A reduced model for droplet dynamics with interfacial viscosity

This paper extends the Maffettone-Minale model to incorporate interfacial shear and dilatational viscosities for describing droplet deformation in shear flow and validates the resulting extended model's accuracy against fully resolved numerical simulations across various capillary and Boussinesq numbers.

The Gist

Imagine a tiny drop of oil floating in a stream of water. If the water starts flowing faster, the drop gets squished and stretched, turning from a perfect sphere into an oval shape. Scientists have long tried to predict exactly how much this drop will stretch using math.

For decades, they used a famous "recipe" (called the Maffettone–Minale model) that works well for clean drops. But in the real world, drops often have a "skin" made of soap, proteins, or other molecules. This skin isn't just a boundary; it has its own thickness and stickiness, known as interfacial viscosity. Think of it like the drop wearing a sticky, stretchy sweater.

This paper introduces a new, upgraded recipe (the Extended Maffettone–Minale or EMM model) that accounts for this sticky sweater. Here is how the authors broke it down:

1. The Two Types of "Stickiness"

The authors realized that the drop's skin resists movement in two different ways, and they needed to measure both:

• Shear Viscosity (The "Rubber Band" effect): Imagine trying to slide your hand across the surface of the drop. If the skin is "shear viscous," it resists that sliding motion, like dragging your hand through honey.
• Dilatational Viscosity (The "Breathing" effect): Imagine the drop trying to expand or shrink its surface area (like a balloon inflating). If the skin is "dilatational viscous," it resists that stretching or shrinking, like a tight, stiff fabric that doesn't want to expand.

The paper uses special numbers (called Boussinesq numbers) to measure how strong these two resistances are compared to the thickness of the drop.

2. The New Recipe (The EMM Model)

The authors took the old, simple math recipe and added new ingredients to handle these two types of stickiness.

• The Goal: They wanted to know: How far can we stretch this new recipe before it stops working?
• The Method: They didn't just guess. They built a super-detailed computer simulation (like a high-definition movie of the drop) that solved every tiny physics rule from scratch. This served as the "truth."
• The Test: They ran the new EMM recipe alongside the super-detailed simulation. They compared the results to see if the simple recipe matched the complex movie.

3. What They Found

The results were surprising and specific:

• When the "Sweater" is Uniform: If the drop's skin resists sliding and stretching equally (a balanced sweater), the new recipe works incredibly well, even when the drop is being stretched quite a bit. It accurately predicts how fast the drop stretches and how long it takes to settle into its final shape.
• When the "Sweater" is Unbalanced: If the skin is very good at resisting sliding but bad at resisting stretching (or vice versa), the simple recipe starts to get a little fuzzy. It still works for gentle flows, but if the flow gets too strong, the recipe becomes less accurate.
• The "Slow-Down" Effect: The most interesting finding was about time. When the drop has both types of stickiness at the same time, it takes much longer to change shape. It's like the drop gets "stuck" in its own skin. The authors found that their new recipe captures this "slow-motion" effect perfectly.
• The Breaking Point: If the drop has almost no resistance to sliding (but high resistance to stretching), it gets stretched so much that it eventually rips apart. The new recipe correctly predicts that this happens earlier in these specific conditions.

4. The Bottom Line

The authors successfully created a simple, fast, and reliable tool to predict how drops with "sticky skins" behave in flowing liquids.

• Why it matters: It saves scientists from having to run massive, slow computer simulations for every single problem.
• The Catch: The tool is very accurate for small-to-medium stretches, especially when the skin's resistance is balanced. If the flow is extremely violent or the skin's resistance is very unbalanced, the tool starts to lose its precision, and you need the heavy-duty computer simulations instead.

In short, they upgraded the "drop calculator" to handle sticky skins, proving it works great for most everyday scenarios, while clearly marking the boundaries where it might need a little help.

Technical Summary

Technical Summary: A Reduced Model for Droplet Dynamics with Interfacial Viscosity

Problem Statement
Understanding droplet deformation in hydrodynamic flows is a central topic in rheology. While the classical Maffettone–Minale (MM) model provides a successful phenomenological description for clean droplets (those without interfacial viscosity) in shear flow, it does not account for the complexities introduced by interfacial viscosity. In many practical scenarios, such as when interfaces are coated with surfactants, polymers, or proteins, interfacial shear ($\mu_s$) and dilatational ($\mu_d$) viscosities become significant. These viscosities introduce extra friction that alters the balance of stresses between bulk fluids, influencing deformation, relaxation, and breakup thresholds. The challenge lies in developing a reduced-order model that incorporates these interfacial effects while retaining the computational efficiency and analytical tractability of the original MM framework.

