Carreau Model

Quick Reference

• Use when: Polymer melts/solutions with smooth Newtonian-to-power-law transition, well-defined zero-shear viscosity

• Parameters: 4 (\(\eta_0\), \(\eta_\infty\), \(\lambda\), \(n\))

• Key equation: \(\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}\)

• Test modes: Flow (steady shear, rotation)

• Material examples: Polymer melts (PE, PP, PS), polymer solutions, food gels, blood analogues, structured liquids

Notation Guide

Symbol	Meaning
\(\eta\)	Apparent (shear) viscosity (Pa·s)
\(\eta_0\)	Zero-shear viscosity (Pa·s); Newtonian plateau at \(\dot{\gamma} \to 0\)
\(\eta_{\infty}\)	Infinite-shear viscosity (Pa·s); Newtonian plateau at \(\dot{\gamma} \to \infty\)
\(\lambda\)	Time constant (s); reciprocal of critical shear rate
\(n\)	Power-law index (dimensionless); \(n < 1\) shear-thinning, \(n > 1\) shear-thickening
\(\dot{\gamma}\)	Shear rate (1/s)
\(\sigma\)	Shear stress (Pa)

Overview

The Carreau model is a four-parameter constitutive equation for generalized Newtonian fluids that describes the smooth transition from a Newtonian plateau at low shear rates to power-law behavior at intermediate shear rates, and optionally to a second Newtonian plateau at very high shear rates. It is one of the most widely used models in polymer rheology and food science.

The model addresses a key limitation of the simple power-law model: the power law predicts infinite viscosity at zero shear rate, which is unphysical. The Carreau model incorporates a finite zero-shear viscosity \(\eta_0\), making it suitable for flow simulations where low-shear regions exist (e.g., center of pipe flow, stagnation points in dies).

Historical Context

Pierre J. Carreau developed this model in 1972 at the University of Wisconsin as part of his work on rheological equations from molecular network theories [1]. The model builds on earlier work by Cross (1965) [2] and was developed in parallel with similar formulations by Yasuda (leading to the Carreau-Yasuda model with an additional parameter).

The Carreau equation has become a standard in:

• Polymer processing simulation (injection molding, extrusion, blow molding)

• Blood rheology studies (blood is mildly shear-thinning)

• Food engineering (sauces, dairy products, doughs)

• Personal care products (shampoos, lotions)

Its popularity stems from providing physically reasonable behavior across all shear rates while maintaining analytical tractability for computational fluid dynamics.

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Physical Foundations

Molecular Interpretation

The Carreau model captures the macroscopic consequences of molecular-scale phenomena in polymer systems:

Low shear rates (Newtonian plateau \(\eta_0\) ):

At rest or very low shear, polymer chains are in their equilibrium conformations. Chain entanglements form a temporary network. The viscosity is determined by the friction of chains moving past each other:

• Entangled chains must reptate (snake) through the entanglement tube

• Network relaxation time is longer than experimental timescale

• Resistance to flow is maximum → constant \(\eta_0\)

Intermediate shear rates (Power-law region):

As shear rate increases past \(\dot{\gamma}_c \approx 1/\lambda\):

• Chains begin to orient and stretch in flow direction

• Entanglement density decreases (disentanglement)

• Chain-chain friction decreases

• Viscosity drops following power law: \(\eta \sim \dot{\gamma}^{n-1}\)

High shear rates (Second Newtonian plateau \(\eta_{\infty}\) ):

At very high shear:

• Chains are fully oriented in flow direction

• Maximum possible disentanglement achieved

• Only solvent (or chain segment) friction remains

• Viscosity approaches constant \(\eta_{\infty}\)

Note: Many polymer systems never reach the high-shear plateau experimentally due to flow instabilities, thermal degradation, or shear banding at extreme rates.

