reduces to the rigid-impact limit when $H_{{m}}/V_{\!{j}} \gt \tau _{\textit{impact}}$ , because the integral then covers the full force history. In the soft-impact regime, where $H_{{m}}/V_{\!{j}} \lt \tau _{\textit{impact}}$ , the upper limit $t=H_{{m}}/V_{\!{j}}$ cuts off the integral, capturing the partial-impulse mechanism proposed above. Based on the above considerations, $I$ is formulated as the right-hand side of (3.11):

(3.12) \begin{equation} I= \frac {\alpha }{m V_{{i}}} \int _{0}^{{{H_{{m}}}/{V_{\!{j}}}}} F(t) \, {\rm d}t. \end{equation}

In the soft-impact limit, (3.11) also yields a simple asymptotic scaling. The derivation is given in Appendix F, where we show that when the integration window samples only the early-time portion of the contact-force history, the partial-impulse model reduces to

(3.13) \begin{equation} V_{\!{j}} / V_{{i}} \propto \textit{Ca}^{-1/4}. \end{equation}

We also clarify that the dimensional analysis itself does not directly yield a power-law relation. Rather, the dimensional analysis only shows the functional relationship $V_{\!{j}}/V_{{i}}=\varPhi (Ca)$ , whereas the $-1/4$ exponent is derived specifically as the soft-limit asymptote of (3.11).

Figure 6.

The relationship between an arbitrary $V_{\!{j}}$ , the left-hand side of (3.11) $(V_{\!{j}}/V_{{i}})$ , and the right-hand side $(I)$ . The black line represents $V_{\!{j}}/V_{{i}}$ . The red, green and blue lines indicate $I$ for $E = 8.1 \times 10^{-1}$ , $5.7 \times 10^{0}$ and $7.0 \times 10^{2}$ MPa, respectively. For each value of $E$ , the numerical solution for $V_{\!{j}}$ is given by the intersection of the black line with the corresponding coloured line. Here, (Rigid-1) denotes the value of $I$ at the numerical solution for $V_{\!{j}}$ in the case $E = 7.0 \times 10^{2}$ MPa, which belongs to the rigid-impact regime. In this case, $I$ is obtained by integrating $F$ over its entire duration. Also, (Soft-2) and (Soft-4) correspond to the values of $I$ at the numerical solutions for $V_{\!{j}}$ when $E = 8.1 \times 10^{-1}$ and $5.7 \times 10^{0}$ MPa, which belong to the soft-impact regime, respectively. Here, the integration window of $F$ is limited to $t = H_{{m}}/V_{\!{j}}$ , and $I$ is calculated from a partial integration of $F$ . Finally, (Soft-1) to (Soft-3) illustrate the relationship between $V_{\!{j}}$ and $I$ , together with the integration window, for $E = 8.1 \times 10^{-1}$ MPa. These results demonstrate that $I$ represents either the total impulse or a partial impulse, depending on the relative magnitudes of $H_{{m}}/V_{\!{j}}$ and $\tau _{\textit{impact}}$ .

Figure 6. Long description

Panel A: A line graph showing the relationships of V_j/V_i and I with V_j for different values of E. The horizontal axis represents V_j (m s^{-1}), and the vertical axes represent V_j/V_i (top) and I (bottom). The black line represents V_j/V_i, while the blue, green, and red lines show the variation of I for E = 7.0 × 10^2 MPa, E = 5.7 × 10^0 MPa, and E = 8.1 × 10^{-1} MPa, respectively. Panel B: Four subplots showing the contact force F over time t for different values of E. The subplots are labeled (Soft-1), (Soft-2), (Soft-3), and (Rigid-1). Each subplot includes annotations for Hm/Vj, I, and impact duration.

3.3.3. Numerical solution of the model equation

We further discuss $I$ by numerically solving the model equation and comparing the results with the experimental data. First, we describe the numerical procedure used to solve (3.11). Since $V_{\!{j}}$ appears on both sides of (3.11), the equation can be solved numerically by treating $V_{\!{j}}$ as an unknown variable. Specifically, the integration time window of $F$ , which is determined by the relative magnitudes of $\tau _{\textit{impact}}$ and $H_{{m}}/V_{\!{j}}$ , is chosen such that the left- and right-hand sides of (3.11) are equal. This procedure yields the numerical solution for $V_{\!{j}}$ .

