Polymer extension at stagnation points governs flow thickening of polymer solutions in ordered porous media

This study resolves the long-standing mystery of flow thickening in polymer solutions by demonstrating that, in ordered porous media, the phenomenon is quantitatively governed by polymer extension at stagnation points, a mechanism distinct from the unsteady flow fluctuations dominant in disordered media.

The Gist

Imagine you are trying to push a crowd of people (a fluid) through a maze. If the people are just walking normally, the harder you push, the faster they move. But what if the people are holding long, stretchy rubber bands?

This is exactly what happens when you push a polymer solution (like a thick liquid with long, string-like molecules) through a porous material (like a sponge or a rock with tiny holes). For over 50 years, scientists have been puzzled by a strange phenomenon: once you push hard enough, the liquid suddenly gets thicker and resists flowing much more than expected. It's like the crowd suddenly decides to link arms and form a giant, immovable wall, even though they were just walking a second ago.

This paper finally explains why that happens, specifically in ordered mazes (where the holes are arranged in a perfect, repeating pattern).

The "Traffic Jam" at the Dead Ends

The researchers discovered that the thickening isn't caused by the liquid rubbing against the walls (friction) or by the liquid getting chaotic and turbulent. Instead, it's all about stagnation points.

Think of a stagnation point as a dead-end street in your maze. When the fluid flows through the maze, it hits these dead ends. The fluid can't go forward, so it has to squeeze sideways. This squeezing action acts like a giant pair of hands grabbing the long, stringy polymer molecules and stretching them out.

• The Analogy: Imagine a crowd of people walking through a hallway. Most of the time, they just walk past each other. But when they hit a dead-end wall, they have to turn around. If they are holding long, stretchy rubber bands, the act of turning and squeezing past the wall stretches those bands tight.
• The Result: Once those rubber bands are stretched tight, they become very hard to move. The liquid effectively turns from a fluid into a stiff, elastic solid right at those dead ends. This creates a massive amount of resistance, making the liquid feel "thicker."

Ordered vs. Disordered Mazes

The paper makes a crucial distinction between two types of mazes:

1. Ordered Mazes (The Focus of this Paper): These are like a perfectly arranged grid of pillars or a stack of identical spheres. In these mazes, the "dead ends" (stagnation points) are predictable and happen in the exact same spots every time. The researchers found that in these perfect mazes, the stretching of the polymers at these dead ends is the only major reason the liquid gets thick. It's a clean, additive effect: more dead ends = more stretching = more resistance.
2. Disordered Mazes: These are like a pile of random rocks. Here, the liquid gets thick for a mix of reasons. While stretching still happens, there is also a lot of chaotic, wiggling motion (instabilities) that adds extra friction. The paper notes that in these messy mazes, the "stretching at dead ends" is still important, but it shares the spotlight with this chaotic wiggling.

How They Proved It

The scientists didn't just guess; they built tiny, transparent 3D mazes and watched the liquid flow through them using high-speed cameras. They also used a special mathematical model to calculate the energy.

They found that if you only counted the friction from the liquid rubbing against the walls, your math would be way off. You would predict the liquid should flow easily. But when they added the "stretching energy" (the cost of pulling those rubber bands tight at the dead ends) into their equation, the math matched the real-world experiments perfectly.

The Bottom Line

For a long time, scientists thought the thickening of these liquids in porous rocks was a mystery or caused by chaotic turbulence. This paper shows that in ordered structures, the secret is simple: The liquid gets thick because the polymers get stretched out at the dead ends of the flow.

It's not about the liquid getting messy; it's about the liquid getting stretched. Just like a rubber band that is easy to move when it's loose but becomes a rigid barrier when pulled tight, these polymer solutions suddenly resist flow when they hit the specific "dead ends" of the porous medium.

