Role of particle volume fraction on particulate suspension droplet evolution, transition and Hysteresis

This study investigates how increasing the particle volume fraction in non-Brownian particulate suspension droplets alters the critical flow rates for dripping-to-jetting transitions, induces a chaotic intermediate regime, widens the hysteresis loop, and reduces both droplet frequency and size distribution.

The Gist

Imagine you are holding a bottle of thick, honey-like liquid. If you tip it slowly, big, heavy drops form and fall one by one. This is the dripping mode. If you squeeze the bottle harder, the liquid shoots out in a long, steady stream that eventually breaks into smaller drops further down. This is the jetting mode.

Scientists have known for a long time how pure liquids (like water or oil) behave in these two modes. But what happens if you mix tiny solid particles into that liquid? Think of adding sand to water, or chocolate chips to melted chocolate. This is what the researchers in this paper studied. They wanted to see how the amount of "stuff" (particles) inside the liquid changes the way it drips or jets.

Here is a simple breakdown of their findings:

1. The "Memory" of the Liquid (Hysteresis)

One of the coolest things they found is that the liquid has a sort of memory.

• The Experiment: They slowly turned up the flow rate (like slowly squeezing the bottle harder) and watched when it switched from dripping to jetting. Then, they slowly turned the flow rate back down and watched when it switched back to dripping.
• The Surprise: The liquid didn't switch back at the exact same point where it switched forward. It was "stubborn." It wanted to stay in the jetting mode even when the flow slowed down, or stay in the dripping mode even when the flow sped up.
• The Particle Effect: When they added more particles, this "stubbornness" (called hysteresis) got much stronger. It was like the liquid had a wider "gray area" where it couldn't decide whether to drip or jet. The more particles you added, the wider this confusing zone became.

2. The Traffic Jam at the Neck

Imagine a drop of liquid hanging from a faucet. Just before it falls, a thin "neck" of liquid forms.

• In pure liquid: The neck gets thinner and thinner until it snaps cleanly.
• With particles: As the neck gets thin, the particles inside start to crowd together. It's like a traffic jam on a narrowing bridge. The particles can't fit through the tiny gap, so they clog the neck.
• The Result: This clogging changes how the drop breaks. Sometimes it snaps early; sometimes it stretches out weirdly. The particles act like speed bumps, disrupting the smooth flow of the liquid.

3. The "Escape Artist" Mechanism

The researchers noticed something funny happening right before a drop falls.

• Sometimes, the tip of the liquid stretches out, forms a bulge, and then pulls back (recoils) instead of snapping off immediately. It's like a rubber band being stretched and then snapping back before it breaks.
• When this happens, the particles inside rush toward the bulge, like people running toward a stage door. This changes the concentration of particles in the final drop.
• With more particles, this "escape and pull back" happens more often, making the drops behave less predictably.

4. Mixing Up the Drop Sizes

In a normal dripping faucet, the drops are usually all roughly the same size. In a jetting stream, they are also uniform but smaller.

• The Finding: When you add particles, the difference between the "big dripping drops" and the "small jetting drops" starts to disappear.
• The Analogy: Imagine a choir where the basses are very deep and the sopranos are very high. If you add a crowd of people in the middle (the particles), the distinct difference between the deep and high voices gets blurred. The drops become more uniform in size, regardless of whether they are dripping or jetting.

The Big Picture

This study is important because particle-laden liquids are everywhere:

• Inkjet printers (ink with pigments).
• Food industry (chocolate, sauces with chunks).
• Medicine (drug suspensions).
• Rocket fuel (solid particles mixed with fuel).

The scientists learned that if you want to control how these liquids break into drops (for example, to make perfect spray nozzles or uniform medicine droplets), you can't just look at the liquid. You have to account for the particles. Adding more particles makes the liquid "stubborn" (wider hysteresis), causes traffic jams at the breaking point, and makes the drop sizes more uniform.

In short: Particles turn a smooth, predictable liquid flow into a chaotic, memory-having, traffic-jam-prone system that behaves very differently depending on whether you are speeding up or slowing down the flow.

Technical Summary

1. Problem Statement

The study investigates the transitional dynamics of non-Brownian particulate Newtonian liquid jets, specifically focusing on how the particle volume fraction ($\phi$) influences the critical flow rates governing the transition between dripping and jetting regimes. While the behavior of pure Newtonian fluids in these regimes is well-established, the addition of particles introduces complex micro-structural interactions, effective viscosity changes, and particle stratification that alter breakup dynamics.

Key gaps addressed include:

• The specific impact of particle volume fraction on the critical inflow velocity where the dripping-jetting transition occurs.
• The nature of hysteresis (path dependence) in the transition when flow rates are increased (forward sweep) versus decreased (reverse sweep).
• The evolution of droplet size distribution and the underlying mechanisms (e.g., tip recoiling, pinch-off escape) during these transitions.