Connection to Polymer Physics

The time constant \(\lambda\) relates to molecular relaxation:

\[\lambda \sim \tau_d \sim \frac{\zeta N^3}{k_B T} \sim \frac{M^3}{\rho RT / M_e}\]

where:

• \(\tau_d\) = reptation (disengagement) time

• \(\zeta\) = monomeric friction coefficient

• \(N\) = degree of polymerization

• \(M\) = molecular weight

• \(M_e\) = entanglement molecular weight

The power-law index \(n\) reflects the molecular weight distribution:

• Narrow distribution (monodisperse): \(n \approx 0.4-0.5\)

• Broad distribution (polydisperse): \(n \approx 0.2-0.3\)

Material Examples with Typical Parameters

Representative Carreau parameters for common materials

Material	\(\eta_0\) (Pa·s)	\(\eta_{\infty}\) (Pa·s)	\(\lambda\) (s)	\(n\)	T (°C)	Ref
LDPE (190°C)	4,200	0	0.55	0.35	190	[3]
HDPE (190°C)	6,500	0	0.42	0.47	190	[3]
Polypropylene (230°C)	2,100	0	0.18	0.38	230	[3]
Polystyrene (200°C)	8,500	0	1.2	0.28	200	[4]
Blood (37°C)	0.056	0.0035	3.3	0.35	37	[5]
Xanthan gum 0.5%	12.5	0.01	8.2	0.22	25	[6]
Polymer solution 5%	0.8	0.001	0.15	0.55	25	[7]

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Governing Equations

Mathematical Formulation

The Carreau viscosity function is:

\[\eta(\dot{\gamma}) = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}\]

Derivation from molecular network theory:

Carreau derived the model by considering the destruction of network junctions under flow [1]:

Step 1: Assume network structure with junctions breaking at rate proportional to \((\lambda\dot{\gamma})^2\)

Step 2: Define structural variable \(\xi\) representing fraction of intact junctions:

\[\xi = [1 + (\lambda\dot{\gamma})^2]^{-1}\]

Step 3: Relate viscosity to network integrity:

\[\eta - \eta_{\infty} = (\eta_0 - \eta_{\infty}) \xi^{(1-n)/2}\]

Step 4: Substitute and simplify:

\[\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}\]

Shear Stress Relation

The shear stress is:

\[\sigma = \eta(\dot{\gamma}) \cdot \dot{\gamma} = \left\{ \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2} \right\} \dot{\gamma}\]

This is a monotonic function of \(\dot{\gamma}\) for \(n > 0\), ensuring stable flow (no shear banding from non-monotonic flow curves).

Limiting Cases

Asymptotic behavior

Regime	Condition	\(\eta(\dot{\gamma})\)	Physical interpretation
Low shear	\(\lambda\dot{\gamma} \ll 1\)	\(\approx \eta_0\)	Newtonian plateau
Critical	\(\lambda\dot{\gamma} = 1\)	\(\approx 0.5(\eta_0 + \eta_{\infty})\) for \(n=0.5\)	Transition point
Power-law	\(1 \ll \lambda\dot{\gamma} \ll \lambda\dot{\gamma}_{max}\)	\(\approx (\eta_0 - \eta_{\infty})(\lambda\dot{\gamma})^{n-1}\)	Shear-thinning
High shear	\(\lambda\dot{\gamma} \gg 1\), \(\eta_{\infty} > 0\)	\(\to \eta_{\infty}\)	Second Newtonian plateau

Special Cases

Newtonian fluid (\(n = 1\)):

\[\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty}) = \eta_0\]

Strong shear-thinning (\(\eta_{\infty} = 0\)):

\[\eta = \eta_0 [1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}\]

This simplified form is commonly used when high-shear data is unavailable.

Power-law approximation (mid-range only):

\[\eta \approx K \dot{\gamma}^{n-1}, \quad K = \eta_0 \lambda^{1-n}\]

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Parameters

Parameters

Name	Symbol	Units	Bounds	Notes
eta0	\(\eta_0\)	Pa·s	\(10^{-3} - 10^{12}\)	Zero-shear viscosity; Newtonian plateau at low rates
eta_inf	\(\eta_{\infty}\)	Pa·s	\(10^{-6} - 10^{6}\)	Infinite-shear viscosity; often set to 0 or small value
lambda_	\(\lambda\)	s	\(10^{-6} - 10^{6}\)	Time constant; \(1/\lambda\) is critical shear rate
n	\(n\)	—	\(0.01 - 1.0\)	Power-law index; < 1 thinning, = 1 Newtonian

