On the deformation of a shear thinning viscoelastic drop in a steady electric field

This study utilizes numerical simulations to characterize the deformation and breakup dynamics of shear-thinning viscoelastic drops in steady electric fields, revealing that while behavior in certain parameter regions resembles Newtonian fluids, others exhibit complex non-monotonic responses to elasticity and distinct shape transitions like multi-lobed or conical formations depending on conductivity and permittivity ratios.

The Gist

Imagine you have a drop of gooey, stretchy fluid—like a mix of honey and rubber bands—sitting in a bath of water. Now, imagine turning on a powerful electric field around it. What happens? Does it stretch out like a piece of taffy? Does it flatten like a pancake? Or does it snap apart into tiny droplets?

This paper is a deep dive into exactly that question, but with a specific focus on viscoelastic drops (fluids that act like both a liquid and a solid) and how they behave when they are shear-thinning (they get thinner and flow easier when you push them hard).

Here is the story of their findings, explained simply.

The Cast of Characters

1. The Drop: Think of this as a "smart" drop of polymer solution (like a thick shampoo or melted plastic). It has two personalities:
   • The Liquid Part: It flows like water.
   • The Rubber Band Part: It has memory. If you stretch it, it wants to snap back.
   • The "Shear-Thinning" Trick: If you push it fast, it gets "lazy" and flows more easily, like ketchup that only comes out when you shake the bottle hard.
2. The Electric Field: Imagine invisible hands pushing on the drop from the top and bottom.
3. The Old Rival (Oldroyd-B): In previous studies, scientists used a simpler model for these drops. Think of the "Old Rival" as a drop made of infinite rubber bands. If you pull it too hard, the rubber bands stretch forever and the math breaks down. The new model in this paper (LPTT) is more realistic: the rubber bands have a limit. They can only stretch so far before they stop getting stiffer.

The Experiment: A Map of Possibilities

The researchers didn't just look at one type of drop. They looked at drops with different electrical properties (how well they conduct electricity vs. how they store electric charge). They mapped these properties onto a "phase map" with six different zones.

Think of these zones as different weather patterns for the drop:

• Some zones are calm: The drop just stretches a little and stays happy.
• Some zones are stormy: The drop stretches, gets weird shapes, and might explode.

The Big Discoveries

1. The "Stretchy" Effect (Deborah Number)

The researchers used a number called the Deborah number (De) to measure how "rubbery" the drop is.

• Low De: The drop acts mostly like a liquid.
• High De: The drop acts mostly like a solid rubber band.

The Surprise: In many cases, making the drop more rubbery (increasing De) actually helps it resist breaking. It's like a strong rubber band holding a fragile balloon together. The electric field tries to tear it apart, but the rubbery nature of the drop fights back, allowing it to survive stronger electric fields than a simple liquid drop would.

2. The "Goldilocks" Zone (Non-Monotonic Behavior)

This is the most fascinating part. In some specific electrical conditions, the drop doesn't just get "more rubbery = more stable." It gets complicated.

Imagine you are trying to stretch a piece of taffy.

• At first, as you make the taffy slightly more elastic, it stretches more easily because it gets "thinner" (shear-thinning) under the stress.
• But then, if you make it too elastic, it becomes a stiff rubber band that refuses to stretch.

The researchers found that the drop's shape follows this "Goldilocks" rule. Sometimes, increasing the elasticity makes the drop deform more at first, and then less as it gets even more elastic. It's a tug-of-war between the drop getting "thinner" to flow and getting "stiffer" to resist.

3. The "Pointy" vs. "Lobed" Shapes

Depending on the electrical settings, the drop changes its outfit:

• The Sausage: It stretches into a long, smooth oval.
• The Multi-Lobed Monster: If the electric field is strong, the drop might pinch in the middle and form two or three distinct blobs (like a dumbbell or a peanut).
• The Pointy Cone: In some zones, the ends of the drop sharpen into tiny needles (cones) before breaking.

The Difference: The "Old Rival" (Oldroyd-B) drops would often snap into these weird shapes and stay there. The new "Realistic" (LPTT) drops were smarter. They would sometimes form a weird shape, realize it was unstable, and snap back into a smooth oval, or they would delay the explosion longer.

