Flow Curve Models¶

This section documents models for steady-state shear flow behavior—the relationship between shear stress and shear rate under continuous deformation.

Quick Reference¶

Model	Parameters	Use Case
Power-Law (Ostwald–de Waele)	2 (K, n)	Simple shear-thinning/thickening fluids
Carreau Model	4 (\(\eta_0\), \(\eta_\infty\), \(\lambda\), n)	Full flow curve with Newtonian plateaus
Carreau–Yasuda Model	5 (+a)	Sharper transition region control
Cross Model	4 (\(\eta_0, \eta_{\infty}\), K, n)	Alternative to Carreau, different transition
Bingham Plastic	2 (\(\sigma_y\), \(\eta_p\))	Simple yield stress fluids
Herschel-Bulkley Model	3 (\(\sigma_y\), K, n)	Yield stress + power-law flow

Overview¶

Flow curve models describe the steady-state relationship \(\sigma = f(\dot{\gamma})\) between shear stress and shear rate. These models are essential for:

Process design: Pump selection, pipe flow calculations
Material characterization: Viscosity classification (ASTM, ISO)
Quality control: Batch-to-batch consistency
Formulation development: Additive effects on flow behavior

Key phenomena captured:

Shear thinning: Decreasing viscosity with increasing shear rate (most polymeric fluids)
Shear thickening: Increasing viscosity (concentrated suspensions, some pastes)
Yield stress: Finite stress required to initiate flow
Newtonian plateaus: Constant viscosity at extreme shear rates

Model Hierarchy¶

Flow Curve Models
│
├── Newtonian Region Models (no yield stress)
│   ├── Power Law (Ostwald-de Waele)
│   │   └── σ = K · γ̇^n
│   │   └── Simple, 2 parameters
│   │   └── No plateaus
│   │
│   ├── Carreau
│   │   └── η = η∞ + (η_0-η∞)[1+(λγ̇)^2]^((n-1)/2)
│   │   └── Both plateaus, smooth transition
│   │
│   ├── Carreau-Yasuda
│   │   └── η = η∞ + (η_0-η∞)[1+(λγ̇)^a]^((n-1)/a)
│   │   └── Adjustable transition sharpness
│   │
│   └── Cross
│       └── η = η∞ + (η_0-η∞)/[1+(Kγ̇)^n]
│       └── Different transition shape
│
└── Yield Stress Models
    ├── Bingham
    │   └── σ = σ_y + η_p · γ̇  (if σ > σ_y)
    │   └── Linear above yield
    │
    └── Herschel-Bulkley
        └── σ = σ_y + K · γ̇^n  (if σ > σ_y)
        └── Power-law above yield

When to Use Which Model¶

Feature	Power Law	Carreau	C-Y	Cross	Bingham	H-B
Shear thinning	✓	✓	✓	✓	✗	✓
Yield stress	✗	✗	✗	✗	✓	✓
Zero-shear plateau	✗	✓	✓	✓	N/A	N/A
High-shear plateau	✗	✓	✓	✓	✗	✗
Transition control	✗	Fixed	✓	Fixed	N/A	N/A
Simple fitting	✓✓	✓	~	✓	✓✓	✓

Decision Flowchart:

Does the material have a yield stress? - Yes → Bingham (linear) or Herschel-Bulkley (power-law) - No → Continue
Do you observe Newtonian plateaus at low and/or high shear rates? - Yes → Carreau, Carreau-Yasuda, or Cross - No → Power Law (limited range)
Is the transition between plateaus sharp or gradual? - Sharp → Carreau-Yasuda (tune parameter a) - Gradual → Carreau or Cross

