Interfacial dynamics induced by impacts across rigid and soft substrates

This study establishes the Cauchy number as a unifying dimensionless parameter that defines the transition between rigid and soft impact regimes in gas-liquid interfacial dynamics, demonstrating that a "partial impulse" framework accurately predicts jet velocity reductions in soft impacts by accounting for the mismatch between contact duration and jet formation time.

The Gist

Imagine you have a test tube filled with a thick, clear liquid (like silicone oil). At the top of the tube, the liquid curves inward, forming a shallow "bowl" shape. Now, imagine dropping this tube onto a surface.

The Classic Scenario: The Hard Floor
If you drop this tube onto a hard floor (like steel or concrete), something dramatic happens. The moment the tube hits the ground, it stops dead in its tracks. Because the liquid inside is still moving downward, it crashes into the bottom of the tube and is forced upward. Because the surface of the liquid was already curved like a bowl, all that upward energy gets focused into a single, tiny point right in the center. The result? A super-fast, needle-thin jet of liquid shoots straight up out of the tube, like a tiny fountain.

Scientists have known for a long time that on hard floors, this happens almost instantly. The floor is so stiff that it stops the tube in a fraction of a millisecond, and the liquid jet forms a split second after the tube has already stopped bouncing.

The New Discovery: The Soft Floor
This paper asks a simple question: What happens if you drop the tube onto a soft surface, like a rubber mat or a sponge?

The researchers dropped the tube onto nine different surfaces, ranging from hard steel to very soft rubber and silicone. They found that as the surface got softer, the liquid jet didn't shoot up as fast. In fact, on the softest surfaces, the jet was much slower and took longer to form.

The "Timing" Analogy: The Runner and the Finish Line
To understand why this happens, the authors use a clever analogy involving timing. They identified two critical moments in the drop:

1. The "Contact Time" (Impact Interval): How long the tube stays touching the floor before bouncing away.
2. The "Jet Formation Time" (Focusing Interval): How long it takes for the liquid to gather its energy and shoot up as a jet.

• On a Hard Floor: The tube hits the floor and bounces off almost instantly. The "Contact Time" is very short. The liquid takes a little longer to form the jet. So, the tube is already bouncing away before the jet is ready. The liquid gets a massive, instant "kick" from the floor, and then it does its own thing to form the jet.
• On a Soft Floor: The floor is squishy. When the tube hits, it sinks in and stays in contact with the floor for a long time. The "Contact Time" is now longer than the time it takes for the jet to form.

The "Partial Impulse" Concept
Here is the big idea: The jet only gets the energy that is delivered to it while it is forming.

Think of it like a runner trying to cross a finish line.

• Hard Floor: The runner gets a huge burst of speed from a starting gun, and then sprints across the finish line. The "kick" is over before the sprint is done, but the runner has the full energy.
• Soft Floor: The runner is trying to sprint, but the starting gun is stuck in the "on" position, pushing them slowly for a long time. By the time the runner reaches the finish line (the moment the jet forms), the "push" from the floor hasn't finished yet. The floor is still squishing and holding the tube back.

Because the tube is still stuck to the soft floor when the jet tries to form, the liquid doesn't get the full "kick" it would have gotten on a hard floor. It only gets a "Partial Impulse"—a fraction of the total energy. The rest of the energy is still being absorbed by the squishy floor, which is still in contact with the tube.

The "Stiffness" Rule
The researchers created a simple rule (using a number called the Cauchy number) to predict when this happens.

• If the floor is stiff enough, the jet gets the full kick, and the speed is predictable.
• If the floor is soft enough (specifically, if the "squishiness" is high compared to the speed of the drop), the jet forms too early, while the floor is still holding it back. This causes the jet to slow down significantly.

In Summary
The paper explains that when you drop a liquid-filled container onto a soft surface, the jet of liquid shoots up slower not because the liquid is weaker, but because the timing is off. The soft floor holds the container down for too long. The jet forms while the container is still being "squeezed" by the floor, so it misses out on the full burst of energy it would have received on a hard floor. The researchers proved this by showing that if you account for this "partial kick," you can perfectly predict how fast the jet will go, whether the floor is steel or rubber.