Parameter Interpretation

eta0 (Zero-Shear Viscosity):

• Physical meaning: Viscosity when polymer chains are in equilibrium state

• Molecular origin: Chain entanglement density and reptation resistance

• Typical ranges:

  • Polymer melts: \(10^3 - 10^7\) Pa·s

  • Polymer solutions: \(10^{-2} - 10^2\) Pa·s

  • Blood: \(0.03 - 0.1\) Pa·s

• Scaling: \(\eta_0 \sim M^{3.4}\) for entangled polymers (\(M > M_c\))

eta_inf (Infinite-Shear Viscosity):

• Physical meaning: Residual viscosity at maximum chain orientation

• Molecular origin: Solvent viscosity + fully aligned chain segment friction

• Typical values: Often set to 0 for melts (no solvent); \(\eta_{\infty} \approx \eta_{\text{solvent}}\) for solutions

• Fitting note: May be poorly determined if high-shear data is limited

lambda (Time Constant):

• Physical meaning: Characteristic relaxation time of the material

• Molecular origin: Longest relaxation time (reptation time for entangled chains)

• Relation to critical shear rate: \(\dot{\gamma}_c = 1/\lambda\)

• Typical ranges:

  • Polymer melts: \(10^{-2} - 10^2\) s

  • Polymer solutions: \(10^{-4} - 10^0\) s

• Scaling: \(\lambda \sim M^{3.4}\) (same as \(\eta_0\))

n (Power-Law Index):

• Physical meaning: Degree of shear-thinning

• Molecular origin: Polydispersity (broad MWD → lower n) and chain flexibility

• Interpretation:

  • \(n = 1\): Newtonian

  • \(n = 0.5\): Moderate thinning (typical for monodisperse melts)

  • \(n = 0.2\): Strong thinning (broad MWD, rigid chains)

  • \(n > 1\): Shear-thickening (rare, cornstarch suspensions)

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Validity and Assumptions

Model Assumptions

1. Generalized Newtonian fluid: Stress depends only on current strain rate (no memory)

2. Isothermal: Temperature is constant (use separate \(\eta_0(T)\) for T-dependence)

3. Simple shear flow: Steady, unidirectional shear (not oscillatory or extensional)

4. Incompressible: Constant density

5. Inelastic: No normal stress differences or elastic recoil

Data Requirements

• Required: Flow curve \(\eta(\dot{\gamma})\) or \(\sigma(\dot{\gamma})\) from steady shear

• Shear rate range: At least 3 decades, ideally 4-5 decades

• Coverage: Should span both plateaus and power-law region

• Recommended: \(\dot{\gamma} = 10^{-2}\) to \(10^{4}\) s\(^{-1}\) for polymers

Limitations

No viscoelasticity:

Cannot predict storage/loss moduli, stress relaxation, creep, or normal stresses. Use Maxwell/Zener for linear viscoelasticity, Oldroyd-B for nonlinear.

No yield stress:

The model predicts \(\sigma \to 0\) as \(\dot{\gamma} \to 0\). Use Herschel-Bulkley or Bingham for yield stress fluids.

No time-dependent behavior:

Cannot capture thixotropy, rheopexy, or startup transients. Use structural kinetics models (DMT, fluidity) for thixotropy.

Temperature dependence not built-in:

Must fit at each temperature or use time-temperature superposition: \(\eta_0(T) = \eta_0(T_r) \cdot a_T\), \(\lambda(T) = \lambda(T_r) \cdot a_T\)

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Regimes and Behavior

Flow Curve Characteristics

A log-log plot of \(\eta\) vs \(\dot{\gamma}\) shows three regions:

1. Zero-shear plateau (\(\dot{\gamma} < 0.1/\lambda\)): Horizontal line at \(\eta_0\)

2. Transition region (\(0.1/\lambda < \dot{\gamma} < 10/\lambda\)): Curved transition

3. Power-law region (\(\dot{\gamma} > 10/\lambda\)): Linear with slope \(n - 1\)

Diagnostic: Plot \(\log\eta\) vs \(\log\dot{\gamma}\):