Methodology
The authors propose an Extended Maffettone–Minale (EMM) model, which generalizes the original MM equation for the evolution of a morphological tensor $S$ (describing the droplet shape as an ellipsoid).

• Theoretical Extension: The EMM model incorporates interfacial viscosities through the Boussinesq numbers, $Bqs = \mu_s/\mu R$ and $Bqd = \mu_d/\mu R$. By referencing perturbative calculations for viscous interfaces (specifically Narsimhan's extension of Rallison's work), the authors derive generalized expressions for the model coefficients $f_1$ and $f_2$. These coefficients, which were previously functions only of the viscosity ratio $\lambda$ for clean droplets, now depend on $\lambda$, $Bqs$, and $Bqd$.
• Numerical Validation: To quantify the accuracy and range of validity of the EMM model, the authors perform a systematic comparison against Fully Resolved Simulations (FRS). The FRS utilize an immersed boundary–lattice Boltzmann method to solve the coupled dynamics of the inner and outer fluids and the deformable interface without assuming small deformations.
• Error Analysis: The study evaluates the mean relative error $\langle \epsilon \rangle_T$ between the time-dependent deformation $D(t)$ predicted by the EMM model and the FRS results across a broad parameter space defined by the capillary number ($Ca$) and the two Boussinesq numbers.

Key Contributions

1. Derivation of Generalized Coefficients: The paper provides explicit analytical expressions for the EMM coefficients $f_1^{(EMM)}$ and $f_2^{(EMM)}$ that account for both shear and dilatational interfacial viscosities. These expressions reduce to the original MM coefficients when interfacial viscosities are zero.
2. Characterization of Transient Dynamics: The study reveals that interfacial viscosity affects the transient deformation time non-linearly. While a droplet with predominantly shear or predominantly dilatational viscosity behaves similarly to a clean droplet regarding relaxation time, a droplet with comparable values of both viscosities experiences a dramatic slow-down in deformation dynamics.
3. Mapping the Domain of Validity: Through extensive numerical testing, the authors delineate the specific regions in the $(Ca, Bqs, Bqd)$ parameter space where the EMM model remains quantitatively accurate.

Results

• Steady-State Deformation: The EMM model accurately predicts the steady-state deformation $D_s$ for a wide range of parameters. The model captures the counter-intuitive trend where increasing dilatational viscosity (with zero shear viscosity) slightly increases deformation, whereas increasing shear viscosity (or having equal shear and dilatational viscosities) reduces deformation.
• Transient Behavior: The model successfully reproduces the transition from monotonic relaxation to oscillatory responses observed in FRS, particularly when $Bqs = Bqd$. The characteristic deformation time increases sharply only when both interfacial viscosities are appreciable.
• Accuracy Limits:
  • For small interfacial viscosities ($Bqs, Bqd \ll 1$) and small capillary numbers ($Ca \le 0.2$), the EMM model reproduces FRS data with errors typically below a few percent.
  • The model's accuracy degrades more rapidly in strongly asymmetric viscosity configurations (where one Boussinesq number dominates) compared to the equiviscous case ($Bqs = Bqd$).
  • As $Ca$ increases (e.g., $Ca = 0.5$), nonlinear couplings in the interfacial stress, which the small-deformation limit of the EMM model approximates, lead to larger discrepancies. At high $Ca$ and low $Bqs$, the model approaches the breakup threshold, where FRS indicate numerical instability due to large deformations.

Significance
The paper establishes the EMM model as a robust and computationally inexpensive tool for predicting droplet deformation in the presence of interfacial viscosity. It demonstrates that the model retains a clear connection to classical small-deformation theories while offering independent control over shear and dilatational effects. The work provides a quantitative map of the model's validity, highlighting that while it is highly effective for moderate deformations and specific viscosity ratios, fully resolved simulations remain indispensable for large deformations or strongly asymmetric interfacial conditions. The authors suggest that this framework could serve as a basis for future extensions to more complex flow configurations and surfactant-laden interfaces, though such generalizations are noted as potential future directions rather than current achievements.