Technical Summary

Technical Summary: Interfacial dynamics induced by impacts across rigid and soft substrates

Problem Statement
This study investigates the impact-induced gas–liquid interfacial dynamics, specifically the formation of focused jets from a concave interface within a liquid-filled container. While the "Pokrovski's experiment" (impact of a falling liquid-filled tube onto a rigid substrate) has been extensively modeled using pressure–impulse theory under the assumption of incompressibility and instantaneous impulse transmission, the behavior of such systems when impacting compliant (soft) substrates remains less understood. The central problem addressed is how substrate deformability alters the impulse transferred to the fluid interface and, consequently, the resulting jet velocity. Previous studies have noted that soft substrates increase contact time, but the specific mechanism by which this affects jet formation—particularly when the contact duration overlaps with the jet formation time—had not been quantitatively unified with rigid-substrate models.

Methodology
The authors conducted a series of drop experiments using a test tube filled with silicone oil (kinematic viscosity $10 \text{ mm}^2/\text{s}$) impacting nine different substrates. The substrates spanned a wide range of elastic moduli ($E$) from $8.1 \times 10^{-1}$ MPa (PDMS elastomers) to $2.0 \times 10^5$ MPa (steel), covering nearly six orders of magnitude.

• Experimental Setup: The impactor (test tube with attached hook) was released via an electromagnet to impact the substrate at velocities ($V_i$) between $0.63$ and $1.1$ m/s. High-speed imaging (10,000 fps) captured the interface deformation, jet formation, and contact duration.
• Key Variables: The study varied the substrate elastic modulus ($E$), impact velocity ($V_i$), and liquid filling height ($H$).
• Timescale Analysis: Two critical timescales were defined and measured: the impact interval ($\tau_{\text{impact}}$), the duration of contact between the container and substrate, and the focusing interval ($\tau_{\text{focusing}}$), the time required for the concave interface to deform and form a visible jet.
• Modeling: The authors developed a theoretical framework combining momentum conservation with an elastic foundation model to describe the contact force history. They introduced the concept of a "partial impulse," where the impulse driving the jet is limited to the time window up to $\tau_{\text{focusing}}$, rather than the total impulse over the full contact duration.

Key Results

1. Jet Velocity Reduction: Experimental results demonstrated that as the substrate elastic modulus ($E$) decreases, the dimensionless jet velocity ($V_j/V_i$) decreases significantly.
2. Cauchy Number Scaling: Through dimensional analysis, the authors showed that all experimental data collapse onto a single curve when $V_j/V_i$ is plotted against the Cauchy number ($Ca = \rho_e V_i^2 / E$), representing the ratio of inertial forces to elastic restoring forces.
3. Regime Identification: The data revealed two distinct regimes separated by a critical Cauchy number of approximately $10^{-4}$:
   • Rigid-impact regime ($Ca < 10^{-4}$): The jet velocity remains nearly constant and aligns with predictions from conventional pressure–impulse theory for rigid substrates. Here, $\tau_{\text{impact}} \ll \tau_{\text{focusing}}$.
   • Soft-impact regime ($Ca > 10^{-4}$): The jet velocity decreases markedly as $Ca$ increases. In this regime, $\tau_{\text{impact}} > \tau_{\text{focusing}}$, meaning the jet forms while the container is still in contact with the substrate.
4. Partial Impulse Mechanism: The reduction in jet velocity in the soft-impact regime is attributed to the "partial impulse" effect. Because the jet forms before the contact ends, the fluid interface is accelerated only by the impulse accumulated up to $\tau_{\text{focusing}}$, not the total impulse delivered over the entire contact duration.
5. Quantitative Agreement: A predictive model derived from the partial impulse framework, utilizing an elastic foundation model for contact force and solving the momentum equation over the effective time window, quantitatively reproduced the experimental jet velocities across both rigid and soft regimes.

Significance and Claims
The paper claims to provide the first quantitative criterion for defining "softness" in impact-driven interfacial flows using the Cauchy number. By identifying the transition at $Ca \approx 10^{-4}$, the study establishes a boundary where classical incompressible, rigid-body impact models fail.

The primary contribution is the introduction of a "partial impulse" framework. This concept unifies the dynamics of impact-driven jetting across both rigid and compliant substrates. The authors argue that when the impact interval exceeds the focusing interval, the assumption of total impulse transmission is invalid; instead, only the impulse delivered within the effective time window for jet formation contributes to interface acceleration. This framework extends the applicability of classical impulse-based models to soft substrates, offering a unified description of interfacial flows that accounts for the temporal overlap between contact mechanics and fluid focusing. The study does not propose new applications but rather provides a fundamental physical explanation for jet velocity reduction in compliant systems, refining the understanding of Pokrovski's experiment in broader material contexts.