4. The "Explosion" Threshold

Every drop has a breaking point.

• Simple drops break easily.
• Rubbery drops can handle a much stronger electric field before breaking.
• The Twist: In some cases, if the drop is too rubbery, it actually breaks sooner than a slightly less rubbery one because of that "Goldilocks" effect mentioned earlier.

Why Does This Matter?

You might wonder, "Who cares about electric drops?"

Actually, this is huge for real life:

• Inkjet Printers: Making sure the ink drops fly straight and don't splatter.
• Oil & Water Separation: Using electricity to separate oil from water in industrial plants.
• Medicine: Creating tiny droplets of medicine for inhalers or targeted drug delivery.
• 3D Printing: Controlling how plastic flows when it's being printed.

The Bottom Line

This paper tells us that when dealing with "smart" fluids (like polymers) in electric fields, you can't just use simple math. You have to account for the fact that these fluids get thinner when pushed and have a limit to how much they can stretch.

The main takeaway: Adding elasticity to a drop usually makes it stronger and harder to break, but it's a delicate balance. Too much elasticity can sometimes make the drop behave unpredictably, stretching in weird ways before finally giving up. Understanding this helps engineers design better printers, cleaner oil, and more precise medical tools.

Technical Summary

Here is a detailed technical summary of the paper "On the deformation of a shear thinning viscoelastic drop in a steady electric field."

1. Problem Statement

The study investigates the electrohydrodynamic (EHD) deformation and breakup dynamics of a viscoelastic drop suspended in a Newtonian fluid under a uniform, steady electric field. While previous research has extensively covered Newtonian drops and Oldroyd-B viscoelastic drops, these models often fail to capture realistic fluid behaviors such as shear-thinning and finite polymer extensibility. The Oldroyd-B model, for instance, assumes infinitely extensible polymer chains, leading to unphysical stress divergence at high extension rates.

This paper addresses these limitations by modeling the drop using the Linear Phan-Thien–Tanner (LPTT) constitutive equation. The LPTT model incorporates:

• Shear-thinning behavior: Viscosity decreases with shear rate.
• Finite extensibility: Polymer chains have a maximum stretch limit, preventing unphysical stress divergence.

The study aims to map the deformation regimes (prolate, oblate, multi-lobed, breakup) across the $(\sigma_r, \epsilon_r)$ phase space (conductivity and permittivity ratios) and analyze the influence of the Deborah number ($De$) and Electric Capillary number ($Ca_E$) on drop stability and morphology.

2. Methodology

• Governing Equations: The flow is governed by the incompressible Navier-Stokes equations coupled with the LPTT constitutive equation for the drop phase and the Maxwell stress tensor for the electric body force. The electric field is modeled using Gauss's and Faraday's laws, accounting for charge convection.
• Numerical Solver: Simulations were performed using Basilisk, an open-source solver.
  • Interface Tracking: Geometric Volume of Fluid (VOF) method.
  • Viscoelasticity: Solved using the log-conformation tensor approach (Fattal and Kupferman) to ensure numerical stability at high $De$.
  • Electric Field: Solved via a coupled scheme for electric potential and charge conservation (López-Herrera et al.).
• Validation: The code was validated against lid-driven cavity flow simulations for LPTT fluids, showing excellent agreement with literature (Dalal et al., 2016).
• Parameter Space:
  • Fixed parameters: Density ratio ($\rho_r=1$), Viscosity ratio ($\mu_r=1$), Reynolds number ($Re=1$), Extensibility parameter ($\epsilon=0.25$), Solvent viscosity ratio ($\beta_i=1/9$).
  • Variable parameters: Conductivity ratio ($\sigma_r$), Permittivity ratio ($\epsilon_r$), $Ca_E$, and $De$.
  • Six characteristic regions of the $(\sigma_r, \epsilon_r)$ phase space were selected: $PR^+_A, PR^+_B, PR^-_A, PR^-_B, OB^+, OB^-$.