First, we examine how the left- and right-hand sides of (3.11) vary as functions of an arbitrary $V_{\!{j}}$ . Figure 6 shows how both sides of (3.11) depend on $V_{\!{j}}$ for different values of $E$ with $H = 20$  mm and $V_{{i}} = 0.63$  m s⁻¹. The $x$ -axis represents an arbitrary $V_{\!{j}}$ , while the $y$ -axis shows $V_{\!{j}}/V_{{i}}$ (black line) and $I$ (red, green and blue lines). The red, green and blue lines correspond to $E = 8.1 \times 10^{-1}$ , $5.7 \times 10^{0}$ and $7.0 \times 10^{2}$ MPa, respectively. For each value of $E$ , the numerical solution for $V_{\!{j}}$ is given by the intersection of the black line with the corresponding coloured line.

The value of $V_{\!{j}}/V_{{i}}$ increases monotonically, with slope determined by $V_{{i}}$ . In contrast, the behaviour of $I$ varies depending on both the force profile $F$ , which is primarily governed by $E$ , and the integration time window, which is determined by the relative magnitudes of $\tau _{\textit{impact}}$ and $H_{{m}}/V_{\!{j}}$ , as illustrated by (Rigid-1) and (Soft-1)–(Soft-4) in figure 6. For $E = 7.0 \times 10^{2}$ MPa, which falls within the rigid-impact regime, $I$ remains constant. In contrast, for $E = 5.7 \times 10^{0}$ and $8.1 \times 10^{-1}$ MPa, both of which belong to the soft-impact regime, $I$ exhibits a constant followed by a decreasing region.

First, we consider the case $E = 7.0 \times 10^{2}$ MPa. As shown in figure 6 (Rigid-1), for $ V_{\!{j}}$ up to 5 m s⁻¹, $H_{{m}}/V_{\!{j}}$ is larger than $\tau _{\textit{impact}}$ , therefore the integration time window of $F$ is $[0, \tau _{\textit{impact}}]$ . In this case, $F$ is integrated over its entire duration. Because the integration time window does not depend on $V_{\!{j}}$ , the value of $I$ remains constant regardless of $V_{\!{j}}$ . Figure 6 (Rigid-1) shows the value of $I$ at which $V_{\!{j}}$ becomes the numerical solution for $E = 7.0 \times 10^{2}$ MPa. When $I$ , obtained by integrating the full force history, equals $V_{\!{j}}/V_{{i}}$ , the corresponding $V_{\!{j}}$ is identified as the numerical solution.

Next, we consider the case $E = 8.1 \times 10^{-1}$ MPa. In the range up to approximately $V_{\!{j}} = 0.6$  m s⁻¹, as shown in figure 6 (Soft-1), $H_{{m}}/V_{\!{j}}$ is larger than $\tau _{\textit{impact}}$ , and therefore, the integration time window of $F$ is $[0, \tau _{\textit{impact}}]$ . In this regime, the entire force history is integrated, thus $I$ takes the same value as in the case $E = 7.0 \times 10^{2}$ MPa discussed above. However, for $V_{\!{j}}$ exceeding approximately 0.6 m s⁻¹, as shown in figure 6 (Soft-2) and (Soft-3), $H_{{m}}/V_{\!{j}}$ becomes smaller than $\tau _{\textit{impact}}$ , so that the integration time window of $F$ is reduced to $[0, H_{{m}}/V_{\!{j}}]$ . Because $F$ is then only partially integrated, the resulting $I$ decreases. Moreover, as seen by comparing figure 6 (Soft-2) and (Soft-3), the integration time window $[0,H_{{m}}/V_{\!{j}}]$ shrinks with increasing $V_{\!{j}}$ , leading to a reduction in $I$ as $V_{\!{j}}$ increases. Figure 6 (Soft-2) shows the value of $I$ at which $V_{\!{j}}$ becomes the numerical solution for $E = 8.1 \times 10^{-1}$ MPa. When $I$ , obtained from the partial integration of $F$ , equals $V_{\!{j}}/V_{{i}}$ , the corresponding $V_{\!{j}}$ is identified as the numerical solution.

