Impact-induced viscoelastic bungee-jumper jets with uniform extension and stress

This study reveals that impact-induced viscoelastic "bungee-jumper" jets exhibit uniform extensional rates and stress distributions despite extreme Deborah and Reynolds numbers, indicating that their complex dynamics can be effectively modeled using spatially uniform constitutive equations, with the Voigt model providing the best agreement.

The Gist

Imagine you are holding a glass syringe filled with a thick, gooey liquid (like a very runny honey mixed with plastic). If you suddenly slam the bottom of the syringe, a thin stream of this liquid shoots out the top.

In a normal liquid like water, this stream would shoot out, get thinner, break apart into droplets, and fly away. But in this experiment, the liquid behaves like a bungee jumper. It shoots up, stretches out to its limit, and then—instead of flying off—it snaps back down toward the syringe, just like a rubber band being pulled and released.

The scientists wanted to understand why this happens and what is happening inside the liquid while it stretches and snaps back. They used high-speed cameras and special light techniques to "see" inside the moving stream.

Here is what they found, explained simply:

1. The "Uniform" Surprise

Usually, when you stretch something complex and fast (like a rubber band being pulled at incredible speed), you expect the stretching to be messy. You might think the top stretches one way and the bottom another, or that the tension is high in some spots and low in others.

However, the researchers discovered something surprising: The entire stream acts like a single, perfect unit.

• Uniform Stretching: Every part of the jet stretches at the exact same speed. It's as if the whole stream is made of a single, perfectly elastic rope.
• Uniform Tension: The "pulling force" (stress) inside the liquid is the same from the bottom to the top. There are no weak spots or tight spots; the tension is evenly distributed.

Even though the liquid is moving incredibly fast and is in a chaotic state, it behaves with a simple, orderly rhythm.

2. The "Bungee" Models

To explain this behavior, the scientists tried fitting the data into different mathematical "toy models" (like trying to describe a car's movement using different physics equations).

• The "Single Spring" Model: Imagine the jet is just a perfect, bouncy spring with no friction. This model worked well for the stickiest, most elastic liquids (the ones that snapped back the strongest). However, it failed for the less sticky liquids because it ignored the "drag" or friction inside the fluid.
• The "Voigt" Model (The Winner): This model is like a spring attached to a shock absorber (a dashpot). It accounts for both the bounciness (elasticity) and the drag (viscosity) of the liquid.
  • The scientists found that this "spring plus shock absorber" model perfectly described the movement of all the liquids they tested, from the less sticky ones to the super-sticky ones.
  • Because the stretching and tension were uniform, they could treat the entire messy, high-speed jet as a single, simple object with uniform properties.

3. Why This Matters (According to the Paper)

The paper explains that this "bungee-jumper" behavior is a rare way to study how thick, elastic liquids react when they are stretched extremely fast. Usually, our standard tools can't measure these extreme conditions.

By proving that these complex, high-speed jets actually follow simple rules (uniform stretching and uniform tension), the researchers showed that we don't need incredibly complicated math to predict how they move. A simple model with uniform coefficients (like the Voigt model) is enough to capture the essence of the motion.

In short: Even though these liquid jets are shooting out at high speeds and behaving in a chaotic, non-equilibrium way, they surprisingly organize themselves into a simple, uniform pattern that can be described by a basic "spring and shock absorber" equation.

Technical Summary

Technical Summary: Impact-Induced Viscoelastic Bungee-Jumper Jets with Uniform Extension and Stress

Problem Statement
The study addresses the dynamics of "bungee-jumper" jets—liquid jets induced by an impulsive force that reach a peak extension and subsequently retract. While the extensional rheology of viscoelastic fluids is critical for next-generation printing technologies (e.g., 3D printing, bio-printing, and food printing), these specific retraction-dominant behaviors occur under strongly extensional, highly nonequilibrium conditions that are difficult to characterize using conventional rheometric tools. Previous research has largely focused on fluids where viscoelastic effects are minor; however, there is a need to understand regimes where viscoelasticity is dominant, particularly to elucidate how complex jet dynamics can be represented by constitutive models.

Methodology
The authors employed a focused liquid jet technique using a glass syringe to generate impact-induced jets from dilute polyethylene oxide (PEO) solutions.

• Experimental Setup: A glass syringe with a nozzle radius of 0.6 mm was subjected to an impulsive acceleration, generating jets with initial velocities ($U'$) of approximately 5 m/s.
• Fluids: Three PEO solutions with different molecular weights (2M, 5M, and 8M) at a concentration of 1.0 wt.% were tested.
• Flow Regime: The experiments covered a wide range of dimensionless numbers, with Deborah numbers ($De$) spanning $2.1 \times 10^1$ to $3.3 \times 10^3$, Reynolds numbers ($Re$) from $2.8 \times 10^1$ to $4.6 \times 10^2$, and capillary numbers ($Ca$) from 1.2 to 5.8.
• Measurement Techniques:
  • High-speed velocimetry: A high-speed camera (30,000 fps) tracked the displacement of segmented volume elements to determine the spatial distribution of velocity.
  • Polarization-based stress imaging: A high-speed polarization camera (20,000 fps) measured phase retardation in a birefringent probe (cellulose nanocrystals dissolved in PEO 5M). Using the stress-optic law, the authors calculated the path-integrated normal stress and corrected for jet diameter to determine birefringence ($\Delta/D_{jet}$).

Key Contributions and Results
The primary contribution of this work is the experimental demonstration that, despite the extreme $De$ and highly nonequilibrium nature of the flow, bungee-jumper jets exhibit two distinct uniform characteristics:

1. Uniform Extensional Rate: High-speed velocimetry revealed that the axial velocity within the jet is linearly distributed as a function of position ($y$). This indicates a nearly uniform extensional rate throughout the jet's flight, a finding that is remarkable given the large Deborah numbers.
2. Uniform Extensional Stress: Polarization imaging showed that the corrected birefringence ($\Delta/D_{jet}$) is nearly constant along the length of the jet (excluding the very tip due to resolution limits). This implies that the extensional stress is spatially uniform during the jetting motion.

These findings suggest that the seemingly complex dynamics of these jets can be effectively represented by constitutive models with spatially uniform coefficients. The authors compared experimental data against four viscoelastic models:

• Voigt Model: This model provided the best agreement with the measured dynamics across all three PEO solutions (2M, 5M, and 8M). The equation of motion, $d^2x/dt^2 + (c/m)dx/dt + (G/m)x = 0$, successfully captured both weak- and strong-elasticity cases.
• Single-Spring Model: This model (representing a perfectly elastic body) only captured the behavior of the 5M and 8M solutions where elasticity dominates. It failed to describe the 2M solution, which requires the inclusion of viscous dissipation.
• Maxwell and FENE–CR Models: While these models could approximately predict the jet behavior, they were less accurate than the Voigt model.

Significance
The paper claims that the simultaneous uniformity of extensional rate and stress allows for a "reduced description" of the jet dynamics using simple constitutive representations. Specifically, the study demonstrates that a relatively simple Voigt element with uniform coefficients can capture the essential viscoelastic behavior of impact-induced jets in extreme extensional regimes. This finding challenges the assumption that such complex, high-strain-rate flows require spatially varying or highly complex constitutive descriptions, offering a simplified framework for modeling viscoelastic jet dynamics in applications like high-speed printing and material processing.