Elastic and Viscous Effects in Viscoelastic Flows: Elucidating the Distinct Roles of the Deborah and Weissenberg Numbers

This paper clarifies the distinct physical roles and interpretations of the Deborah and Weissenberg numbers in characterizing the competition between elastic and viscous effects in viscoelastic flows by analyzing both an unsteady planar flow solution and a numerical simulation of flow between rotating coaxial cylinders using the Oldroyd-B model.

The Gist

Imagine you are stirring a pot of soup. If it's just water or broth, it flows smoothly and stops moving the moment you stop stirring. But if you add a huge amount of cornstarch or honey, the liquid becomes "sticky" and "stretchy." It doesn't just flow; it remembers being stretched and tries to snap back. This is what scientists call a viscoelastic fluid—a substance that acts like a mix between a liquid (viscous) and a rubber band (elastic).

For decades, scientists have used two special "scorecards" (numbers) to predict how these tricky fluids will behave: the Deborah Number and the Weissenberg Number.

This paper argues that for a long time, people have been using these scorecards incorrectly. They thought one number could tell the whole story, but the authors show that you need two different numbers to get it right.

Here is the breakdown of their discovery using simple analogies:

1. The Two Scorecards

• The Deborah Number (De): Think of this as a speedometer. It measures how fast you are moving the fluid compared to how fast the fluid "relaxes" (snaps back).
  • The Problem: The authors say this number is like checking how fast a car is going without knowing if the engine is even on. You can have a high speed (high De), but if the engine is off (no elasticity), the car isn't actually doing anything special.
• The Weissenberg Number (Wi): Think of this as a stress meter. It measures how much the fluid is actually being stretched and how hard it's pushing back.
  • The Good News: This number actually cares if the "engine" (the elastic properties) is on. If the fluid has no elasticity, this number drops to zero, which makes sense.

2. The "Rubber Band" Analogy

Imagine you have two identical-looking jars of liquid.

• Jar A is pure water.
• Jar B is water mixed with a tiny bit of invisible rubber bands.

If you stir both jars at the exact same speed, the Deborah Number (the speedometer) says they are identical. It sees the same speed and the same time it takes to stir.

But if you stop stirring:

• Jar A stops immediately.
• Jar B wiggles and snaps back because of the rubber bands.

The authors realized that the old "speedometer" (Deborah) couldn't tell the difference between the water and the rubber-band water. It failed because it ignored the amount of rubber (the elastic modulus, or $G$) in the jar.

3. The New Solution: The "Elasticity Score" ($\vartheta_e$)

The authors propose a new, better way to look at things. They suggest we need a score that combines:

1. How stretchy the fluid is (The rubber bands).
2. How fast we are moving it (The speed).

They created a new parameter called $\vartheta_e$ (theta-e).

• Analogy: If the Deborah number is just the speed of a race car, $\vartheta_e$ is the horsepower of the engine.
• In their experiments (stirring fluids between flat plates and spinning cylinders), they found that the "overshoot" (when the fluid swings past its target before settling down) was directly linked to this new $\vartheta_e$ score.
• If you have a high speed but zero rubber bands, the fluid acts like water.
• If you have rubber bands, the fluid acts like a spring, and the "overshoot" gets bigger.

4. Why This Matters

In the past, scientists might have looked at a complex flow (like blood flowing through a vein or plastic being molded) and said, "The Deborah number is high, so the fluid will act very elastically."

This paper says: "Wait! Check the rubber band content first!"

If the fluid is very dilute (very few rubber bands), even a high Deborah number won't make it act elastic. You need to know the Weissenberg number (to see the stress) and this new $\vartheta_e$ (to see the material's inherent tendency to be elastic).

The Takeaway

To understand how a stretchy fluid behaves, you can't just look at how fast you are moving it. You have to ask:

1. How fast are we moving? (Deborah/Weissenberg kinematic)
2. How much "spring" is actually in the fluid? (Elastic modulus/Weissenberg stress)

The authors conclude that by using the right combination of these numbers, we can finally predict exactly how these complex fluids will wiggle, snap, and flow, leading to better designs for everything from medical devices to industrial manufacturing.

Technical Summary

1. Problem Statement

The interpretation of dimensionless numbers in viscoelastic fluid mechanics, specifically the Deborah number ($De$) and the Weissenberg number ($Wi$), is often ambiguous and context-dependent.

• The Core Issue: While these numbers are ubiquitous in theoretical and experimental research, there is a lack of consensus on whether they adequately characterize the competition between elastic and viscous effects.
• The Specific Gap: Standard definitions often fail to distinguish between a fluid with high elasticity and one with negligible elasticity if the relaxation time ($\lambda$) and flow kinematics remain constant. Specifically, the paper questions whether $De$ (defined as $\lambda/t_{sc}$) and the kinematic Weissenberg number ($W = \lambda\dot{\gamma}$) can effectively quantify elastic forces when the elastic modulus ($G$) approaches zero.
• Objective: To clarify the physical significance of these parameters and determine which dimensionless groups truly govern the magnitude of elastic effects in viscoelastic flows.