2. Methodology

Experimental Setup:

• Fluid System: Neutrally buoyant suspensions created using an aqueous 22% glycerol solution as the background fluid and solid particles.
• Geometry: A hypodermic needle with a nozzle diameter ($D_n$) to particle diameter ($D_p$) ratio of 20 ($D_n/D_p = 20$). This ratio was chosen to prevent nozzle clogging while allowing high volume fractions.
• Volume Fractions ($\phi$): Experiments were conducted across a range of volume fractions: 0% (pure fluid), 2%, 5%, 10%, 15%, 20%, 25%, 30%, and 35%.
• Flow Control: A syringe pump (New Era Pump Systems) precisely controlled the flow rate. The dimensionless inflow velocity ($U_m$, related to the Weber number) was varied between 0.409 and 0.927.
• Sweep Protocol: To capture hysteresis, experiments were performed in two directions:
  • Forward Sweep: Increasing flow rate from $U_m = 0.409$ to $0.927$.
  • Reverse Sweep: Decreasing flow rate from $U_m = 0.927$ to $0.409$.
• Imaging & Analysis: High-speed shadowgraphy (Photron Fastcam Nova SA9) captured droplet evolution. Image processing (Canny edge detection, dilatation) was used to track:
  • Jet tip location ($L$).
  • Pinch-off length ($L_b$).
  • Droplet diameter ($D_d$).
• Dimensionless Parameters: The study utilized the Kapitza number ($Ka$), Weber number ($We$), Bond number ($Bo$), and Ohnesorge number ($Oh$) to characterize the flow.

3. Key Contributions

1. Quantification of Hysteresis Widening: The paper demonstrates that increasing particle volume fraction significantly widens the hysteresis loop between dripping and jetting regimes.
2. Identification of Chaotic Transition Regimes: It reveals that for higher volume fractions, the transition from dripping to jetting does not occur abruptly but passes through a chaotic dripping regime characterized by recurrent escape of the pinch-off mechanism.
3. Mechanism of Tip Recoiling and Particle Stratification: The study details how tip recoil and capillary wave propagation induce particle stratification, leading to local volume fraction variations that alter effective viscosity and breakup dynamics.
4. Droplet Size Distribution Convergence: It establishes that as $\phi$ increases, the distinct difference in droplet sizes between dripping and jetting regimes diminishes, leading to a more uniform size distribution.

4. Key Results

A. Transition Dynamics and Hysteresis

• Forward vs. Reverse Sweep:
  • Pure Fluid ($\phi=0$): Exhibits a standard hysteresis loop where the transition from dripping to jetting occurs at a higher velocity than the reverse transition (jetting to dripping).
  • With Particles: As $\phi$ increases, the dripping-to-jetting transition occurs at lower flow rates (earlier) during the forward sweep. Conversely, the jetting-to-dripping transition is delayed to lower flow rates during the reverse sweep.
  • Result: This asymmetry causes the hysteresis loop to widen significantly with increasing particle loading.
• Chaotic Regime: At high volume fractions (e.g., $\phi = 30\%$), the transition is not a simple jump but involves a chaotic dripping phase. Poincaré plots of pinch-off lengths show a "U-shaped" pattern indicative of chaos, similar to period-doubling routes observed in Newtonian fluids but induced here by particle interactions.
• Stabilization: Higher particle fractions stabilize the fluid column, reducing the jet length during the dripping-to-jetting transition due to increased effective viscosity.

B. Pinch-off Mechanisms and Tip Evolution

• Tip Recoiling and Bulging: Upon droplet separation, the jet tip retracts, forming a bulge. In particle-laden jets, this process involves:
  • Capillary Wave Propagation: Waves travel upstream, creating relative velocities between fluid and particles.
  • Particle Stratification: Particles migrate toward the tip bulge, increasing the local volume fraction and effective viscosity in that region.
  • Escape of Pinch-off: The tip oscillation can cause a "recurrent escape" of the pinch-off mechanism, where the neck elongates rather than breaking immediately. This leads to the formation of long particulate filaments.
• Clogging and Early Breakup: At very high volume fractions ($\phi \geq 30\%$), particles can clog the neck region, leading to earlier breakup and interface corrugation, contrasting with the smooth breakup of pure fluids.

C. Droplet Size Distribution

• Frequency and Size: As $\phi$ increases, the frequency of droplet pinch-off decreases, and the average droplet size increases.
• Distribution Convergence: In pure fluids, there is a clear distinction in droplet size between dripping (larger, irregular) and jetting (smaller, uniform) regimes. As $\phi$ increases (specifically at 25% and 30%), this distinction narrows. The droplet size distribution in both regimes becomes more uniform and concentrated.
• Theoretical Insight: The authors propose a modified wave number equation (Eq. 3.2) incorporating the Maron-Pierce relation for effective viscosity. This suggests that higher local viscosity (due to particle accumulation) increases the Ohnesorge number, damping capillary waves and stabilizing the jet tip, thereby reducing the variability in droplet sizes.

5. Significance

This research provides critical insights for industries dealing with particle-laden fluids, such as pharmaceuticals, food processing, and propellant manufacturing.

• Process Control: Understanding the widened hysteresis loop is vital for designing microfluidic devices and spray nozzles where precise control over regime transitions (dripping vs. jetting) is required. The presence of particles makes the system more "memory-dependent," requiring careful management of flow rate history to avoid unintended regime shifts.
• Droplet Uniformity: The finding that high volume fractions lead to more uniform droplet sizes in both regimes suggests that particle loading can be used as a tuning parameter to achieve specific droplet size distributions in spray applications.
• Fundamental Physics: The study bridges the gap between macroscopic fluid dynamics and microscopic particle interactions, highlighting how particle stratification and effective viscosity modulation drive complex non-linear behaviors like chaos and hysteresis in suspension jets.