Extensional Flow Simulator: Elongational Viscosity & Polymer Stretching

simulator intermediate ~10 min
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Wi = 5, Tr ≈ 850 — strong strain-hardening

At ε̇ = 10/s and λ_r = 0.5 s (Wi = 5), the FENE-P model predicts a Trouton ratio of approximately 850 — the extensional viscosity is hundreds of times the shear viscosity, illustrating dramatic strain-hardening from polymer chain uncoiling.

Formula

Wi = λ_r × ε̇ (Weissenberg number)
Tr = η_E / η₀ (Trouton ratio)
f(R) = L² / (L² − R²) (FENE spring law)

Stretching vs. Shearing

While shear flow slides fluid layers past each other, extensional flow stretches material elements along one direction and compresses them along others. This seemingly simple difference has profound consequences for polymer solutions: in shear, flexible chains tumble and partially align but remain largely coiled; in extension, chains are pulled from both ends and can unravel dramatically. The result is extensional viscosities that can exceed shear viscosities by factors of 1000 or more — a phenomenon with enormous implications for processing.

The Coil-Stretch Transition

De Gennes predicted that at a critical strain rate, polymer chains would undergo an abrupt conformational change from random coils to nearly fully extended states. This coil-stretch transition occurs when the Weissenberg number Wi = λ_r × ε̇ exceeds approximately 0.5 — the extensional deformation rate overwhelms the chain's relaxation. The simulation visualizes this transition as a sharp rise in extensional viscosity and an animated representation of chain conformation.

Filament Thinning

One of the most elegant ways to measure extensional properties is the capillary breakup extensional rheometer (CaBER). A fluid bridge is stretched between two plates, and the midpoint diameter is tracked as the filament thins under capillary pressure. For polymer solutions, the filament thins exponentially rather than linearly, with a time constant equal to three times the relaxation time. This simulation shows the filament profile and diameter evolution in real time as you adjust polymer parameters.

Processing Impact

Extensional flow dominates many industrial processes: converging flows into fiber-spinning dies, inflation of polymer film bubbles, atomization of inkjet droplets, and turbulent drag reduction in pipelines. The strain-hardening behavior of polymer solutions prevents filament breakup during fiber spinning (producing uniform fibers) and suppresses droplet formation during spraying. Understanding and controlling extensional rheology is essential for designing processes that exploit — or avoid — these dramatic viscoelastic effects.

FAQ

What is extensional flow?

Extensional (elongational) flow is deformation where fluid elements are stretched along one axis and compressed along others, without any shearing. It occurs in fiber spinning, film blowing, converging channel flows, and inkjet printing. For polymeric fluids, extensional flow produces fundamentally different behavior than shear flow.

What is the Trouton ratio?

The Trouton ratio Tr = η_E / η₀ compares extensional viscosity to zero-shear viscosity. For Newtonian fluids, Tr = 3 (uniaxial extension). For polymer solutions above the coil-stretch transition, Tr can reach 1000 or more, reflecting the dramatic strain-hardening that occurs when polymer chains unravel in extensional flow.

What is the coil-stretch transition?

At a critical Weissenberg number (Wi = λ_r × ε̇ ≈ 0.5), the extensional flow rate exceeds the polymer chain's ability to relax back to its coiled state. The chain undergoes a sharp conformational transition from a random coil to a nearly fully extended state, causing a sudden jump in extensional viscosity.

What is the FENE-P model?

FENE-P (Finitely Extensible Nonlinear Elastic, Peterlin approximation) is a constitutive model that accounts for the finite maximum stretch of polymer chains. Unlike simpler models that allow infinite extension, FENE-P caps the chain stretch at L (maximum extensibility), producing physically realistic extensional viscosity plateaus at high strain rates.

Sources

Embed

<iframe src="https://homo-deus.com/lab/rheology/extensional-flow/embed" width="100%" height="400" frameborder="0"></iframe>
View source on GitHub