Technical Summary

Problem Statement
Viscoelastic polymer solutions flowing through porous media at low Reynolds numbers exhibit a phenomenon known as "flow thickening," where the macroscopic flow resistance (apparent viscosity, $\eta_{app}$) increases abruptly above a threshold flow rate. This behavior contrasts with the typical shear-thinning rheology of these fluids in bulk and with their behavior in disordered media where other mechanisms may dominate. Despite over half a century of study, a quantitative mechanistic description linking pore-scale dynamics to this macroscopic thickening has remained elusive. Previous work identified elastic flow instabilities and the resulting unsteady flow fluctuations as significant contributors to flow resistance in disordered media, but these mechanisms progressively underpredict the thickening observed in ordered porous media, particularly at higher flow rates. A quantitative link between pore-scale polymer extension and macroscopic flow thickening in ordered media has been lacking.

Methodology
The authors developed a theoretical model based on a mechanical energy balance that separates viscous dissipation into three distinct components:

1. Mean shear dissipation: Governed by the bulk shear viscosity.
2. Unsteady viscous dissipation: Arising from elastic flow instabilities (elastic turbulence), captured by the fluctuating component of the rate-of-strain tensor.
3. Extensional dissipation: A new term accounting for the energy cost of stretching polymer chains, quantified via an apparent extensional viscosity.

To validate this model, the authors conducted experiments in controlled ordered porous media with two distinct geometries:

• 2D Millifluidic Arrays: Hexagonally arranged pillars fabricated via stereolithographic 3D printing, configured in both 'staggered' and 'aligned' arrangements.
• 3D Consolidated Sphere Packings: Fused silica spheres with controlled necks, fabricated via selective laser-induced etching, arranged in simple cubic (SC) and body-centered cuboid (BC) configurations.

The fluid used was a dilute (0.5$c^*$) partially hydrolyzed polyacrylamide (HPAM) solution in a viscous solvent. The experimental setup combined macroscopic pressure drop measurements (to determine $\eta_{app}$) with pore-scale flow visualization using confocal microscopy and particle image velocimetry (PIV). The PIV data allowed for the calculation of Lagrangian trajectories to determine the accumulated Hencky strain experienced by fluid elements. These strain distributions were mapped to local Trouton ratios using transient extensional rheology data obtained from capillary break-up rheometry (CaBER).

Key Contributions and Results

1. Quantitative Model for Ordered Media: The authors derived a modified Darcy's law (Eq. 1) that quantitatively links pore-scale flow fields and fluid rheology to macroscopic flow thickening. The model successfully predicts the measured apparent viscosity in both 2D and 3D ordered geometries across a wide range of Weissenberg numbers ($Wi$).
2. Dominance of Stagnation Points: The study identifies that in ordered porous media, flow thickening is governed primarily by polymer extension at stagnation points, rather than by pore throat contractions or unsteady flow fluctuations alone. While elastic instabilities contribute near the onset of thickening, the extensional contribution becomes the dominant factor at higher flow rates.
3. Stagnation Point Additivity: The authors demonstrated that the macroscopic extensional resistance scales linearly with the number of stagnation points ($n_{SP}$) per unit cell. By modeling a single stagnation point using an optimized cross-slot rheometer (OSCER) and multiplying by the geometrically determined $n_{SP}$, they derived a second predictive model (Eq. 2) that accurately recovers flow resistance without adjustable fitting parameters for ordered geometries.
4. Extension to Disordered Media: The framework was tested on disordered bead packings and crushed glass packings. While the extensional contribution remains necessary to explain flow thickening in these media, the "resistance-per-stagnation-point" model requires $n_{SP}$ to be treated as an effective fitting parameter rather than a geometric constant, reflecting the heterogeneity and complex flow topology of disordered systems.

Significance
The paper claims to provide the first quantitative, mechanistic link between pore-scale polymer extension and macroscopic flow thickening in ordered porous media. By isolating the extensional contribution at stagnation points, the work resolves a long-standing mystery regarding the underprediction of flow resistance by models relying solely on shear viscosity or elastic instability-induced fluctuations. The authors state that this framework provides a foundation for predicting and controlling such flows in applications including chemical production, additive manufacturing, separation processes, energy production (e.g., enhanced oil recovery), and environmental remediation. The work emphasizes that while the specific mechanisms differ between ordered and disordered media, the interplay of shear, elastic instability, and extension is central to understanding viscoelastic flow in porous structures.