Finally, by examining the values of $I$ at which $V_{\!{j}}$ becomes the numerical solution for each value of $E$ , as shown in figure 6 (Rigid-1), (Soft-2) and (Soft-4), we find that $I$ represents either the total impulse or a partial impulse depending on the relative magnitudes of $ H_{{m}}/V_{\!{j}}$ and $\tau _{\textit{impact}}$ . Thus by treating $V_{\!{j}}$ as an unknown variable and determining the integration time window such that $V_{\!{j}}/V_{{i}}$ equals $I$ for each value of $E$ , the numerical solution of (3.11) can be obtained.

3.3.4. Mechanism of jet velocity reduction

We validate (3.11) by comparing its numerical solutions with the experimental results. The red solid line in figure 3(b) represents the results obtained by numerically solving (3.11) using an explicit Runge–Kutta method combined with trapezoidal integration. Equation (3.11) captures both the small variation of $V_{\!{j}}/V_{{i}}$ in the rigid-impact regime and its decrease in the soft-impact regime. This shows that $I$ is well modelled. In addition, the black solid line in figure 3(b) represents (3.13), the soft-limit asymptote of (3.11). Since the $-1/4$ scaling is obtained in the soft-impact limit for sufficiently large $\textit{Ca}$ , the slopes obtained from the experimental data and the partial-impulse model in the range $10^{-4}\lesssim Ca \lesssim 10^{-2}$ do not exactly match this scaling. However, figure 3(b) shows that in the limit $\textit{Ca} \gg 1$ , (3.11) asymptotically approaches the slope $-1/4$ .

Here, we discuss the mechanism of the decrease in $V_{\!{j}}/V_{{i}}$ in the soft-impact regime by focusing on $I$ , as calculated from (3.11). The shaded regions in figure 5(b) represent $I$ for each $E$ obtained from (3.11) with $H = 20$  mm and $V_{{i}} = 0.63$  m s⁻¹. In the rigid-impact regime ( $E = 7.0 \times 10^{2}$ – $2.0 \times 10^{5}$ MPa), the total impulse transferred from the substrate to the container is used to drive the concave gas–liquid interface. In contrast, in the soft-impact regime ( $E = 8.1 \times 10^{-1}$ – $1.9 \times 10^{1}$ MPa), only the partial impulse is used to drive the concave gas–liquid interface compared with the rigid-impact regime. In the soft-impact regime, the impact interval becomes longer than the focusing interval, so the effective impulse time window that accelerates the fluid into a jet becomes relatively short. As a result, fluid motion is driven by only the partial impulse compared with the rigid-impact regime, leading to a decrease in $V_{\!{j}}/V_{{i}}$ . This is the mechanism responsible for the reduction of $V_{\!{j}}/V_{{i}}$ in the soft-impact regime.

In this study, we have introduced a new framework in which only the portion of the total impulse exerted by the substrate on the container that is actually transmitted to the fluid within the time window effective for jet formation is assumed to drive the jet. This framework enables a unified description of impact-driven jet phenomena across a wide range of substrate stiffness, including both rigid and soft substrates, which have been difficult to treat within conventional pressure-impulse theories.

Finally, the present results suggest a broader perspective on impact-induced interfacial flows involving compliant or deformable boundaries. Rapid acceleration of liquids near boundaries appears in a wide range of short-time impact phenomena, including wave impact and ship slamming (Cooker & Peregrine Reference Cooker and Peregrine1995; Peregrine Reference Peregrine2003; Kapsenberg Reference Kapsenberg2011; Dias & Ghidaglia Reference Dias and Ghidaglia2018), water entry of solid objects (Thoroddsen et al. Reference Thoroddsen, Etoh, Takehara and Takano2004; Grumstrup et al. Reference Grumstrup, Keller and Belmonte2007; Truscott et al. Reference Truscott, Epps and Belden2014; Rabbi et al. Reference Rabbi, Mortensen, Kiyama and Truscott2024), and droplet impact on solid surfaces (Josserand & Thoroddsen Reference Josserand and Thoroddsen2016; Josserand et al. Reference Josserand, Ray and Zaleski2016; Philippi et al. Reference Philippi, Lagrée and Antkowiak2016, Reference Philippi, Antkowiak and Lagrée2018). Although the specific asymptotic exponent obtained in this study depends on the contact-force law and hence on the impact geometry and substrate mechanics, the partial-impulse viewpoint provides a way to connect contact dynamics with subsequent interfacial focusing. This perspective may therefore be useful for interpreting other impact-driven flows in which the effective time window of momentum transfer is limited by the response of a deformable boundary.