• Slope = 0: Newtonian plateau

• Slope = n - 1: Power-law region (−0.5 to −0.8 typical)

Stress vs Strain Rate

The stress \(\sigma = \eta \dot{\gamma}\) is always monotonically increasing with \(\dot{\gamma}\):

• Low \(\dot{\gamma}\): \(\sigma \approx \eta_0 \dot{\gamma}\) (slope = 1 on log-log)

• High \(\dot{\gamma}\): \(\sigma \approx K \dot{\gamma}^n\) where \(K = \eta_0 \lambda^{1-n}\) (slope = n)

This monotonicity ensures flow stability (no constitutive instabilities).

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What You Can Learn

This section explains how to interpret fitted Carreau parameters to gain insights about your material’s molecular structure and processing behavior.

Parameter Interpretation

eta0 (Zero-Shear Viscosity):

The zero-shear viscosity reveals the entanglement state and molecular weight:

• Low values (< 100 Pa·s): Short chains, below entanglement threshold \(M_c\), or dilute solution

• Moderate values (100-10,000 Pa·s): Typical processing-grade polymers, well-entangled

• High values (> 10,000 Pa·s): Ultra-high MW, high entanglement density, or very low temperature

For graduate students: Use the scaling \(\eta_0 \sim M^{3.4}\) (valid for \(M > 2M_e\)) to estimate molecular weight. Compare with GPC data to validate. The time constant \(\lambda\) shares the same molecular weight scaling, connecting both parameters to the reptation framework.

For practitioners: \(\eta_0\) directly controls low-shear processes (gravity-driven flow, low-speed filling, sag resistance). Target \(\eta_0\) based on process requirements—higher for vertical coatings, lower for rapid filling.

eta_inf (Infinite-Shear Viscosity):

The high-shear viscosity plateau indicates residual friction:

• eta_inf ≈ 0: Complete structural breakdown, pure melt behavior

• eta_inf > 0: Solvent or matrix contribution remains (polymer solutions)

For graduate students: For solutions, \(\eta_{\infty}\) approximates the solvent viscosity plus a minor hydrodynamic contribution from fully aligned chains.

For practitioners: \(\eta_{\infty}\) controls high-speed operations like spray atomization and fast coating. Lower values enable easier processing at high shear rates.

lambda (Time Constant):

The relaxation time identifies the critical shear rate for structural response:

• Short \(\lambda\) (<0.1 s): Fast-relaxing, good for high-speed processing

• Long \(\lambda\) (>10 s): Slow-relaxing, melt memory effects important, potential for elastic instabilities

For graduate students: \(\lambda\) approximates the terminal relaxation time \(\tau_d\). The Deborah number \(De = \lambda \dot{\gamma}\) indicates elastic vs viscous dominance: \(De > 1\) gives elastic effects, \(De < 1\) gives viscous flow.

For practitioners: Processes with \(\dot{\gamma} > 1/\lambda\) operate in the shear-thinning region, reducing pumping power. The critical shear rate \(1/\lambda\) marks the onset of significant viscosity reduction.

n (Power-Law Index):

The flow index reflects molecular weight distribution breadth:

• n close to 1 (0.7-0.9): Narrow MWD, nearly Newtonian behavior, monodisperse

• n moderate (0.4-0.6): Typical commercial polymers, moderate polydispersity

• n low (0.2-0.4): Broad MWD, strong shear-thinning, or branched architectures

For graduate students: The empirical correlation \(n \approx 1 - 0.3 \cdot \text{PDI}\) (where PDI = Mw/Mn) provides rough MWD estimates. For branched polymers, lower \(n\) reflects additional relaxation modes from long-chain branching.

For practitioners: Low \(n\) means easier flow at high shear (injection molding, extrusion) but potential for flow marks and non-uniform cooling. High \(n\) gives more uniform velocity profiles and better coating consistency.