3. Key Contributions

1. First Systematic Study of LPTT Drops in EHD: This work extends the understanding of EHD deformation from Oldroyd-B fluids to the more physically realistic LPTT model, highlighting the critical role of finite extensibility and shear-thinning.
2. Phase Map Analysis: The study categorizes drop behavior across six distinct regions of the conductivity-permittivity phase space, identifying how viscoelasticity alters stability boundaries compared to Newtonian and Oldroyd-B fluids.
3. Non-Monotonic Deformation Trends: The paper reveals complex, non-monotonic dependencies of deformation and breakup time on the Deborah number in specific regimes ($PR^+_B$ and $OB^-$), a phenomenon not captured by simpler models.
4. Comparative Dynamics: A direct comparison between LPTT and Oldroyd-B drops demonstrates that finite extensibility significantly delays breakup and alters steady-state shapes, particularly at high electric stresses.

4. Key Results

A. Regions with Negligible Viscoelastic Effects ($PR^-_A, PR^-_B, OB^+$)

In these regions, the first- and second-order deformation coefficients have the same sign. The LPTT drops exhibit deformation dynamics that negligibly deviate from Newtonian behavior. Stable shapes are attained even at high $Ca_E$, and viscoelasticity has minimal impact on the critical breakup threshold.

B. Region $PR^+_A$ (Prolate Deformation & Breakup)

• Behavior: Drops deform into prolate spheroids below a critical $Ca_E$ and transition to multi-lobed shapes or breakup above it.
• Viscoelastic Effect: Increasing $De$ opposes deformation, raising the critical $Ca_E$ required for breakup.
• Breakup Morphology:
  • Low $De$: Tendency for multi-lobed breakup (e.g., 3-lobed).
  • High $De$: Breakup transitions to fewer lobes (e.g., 2-lobed) but occurs at delayed times.
  • Comparison with Oldroyd-B: LPTT drops are more stable. While Oldroyd-B drops form stable multi-lobed shapes at high $De$, LPTT drops tend to break up at high $De$ unless the field is moderate.

C. Region $PR^+_B$ (Pointed Ends & Conical Shapes)

• Behavior: Drops form prolate shapes that develop conical (pointed) ends beyond a critical $Ca_E$.
• Non-Monotonic Deformation: The steady-state deformation exhibits a non-monotonic dependence on $De$:
  • Deformation initially increases with $De$ (due to strain hardening).
  • Deformation decreases at higher $De$ (due to stress saturation/finite extensibility).
• Critical $Ca_E$: The threshold for forming pointed ends also varies non-monotonically with $De$.
• Comparison with Oldroyd-B: LPTT drops show larger deformation than Oldroyd-B drops at intermediate to high $De$. Oldroyd-B drops show monotonic decrease in deformation with $De$, whereas LPTT drops show the non-monotonic trend described above.

D. Region $OB^-$ (Oblate Deformation & Breakup)

• Behavior: Drops deform into oblate spheroids below critical $Ca_E$ and breakup above it.
• Non-Monotonic Deformation: Similar to $PR^+_B$, the magnitude of oblate deformation increases initially with $De$ (strain hardening) and decreases at higher $De$ (stress saturation).
• Stability: LPTT drops are significantly more stable than Oldroyd-B drops.
  • LPTT drops do not breakup up to higher $Ca_E$ values at high $De$.
  • Oldroyd-B drops breakup earlier and exhibit oscillatory behaviors or dimple formation before recovery.
• Breakup Time: For LPTT drops, breakup time generally increases with $De$ (retardation), whereas Oldroyd-B drops breakup faster at high elasticity.

5. Significance and Conclusion

This study highlights the complex interplay between viscoelasticity and electric stresses. The inclusion of finite extensibility (LPTT model) fundamentally changes the prediction of drop stability compared to the Oldroyd-B model:

• Stability: LPTT drops generally exhibit greater resistance to breakup at high Deborah numbers due to stress saturation, whereas Oldroyd-B models often predict earlier breakup or unphysical divergence.
• Morphology: The transition from stable shapes to breakup or pointed ends is highly sensitive to the specific viscoelastic model used.
• Applications: These findings are crucial for the controlled manipulation of complex fluids (e.g., polymer solutions, biological fluids) in microfluidics, inkjet printing, and electrospinning, where accurate prediction of breakup thresholds and drop shapes is essential for process optimization.

The work establishes that assuming infinite extensibility (Oldroyd-B) can lead to significant errors in predicting the stability limits and deformation magnitudes of viscoelastic drops in strong electric fields.