Kinematic and rheological equivalence of steady shearing and planar extensional flows

This paper establishes a deep rheological equivalence between steady shearing and planar extensional flows by deriving an effective extension rate that removes rotational effects, thereby enabling the reconstruction of planar extensional viscosity from steady shear measurements for complex fluids.

The Gist

Imagine you are trying to understand how a complex fluid (like a thick polymer solution, honey mixed with rubber, or even ketchup) behaves when you pull it apart. Scientists call this extensional flow. It's crucial for understanding how plastics are molded, how ink sprays from a pen, or how blood flows through narrow capillaries.

However, there's a major problem: Pulling a fluid apart in a perfectly controlled, steady way is incredibly hard to do in a lab. It's like trying to stretch a piece of taffy perfectly straight without it snapping or curling up.

On the other hand, shearing a fluid (sliding layers of it past each other, like spreading butter on toast) is easy. We have machines that do this all the time.

For decades, scientists assumed these two actions—sliding (shear) and pulling (extension)—were totally different worlds. They thought you couldn't predict how a fluid would react to being pulled just by watching it get slid.

The Big Discovery:
This paper, written by Nicholas King and Gareth McKinley, says: "Actually, they are the same thing, just viewed from a different angle."

They found a mathematical "magic trick" that lets us take the easy data from a sliding experiment and perfectly reconstruct what would happen if we pulled the fluid apart.

The Analogy: The Spinning Skater vs. The Stretching Band

To understand how they did this, let's use two analogies:

1. The Spinning Skater (The Problem with Shear)
Imagine a figure skater spinning on ice.

• The Slide: The skater is moving forward (this is the flow).
• The Spin: But they are also spinning wildly.
• The Issue: If you want to know how much the skater's arms are stretching outward due to the spin, it's hard to measure because the whole body is rotating. The "stretching" is hidden inside the "spinning."

In a fluid, when you shear it (slide it), the fluid elements don't just stretch; they also rotate. This rotation messes up the measurement of how much the material is actually being pulled apart.

2. The Stretching Band (The Goal of Extension)
Now, imagine a rubber band being pulled straight.

• There is no spinning.
• The material is just stretching in one direction and getting thinner in the other.
• This is "pure extension."

The "Magic Trick" (The Solution)
The authors realized that the stretching happening in the "Spinning Skater" (shear) is actually identical to the stretching in the "Stretching Band" (extension), IF you can mathematically cancel out the spin.

They invented a concept called the "Effective Extension Rate."
Think of it like a noise-canceling headphone for fluids.

• The fluid is screaming with two noises: "Stretching" and "Spinning."
• The authors built a mathematical filter that subtracts the "Spinning" noise.
• What's left is the pure "Stretching" signal.

How It Works in Practice

1. The Easy Test: You put the fluid in a standard machine and slide it (shear). You measure two things:
   • How thick it is (Viscosity).
   • How much it pushes back sideways (Normal Stress).
2. The Math Filter: You take those numbers and run them through their new formula. This formula acts like a translator, stripping away the rotation component.
3. The Result: The output tells you exactly how the fluid would behave if you had pulled it apart (extension), even though you never actually pulled it apart!

Why This Matters

• No More Hard Experiments: You don't need expensive, finicky machines that try to stretch fluids perfectly. You can use the standard, easy machines everyone already has.
• Universal Truth: They proved this works for many different types of fluids, from simple mathematical models to real-world polymer solutions (like the polyisobutylene they tested).
• New Insights: By looking at the "shear" data through this new lens, they could see hidden physics, like how polymer chains suddenly snap from a tangled ball into a straight line (the "coil-stretch transition"), which was previously hard to spot.

The Bottom Line

This paper is like discovering that you can learn how a car drives on a straight highway just by watching it drive in a circle, provided you know how to subtract the turning motion.

It reveals that sliding and pulling are kinematic twins. By removing the "spin" from the "slide," we can finally predict how complex fluids will behave when they are stretched, opening up new ways to design better materials, medicines, and industrial processes without needing to build impossible machines.

Technical Summary

1. Problem Statement

Complex fluid rheology is typically characterized using two distinct flow fields: simple shear and pure extension.

• The Challenge: While steady shear flows are easily generated in standard torsional rheometers to measure shear viscosity ($\eta$) and normal stress differences ($N_1, N_2$), generating steady planar extensional flows is experimentally difficult. Most existing devices produce transient or inhomogeneous extensional flows.
• The Gap: Consequently, data for steady planar extensional viscosity ($\eta_{P1}$) is sparse. It is generally assumed that shear and planar extension probe fundamentally different microstructural responses (e.g., coil-stretch transitions) and that one cannot be predicted from the other.
• The Question: Is there a fundamental kinematic or rheological mapping that allows the reconstruction of steady planar extensional behavior using only data obtained from steady shear experiments?

2. Methodology

The authors leverage kinematic equivalence between simple shear and planar extension, grounded in the invariants of frame-invariant tensors (strain rate $\dot{\boldsymbol{\gamma}}$ and Finger strain $\mathbf{B}$).