Material Classification

Material Classification from Carreau Parameters

Parameter Pattern	Material Type	Typical Materials	Processing Implications
High \(\eta_0\), low \(n\)	High-MW, broad MWD polymer	UHMWPE, broad-MWD PP	Strong die swell, long residence times, difficult extrusion
Low \(\eta_0\), high \(n\)	Low-MW, narrow MWD polymer	Oligomers, low-MW lubricants	Newtonian-like, easy processing, low melt strength
Long \(\lambda\), low \(n\)	High elasticity polymer	Branched LDPE, ionomers	Melt fracture risk at high rates, good for blow molding
Short \(\lambda\), moderate \(n\)	Linear commodity polymer	LLDPE, linear PP	Good processability window, stable extrusion
Low \(\eta_0/\eta_{\infty}\) ratio	Dilute solution	Polymer in good solvent	Minimal shear-thinning benefit

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Experimental Design

When to Use Carreau Model

Use this model when:

• Material shows clear zero-shear plateau

• Smooth transition to power-law behavior

• No yield stress (material flows at all stresses)

• Inelastic approximation acceptable

Consider alternatives when:

• Sharper transition: Carreau-Yasuda (adds \(a\) parameter)

• Different functional form: Cross model (denominator instead of power)

• Yield stress present: Herschel-Bulkley, Bingham

• Viscoelasticity needed: Maxwell, Oldroyd-B

• Thixotropy observed: DMT, fluidity models

Recommended Test Protocol

Steady Shear Flow Curve (Rotational Rheometry)

Step 1: Sample preparation

• Melt polymers above \(T_g + 50°C\), anneal 5-10 min

• Load fresh sample for each complete sweep (avoid shear history)

• Use 25 mm parallel plates with 1 mm gap (typical)

Step 2: Thermal equilibration

• Equilibrate at test temperature for 10-15 min

• Verify temperature uniformity (< 0.5°C variation)

Step 3: Flow curve measurement

• Sweep shear rate: \(10^{-2}\) to \(10^{3}\) s\(^{-1}\) (or instrument limit)

• Log spacing: 5-10 points per decade

• Measurement time: Auto (until stress steady) or 30 s minimum

• Direction: Ascending preferred (avoid thixotropic artifacts)

Step 4: Data quality checks

• Torque > minimum specification (typically 0.1 µNm)

• No slip (compare with serrated plates if suspect)

• No edge fracture (visual inspection, stress drop)

Sample Preparation

Polymer melts:

• Compression mold at \(T > T_m + 20°C\)

• Cool slowly to avoid residual stress

• Discs: 25 mm diameter, 1-2 mm thick

• Check for bubbles (reduce if present)

Polymer solutions:

• Dissolve with gentle stirring (avoid degradation)

• Filter (0.45 µm) to remove aggregates

• Degas if bubbles present

• Fresh preparation for each measurement day

Blood and biological fluids:

• Use anticoagulant (EDTA, heparin)

• Measure within 4 hours of collection

• Temperature control critical (37 ± 0.1°C)

• Use Couette geometry to minimize sample volume

Common Experimental Artifacts

Troubleshooting experimental issues

Artifact	Symptom	Solution
Wall slip	\(\eta\) artificially low, gap-dependent	Serrated plates, reduce gap, Mooney analysis
Sample degradation	\(\eta\) drifts during measurement	Reduce temperature, inert atmosphere, antioxidants
Edge fracture	Sudden stress drop at high \(\dot{\gamma}\)	Cone-plate geometry, reduce strain, add edge sealant
Inertia effects	Upturn at high \(\dot{\gamma}\)	Correct with inertia routine, limit max rate
Secondary flow	Anomalous behavior at high \(\dot{\gamma}\)	Verify Taylor number, use narrower gaps

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Computational Implementation

JAX Vectorization

RheoJAX implements the Carreau model with full JAX vectorization:

from rheojax.core.jax_config import safe_import_jax
jax, jnp = safe_import_jax()

@jax.jit
def carreau_viscosity(gamma_dot, eta0, eta_inf, lambda_, n):
    """Vectorized Carreau viscosity computation.