Material Examples¶

Material	Typical Model	Key Parameters	Industry
Polymer solutions	Carreau	\(\eta_0\) = 1-100 Pa·s, n = 0.3-0.7	Plastics, coatings
Polymer melts	Carreau-Yasuda	\(\eta_0\) = \(10^3-10^5\) Pa·s, a = 2	Extrusion, injection
Blood	Carreau	\(\eta_0\) ≈ 50 mPa·s, n ≈ 0.4	Biomedical
Paints	Cross or H-B	\(\sigma_y\) = 0.5-10 Pa	Coatings
Toothpaste	Herschel-Bulkley	\(\sigma_y\) = 10-100 Pa	Personal care
Drilling mud	Herschel-Bulkley	\(\sigma_y\) = 5-50 Pa, n = 0.5-0.8	Oil & gas
Ketchup	Herschel-Bulkley	\(\sigma_y\) ≈ 15 Pa	Food
Concrete	Bingham	\(\sigma_y\) = 10-100 Pa	Construction

Key Parameters¶

Parameter	Symbol	Units	Physical Meaning
Zero-shear viscosity	\(\eta_0\)	Pa·s	Viscosity at rest (Newtonian plateau)
Infinite-shear viscosity	\(\eta_\infty\)	Pa·s	High-rate limit (often ≈ 0)
Consistency index	K	Pa·s^n	Power-law prefactor (magnitude)
Flow index	n	—	n < 1: thinning, n > 1: thickening
Relaxation time	\(\lambda\)	s	Onset of shear thinning (1/\(\lambda\))
Yield stress	\(\sigma_y\)	Pa	Stress to initiate flow
Yasuda parameter	a	—	Transition sharpness (a = 2 gives Carreau)

Quick Start¶

Herschel-Bulkley (yield stress fluid):

from rheojax.models import HerschelBulkley
import numpy as np

model = HerschelBulkley()
gamma_dot = np.logspace(-2, 2, 50)

# Fit to flow curve data
model.fit(gamma_dot, stress_data, test_mode='flow_curve')

# Extract yield stress
sigma_y = model.parameters.get_value('sigma_y')
print(f"Yield stress: {sigma_y:.1f} Pa")

Carreau model (full flow curve):

from rheojax.models import Carreau

model = Carreau()
model.fit(gamma_dot, viscosity_data, test_mode='flow_curve')

# Get zero-shear viscosity and critical shear rate
eta_0 = model.parameters.get_value('eta_0')
lambda_param = model.parameters.get_value('lambda')
gamma_dot_c = 1 / lambda_param  # Critical shear rate

Bayesian parameter estimation:

# Bayesian inference with NLSQ warm-start
result = model.fit_bayesian(
    gamma_dot, data,
    test_mode='flow_curve',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,
    seed=42
)

# Yield stress uncertainty
intervals = model.get_credible_intervals(result.posterior_samples)
print(f"σ_y: [{intervals['sigma_y'][0]:.1f}, {intervals['sigma_y'][1]:.1f}] Pa")

Model Documentation¶

Simple Models:

Generalized Newtonian Models:

Yield Stress Models:

Herschel-Bulkley Model

References¶

Bird, R.B., Armstrong, R.C., & Hassager, O. (1987). Dynamics of Polymeric Liquids, Vol. 1, 2nd ed. Wiley. ISBN: 978-0471802457.
Carreau, P.J. (1972). “Rheological equations from molecular network theories.” Trans. Soc. Rheol., 16, 99-127. https://doi.org/10.1122/1.549276
Cross, M.M. (1965). “Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems.” J. Colloid Sci., 20, 417-437.
Herschel, W.H. & Bulkley, R. (1926). “Konsistenzmessungen von Gummi-Benzollösungen.” Kolloid-Z., 39, 291-300.
Yasuda, K., Armstrong, R.C., & Cohen, R.E. (1981). “Shear flow properties of concentrated solutions of linear and star branched polystyrenes.” Rheol. Acta, 20, 163-178. https://doi.org/10.1007/BF01513059
Barnes, H.A., Hutton, J.F., & Walters, K. (1989). An Introduction to Rheology. Elsevier. ISBN: 978-0444871404.
Macosko, C.W. (1994). Rheology: Principles, Measurements, and Applications. Wiley-VCH. ISBN: 978-0471185758.