2. Methodology

The authors employ a multi-faceted approach combining theoretical derivation, analytical solutions, and numerical simulations:

• Theoretical Framework:

  • The study focuses primarily on the Oldroyd-B model, chosen for its ability to capture essential elastic-viscous competition while remaining mathematically tractable.
  • The analysis is extended to other constitutive models (FENE-P, Giesekus, and linear Phan-Thien-Tanner) to test the generalizability of the findings.
  • Non-dimensionalization: The governing momentum and constitutive equations are non-dimensionalized using characteristic velocity ($U$) and length ($\ell$) scales. This derivation reveals the interplay between the Reynolds number ($Re$), the ratio of solvent to total viscosity ($\beta$), and a new elasticity parameter ($\zeta$).

• Analytical Solution (Parallel Plates):

  • The authors derive an exact analytical solution for the start-up Couette flow (sudden motion of a plate) of an Oldroyd-B fluid between parallel plates.
  • This allows for the precise calculation of the velocity overshoot ($\delta$), defined as the relative difference between the maximum transient velocity and the steady-state velocity. This overshoot serves as the primary metric for elastic strength.

• Numerical Simulations (Taylor-Couette Flow):

  • Finite-element simulations (using COMSOL) are performed for the start-up flow of an Oldroyd-B fluid between rotating coaxial cylinders.
  • Grid independence was verified, ensuring the results accurately reflect the physics rather than numerical artifacts.

• Parameter Variation:

  • The study systematically varies the relaxation time ($\lambda$), the elastic modulus ($G$), and the solvent viscosity ($\mu_s$) to observe how different dimensionless groups correlate with the observed elastic overshoot.

3. Key Contributions and Definitions

The paper introduces and validates a new intrinsic fluid parameter and re-evaluates existing definitions:

• Critique of $De$ and $W$:

  • The authors demonstrate that the Deborah number ($De = \lambda/t_{sc}$) and the kinematic Weissenberg number ($W = \lambda\dot{\gamma}$) are insufficient to characterize elastic effects.
  • Reasoning: Both parameters depend only on $\lambda$ and flow kinematics. They remain finite even if the elastic modulus $G \to 0$ (i.e., in a dilute solution where elastic forces vanish). Therefore, they cannot distinguish between a highly elastic fluid and a Newtonian fluid with the same relaxation time.

• Critique of the Stress-Based $Wi$:

  • The stress-based Weissenberg number ($Wi = (\tau_{xx} - \tau_{yy})/\tau_{xy}$) correctly vanishes as $G \to 0$. However, the authors show that for a fixed $Wi$, the magnitude of the elastic response (overshoot) still depends on $\lambda$. Thus, $Wi$ alone is not a complete descriptor.

• Introduction of $\vartheta_e$:

  • The authors derive a new dimensionless parameter, $\vartheta_e$, by combining the stress-based $Wi$ and a geometric elasticity parameter $\Gamma$.
  • Definition: $\vartheta_e = \sqrt{\frac{Wi \cdot \Gamma}{2}} = \frac{G\lambda}{\sqrt{\mu_s(\mu_s + G\lambda)}}$.
  • Significance: $\vartheta_e$ depends only on material properties ($\lambda, G, \mu_s$) and is independent of flow conditions ($U, \ell$). It represents an intrinsic property of the fluid quantifying its tendency to exhibit viscoelastic effects.

4. Key Results

• Parallel Plate Flow (Analytical):

  • The velocity overshoot ($\delta$) was found to be independent of the kinematic Weissenberg number ($W$) when $G$ varies.
  • Conversely, $\delta$ increases almost linearly with the new parameter $\vartheta_e$.
  • Contour plots of $\delta$ vs. $(\lambda, G)$ showed that curves of constant $\vartheta_e$ align perfectly with curves of constant overshoot, confirming $\vartheta_e$ as the governing parameter for the magnitude of elastic effects.

• Concentric Cylinder Flow (Numerical):

  • Simulations confirmed the analytical findings. Even with a fixed low $W$, varying $\Gamma$ (and thus $\vartheta_e$) caused the flow to transition from Newtonian-like behavior (weak/no overshoot) to strong elastic oscillations.
  • This proves that $W$ alone does not control the strength of elastic effects; the material property $\vartheta_e$ is required.

• Generalizability:

  • The analysis was extended to FENE-P, Giesekus, and PTT models. In all cases, setting $G \to 0$ eliminated elastic forces regardless of $\lambda$. This confirms that any valid metric for elastic importance must explicitly depend on $G$.

5. Significance and Conclusions

• Resolution of Ambiguity: The paper resolves the long-standing ambiguity regarding the roles of $De$ and $Wi$. It establishes that $De$ and kinematic $W$ characterize the timescale of the flow relative to the polymer, but they do not quantify the magnitude of elastic forces.
• Practical Guidelines:
  • To characterize the intrinsic viscoelasticity of a fluid, researchers should use $\vartheta_e$.
  • To estimate the elastic effects in a specific flow, a combined approach is recommended: use $\vartheta_e$ to define the material's potential and $Wi$ (stress-based) to estimate the actual stress levels generated in that specific geometry.
• Experimental Implication: Researchers cannot assume that two fluids with the same $De$ or $W$ will exhibit similar elastic behaviors if their elastic moduli ($G$) differ. Proper characterization requires knowledge of both the relaxation time and the elastic modulus (or concentration).

In summary, the authors argue that elastic effects are determined by the product of relaxation time and elastic modulus, not relaxation time alone. The parameter $\vartheta_e$ provides a rigorous, flow-independent metric for this property, offering a more robust framework for analyzing viscoelastic flows than traditional definitions of $De$ and $Wi$.