    Parameters
    ----------
    gamma_dot : array_like
        Shear rate values
    eta0 : float
        Zero-shear viscosity
    eta_inf : float
        Infinite-shear viscosity
    lambda_ : float
        Time constant
    n : float
        Power-law index

    Returns
    -------
    eta : array_like
        Viscosity values at given shear rates
    """
    factor = jnp.power(1 + jnp.square(lambda_ * gamma_dot), (n - 1) / 2)
    return eta_inf + (eta0 - eta_inf) * factor

Key optimizations:

• JIT compilation for 10-100x speedup

• Vectorization over shear rate arrays

• Automatic differentiation for sensitivity analysis

Numerical Stability

The model is numerically stable across typical parameter ranges:

• Large \(\lambda\dot{\gamma}\): Factor approaches \((\lambda\dot{\gamma})^{n-1}\), no overflow for \(n > 0\)

• Small \(\lambda\dot{\gamma}\): Factor approaches 1, stable

• Edge case \(\eta_0 = \eta_{\infty}\): Returns constant viscosity (correct)

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Fitting Guidance

Parameter Initialization

Method 1: From flow curve features

Step 1: Estimate \(\eta_0\) from low-shear plateau

\(\eta_0 \approx \eta(\dot{\gamma} \to 0)\)

Step 2: Estimate \(\eta_{\infty}\) from high-shear plateau (if visible)

\(\eta_{\infty} \approx \eta(\dot{\gamma} \to \infty)\), or set to 0

Step 3: Find transition point where \(\eta\) drops to \(0.5(\eta_0 + \eta_{\infty})\)

\(\lambda \approx 1 / \dot{\gamma}_{1/2}\)

Step 4: Estimate \(n\) from power-law slope

Fit \(\log\eta = \log K + (n-1)\log\dot{\gamma}\) in mid-range

Method 2: Using derivative

# Numerical derivative on log scale
d_log_eta = np.gradient(np.log(eta), np.log(gamma_dot))
n_estimate = 1 + np.min(d_log_eta)  # Most negative slope

Optimization Algorithm Selection

RheoJAX default: NLSQ (GPU-accelerated)

• Recommended for Carreau (4 parameters)

• Converges in < 100 iterations typically

• 5-270x faster than scipy.optimize

Bounds (recommended):

• \(\eta_0\): [1e-2, 1e10] Pa·s

• \(\eta_{\infty}\): [0, \(0.1 \cdot \eta_0\)] Pa·s

• \(\lambda\): [1e-6, 1e4] s

• \(n\): [0.1, 1.0]

Bayesian inference (NUTS):

• Use when uncertainty quantification needed

• Warm-start from NLSQ for efficiency

• Priors: Log-normal for \(\eta_0, \eta_{\infty}, \lambda\); Beta for \(n\)

Troubleshooting

Fitting diagnostics

Problem	Diagnostic	Solution
\(\eta_0\) poorly determined	Low-shear data missing or noisy	Extend to lower shear rates, more averages
\(\eta_{\infty}\) at upper bound	High-shear plateau not reached	Fix \(\eta_{\infty} = 0\) or extend \(\dot{\gamma}\) range
\(\lambda\) outside data range	Transition not captured	Adjust shear rate sweep to bracket \(1/\lambda\)
\(n\) near 1 (Newtonian)	Material barely shear-thins	Verify non-Newtonian behavior; use simpler model
Poor fit quality	Systematic residuals	Try Carreau-Yasuda (adds transition sharpness)

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Usage

Basic Example

import numpy as np
from rheojax.models import Carreau

# Shear rate sweep data
gamma_dot = np.logspace(-2, 4, 100)
eta_data = experimental_viscosity(gamma_dot)

# Create and fit model
model = Carreau()
model.fit(gamma_dot, eta_data, test_mode='rotation')

# Extract parameters
eta0 = model.parameters.get_value('eta0')
eta_inf = model.parameters.get_value('eta_inf')
lambda_ = model.parameters.get_value('lambda_')
n = model.parameters.get_value('n')

print(f"Zero-shear viscosity: {eta0:.1f} Pa·s")
print(f"Time constant: {lambda_:.3f} s")
print(f"Power-law index: {n:.3f}")