• Kinematic Analysis:

  • Both flows reside on the same invariant lines in the flow classification map, making them kinematically indistinguishable if rotational components are removed.
  • In simple shear, the strain rate tensor eigenvectors are fixed at $45^\circ$ to the flow direction, while the stress tensor eigenvectors rotate depending on the material's constitutive response.
  • In planar extension, the strain rate and stress eigenvectors are coincident and fixed.

• Derivation of "Effective Extension Rate" ($\dot{\epsilon}_{eff}$):

  • The authors define a new variable, $\dot{\epsilon}_{eff}$, by projecting the imposed deformation rate of the shear flow onto the principal axes of the resulting stress tensor.
  • This projection mathematically removes the rotational (vorticity) component inherent in shear flow, isolating the pure stretching component experienced by a material element.
  • The formula derived is:
    $$\dot{\epsilon}_{eff} = \frac{\dot{\gamma} \sigma_{12}}{\Delta \sigma} = \frac{\dot{\gamma} \sigma_{12}}{\sqrt{N_1^2 + 4\sigma_{12}^2}}$$
    Where $\dot{\gamma}$ is the shear rate, $\sigma_{12}$ is the shear stress, and $\Delta \sigma$ is the principal stress difference.

• Definition of Composite Viscosity:

  • Using $\dot{\epsilon}_{eff}$, they define a composite planar extensional viscosity function, $\eta_{[P1]}$, which maps shear data to extensional behavior:
    $$\eta_{[P1]} \equiv \frac{\sigma_1 - \sigma_2}{\dot{\epsilon}_{eff}} = \frac{N_1^2 + 4\sigma_{12}^2}{\dot{\gamma}\sigma_{12}} = 4\eta(\dot{\gamma})\left[1 + \left(\frac{N_1}{2\sigma_{12}}\right)^2\right]$$
  • This equation relies only on steady shear measurements ($\eta$ and $N_1$).

• Validation:

  • Theoretical: The method was tested against phenomenological models (Linear PTT, UCM) and microscopically motivated models (Rolie-Poly for entangled melts).
  • Experimental: Experiments were conducted on a 2 wt% polyisobutylene (PIB) solution in paraffin oil using a cone-and-plate rheometer. The derived $\eta_{[P1]}$ was compared against theoretical predictions for planar extension.

3. Key Contributions

1. Proof of Equivalence: The paper establishes that for any materially objective constitutive equation, steady planar extensional viscosity is completely determined by steady shear viscosity and the first normal stress difference.
2. Removal of Rotation: By introducing the effective extension rate, the authors successfully decouple the stretching physics from the rotational physics of shear flow, revealing that the two flows are kinematically equivalent when viewed in the stress principal frame.
3. Experimental Accessibility: The work provides a practical route to determine steady planar extensional viscosity curves for fluids that are too viscous for microfluidic extensional devices or too weak for filament stretching, using only standard shear rheometry.
4. Generalization of Cox-Merz: The authors draw a parallel to the Cox-Merz rule, suggesting that just as linear viscoelasticity predicts nonlinear shear, nonlinear shear data can now predict nonlinear extensional behavior.

4. Results

• Model Validation:
  • For the Linear PTT model, the composite function $\eta_{[P1]}$ perfectly reconstructed the steady planar extensional viscosity curve, including the saturation behavior at high strains.
  • For the Rolie-Poly model (entangled polymers), the method accurately captured the complex non-monotonic behavior: an initial decrease due to chain orientation, followed by a sharp increase (coil-stretch transition) at a critical Weissenberg number, and eventual saturation due to finite extensibility.
• Experimental Confirmation:
  • Using the 2 wt% PIB solution, the authors calculated $\eta_{[P1]}$ from shear data.
  • The resulting curve matched the theoretical prediction for planar extension perfectly.
  • The Trouton ratio ($Tr = \eta_{P1}/\eta_0$) was observed to increase from the Newtonian limit of 4 to approximately 50 at high effective extension rates, confirming the strain-stiffening behavior typical of viscoelastic polymers.
• Asymptotic Behavior:
  • In the Newtonian limit ($N_1 \to 0$), the equation correctly reduces to the Trouton ratio of 4.
  • At high shear rates, the effective extension rate decreases relative to the shear rate, reflecting the alignment of stress eigenvectors with the flow direction.

5. Significance

• Overcoming Experimental Limitations: This work eliminates the need for specialized, difficult-to-implement planar extensional rheometers for many complex fluids. Researchers can now infer extensional properties from standard, high-precision shear data.
• Deepening Physical Understanding: It challenges the dogma that shear and extension are fundamentally distinct deformation histories. Instead, it shows they are different projections of the same underlying kinematic state, distinguished only by the presence of rotation.
• Industrial Application: Since many industrial processes (filament breakup, jetting, coating) involve mixed kinematics where extension dominates, this method allows for better prediction of fluid behavior in these processes using existing rheological databases.
• Future Directions: The authors suggest re-evaluating historical shear data to extract extensional properties and applying this framework to other complex flows previously thought to be distinct.

In summary, King and McKinley demonstrate that the "missing link" between shear and planar extension is the rotational component of the flow. By mathematically removing this rotation, they unify the rheological description of these two flow types, offering a powerful new tool for characterizing complex fluids.