# Predict at new shear rates
gamma_dot_new = np.logspace(-3, 5, 200)
eta_pred = model.predict(gamma_dot_new)

Bayesian Inference

from rheojax.models import Carreau

model = Carreau()
model.fit(gamma_dot, eta_data, test_mode='rotation')

# Bayesian with NLSQ warm-start
result = model.fit_bayesian(
    gamma_dot, eta_data,
    test_mode='rotation',
    num_warmup=1000,
    num_samples=2000
)

# Credible intervals
intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95)
for param, (low, high) in intervals.items():
    print(f"{param}: [{low:.3g}, {high:.3g}]")

Stress Prediction

import numpy as np

# Predict shear stress (sigma = eta * gamma_dot)
stress = model.predict(gamma_dot, test_mode='rotation') * gamma_dot

# For simulation: apparent viscosity at specific rates
process_rates = np.array([100, 1000, 10000])  # Typical processing rates (s^-1)
process_viscosities = model.predict(process_rates, test_mode='rotation')

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Model Comparison

When to Use Carreau vs Alternatives

If you observe…	Consider…	Because…
Sharp transition	Carreau-Yasuda	Extra parameter \(a\) controls transition shape
Different functional form fits better	Cross	\(\eta = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{1 + (\lambda\dot{\gamma})^m}\)
Yield stress (no flow at low stress)	Herschel-Bulkley	\(\sigma = \sigma_y + K\dot{\gamma}^n\)
Need viscoelastic properties	Maxwell + Cox-Merz	Fit oscillatory, predict viscosity via Cox-Merz rule

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See Also

• Carreau–Yasuda Model — adds Yasuda exponent \(a\) for sharper transitions

• Cross Model — alternative sigmoidal form with denominator exponent

• Power-Law (Ostwald–de Waele) — local approximation of Carreau mid-rate region

• Herschel-Bulkley Model — for yield stress fluids

• Mastercurve (Time-Temperature Superposition) — time-temperature superposition for T-dependence

• ../../examples/flow/carreau_fitting — complete fitting tutorial

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API References

• Module: rheojax.models

• Class: rheojax.models.Carreau

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References

[1] (1,2)

Carreau, P. J. “Rheological equations from molecular network theories.” Transactions of the Society of Rheology, 16, 99-127 (1972). https://doi.org/10.1122/1.549276

[2]

Cross, M. M. “Rheology of non-Newtonian fluids: a new flow equation for pseudoplastic systems.” Journal of Colloid Science, 20, 417-437 (1965). https://doi.org/10.1016/0095-8522(65)90022-X

[3] (1,2,3)

Dealy, J. M. & Wissbrun, K. F. Melt Rheology and Its Role in Plastics Processing. Van Nostrand Reinhold, New York (1990).

[4]

Ferry, J. D. Viscoelastic Properties of Polymers, 3rd Edition. Wiley, New York (1980).

[5]

Thurston, G. B. “Viscoelasticity of human blood.” Biophysical Journal, 12, 1205-1217 (1972). https://doi.org/10.1016/S0006-3495(72)86156-3

[6]

Barnes, H. A., Hutton, J. F. & Walters, K. An Introduction to Rheology. Elsevier, Amsterdam (1989).

[7]

Larson, R. G. Constitutive Equations for Polymer Melts and Solutions. Butterworths, Boston (1988).

[8]

Macosko, C. W. Rheology: Principles, Measurements, and Applications. Wiley-VCH, New York (1994).

[9]

Osswald, T. A. & Menges, G. Materials Science of Polymers for Engineers, 3rd Edition. Hanser, Munich (2012).

[10]

Morrison, F. A. Understanding Rheology. Oxford University Press (2001).

[11]

Bird, R. B., Armstrong, R. C. & Hassager, O. Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics, 2nd Edition. Wiley, New York (1987).

Further Reading

• Yasuda, K. “Investigation of the analogies between viscometric and linear viscoelastic properties of polystyrene fluids.” PhD Thesis, MIT (1979). [Original Carreau-Yasuda derivation]

• Tanner, R. I. Engineering Rheology, 2nd Edition. Oxford University Press (2000). [Comprehensive treatment of flow models]