Power-Law (Ostwald–de Waele)

Quick Reference

• Use when: Linear log-log flow curves, mid-range shear rates, quick characterization

• Parameters: 2 (\(K\), \(n\))

• Key equation: \(\sigma = K \dot{\gamma}^n\)

• Test modes: Flow curve (Steady Shear)

• Material examples: Polymer melts, paints, shampoo, sauces, drilling fluids

Overview

The Power-Law (or Ostwald–de Waele) model is the simplest and most widely used description of non-Newtonian flow. It assumes that shear stress scales as a power of shear rate. While it lacks the physical realism of identifying zero- and infinite-shear viscosity plateaus (unlike Carreau or Cross models), it provides an excellent empirical fit for the intermediate shear rate region where most processing and applications occur.

Notation Guide

Symbol	Meaning
\(\sigma\)	Shear stress (Pa)
\(\dot{\gamma}\)	Shear rate (s-1)
\(K\)	Consistency index (Pa·sn). Viscosity magnitude at \(\dot{\gamma}=1\).
\(n\)	Flow index (dimensionless). Slope of log-log flow curve.
\(\eta\)	Apparent viscosity (Pa·s)

Physical Foundations

Why “Power-Law”?

For many complex fluids, the microscale structure reorganizes under flow in a way that creates a self-similar response. This leads to a scaling law:

\[\sigma \propto \dot{\gamma}^n \implies \log(\sigma) = n \log(\dot{\gamma}) + \log(K)\]

1. Shear-Thinning ( \(n < 1\) ): * Microstructure: Polymer chain alignment, disentanglement, or breakdown of particle aggregates. * Analogy: “Traffic organizing into lanes” – resistance drops as flow speeds up.

2. Shear-Thickening ( \(n > 1\) ): * Microstructure: Hydrodynamic clustering, jamming, or formation of force chains (common in cornstarch/water). * Analogy: “Crowd panic” – jamming occurs as everyone tries to move faster.

Limitations

The Power-Law has no intrinsic time scale and predicts unphysical behavior at extremes: * Low Shear Limit: \(\eta \to \infty\) (for \(n<1\)). Real fluids have a Newtonian plateau \(\eta_0\). * High Shear Limit: \(\eta \to 0\) (for \(n<1\)). Real fluids have a solvent plateau \(\eta_\infty\).

Governing Equations

Constitutive Equation

\[\sigma = K \dot{\gamma}^n\]

Apparent Viscosity

\[\eta(\dot{\gamma}) = \frac{\sigma}{\dot{\gamma}} = K \dot{\gamma}^{n-1}\]

Parameters

Parameters

Name	Symbol	Units	Description
K	\(K\)	Pa·sn	Consistency Index. Measures the “thickness” of the fluid.
n	\(n\)		Flow Index. \(n=1\) (Newtonian), \(n<1\) (Thinning), \(n>1\) (Thickening).

Material Behavior Guide

Typical Parameter Ranges

Material Class	n	K (Pa·sn)	Notes
Polymer Melts	0.3 - 0.7	1k - 50k	Strongly thinning in processing range.
Paints (Latex)	0.4 - 0.6	10 - 100	Thinning for brush application.
Foods (Sauces)	0.2 - 0.5	5 - 50	e.g., Ketchup, Mayo.
Dilute Solutions	0.8 - 0.95	0.01 - 0.1	Weakly thinning.
Cornstarch/Water	1.5 - 2.0	0.1 - 10	Shear thickening (dilatant).

Validity and Assumptions

When the Power-Law Applies

The Power-Law model is valid when:

1. Linear log-log region: The \(\log(\eta)\) vs \(\log(\dot{\gamma})\) plot is linear over the shear rate range of interest.

2. Mid-range shear rates: Data span the power-law region, avoiding zero-shear and infinite-shear plateaus (typically 1–1000 s^-1 for most materials).

3. Steady-state flow: The material has reached equilibrium at each shear rate (no time-dependent effects like thixotropy).

4. Isothermal conditions: Temperature is constant throughout the measurement.

When to Use a Different Model

Model Selection Guide

Observation	Issue	Alternative Model
Curvature at low \(\dot{\gamma}\)	Zero-shear plateau visible	Carreau Model, Cross Model
Curvature at high \(\dot{\gamma}\)	Infinite-shear plateau	Carreau–Yasuda Model, Cross Model
Stress intercept at \(\dot{\gamma}=0\)	Material has yield stress	Herschel-Bulkley Model, Bingham Plastic
Time-dependent response	Thixotropy/aging	Fluidity models, DMT

What You Can Learn

This section explains how to translate fitted Power-Law parameters into material insights and actionable knowledge for both research and industrial applications.

Parameter Interpretation

Flow Index (n):

The flow index reveals the degree of non-Newtonian behavior:

• n = 1.0: Newtonian fluid with constant viscosity. The consistency index equals the Newtonian viscosity.

• 0.5 < n < 1.0: Mildly shear-thinning. Common in dilute polymer solutions where chain extension provides some alignment under flow.

• 0.2 < n < 0.5: Strongly shear-thinning. Indicates significant microstructural reorganization—polymer chain disentanglement, aggregate breakdown, or particle alignment.

• n < 0.2: Extremely shear-thinning. Often seen in highly concentrated suspensions or systems with strong interparticle attractions.

• n > 1.0: Shear-thickening (dilatant). Indicates hydrodynamic clustering, order-disorder transitions, or jamming phenomena.

For graduate students: The flow index relates to microstructural dynamics. For polymer melts, \(n \approx 1/(1 + 2a)\) where \(a\) is the tube model constraint release parameter. For suspensions, \(n\) decreases with increasing volume fraction as crowding amplifies thinning.

For practitioners: Target \(n \approx 0.4-0.6\) for brushable coatings (easy application, minimal dripping). For injection molding, lower \(n\) reduces pressure drop in runners. Values \(n > 1\) signal potential processing issues (e.g., die swell instability).

Consistency Index (K):

The consistency index sets the overall viscosity level:

• Physical meaning: \(K\) equals the apparent viscosity at \(\dot{\gamma} = 1\) s^-1 (only for \(n=1\)).

• Concentration dependence: For polymer solutions, \(K \propto c^{[\eta]M_w}\) where \(c\) is concentration and \([\eta]\) is intrinsic viscosity.

• Temperature sensitivity: \(K\) follows Arrhenius behavior: \(K(T) = K_0 \exp(E_a/RT)\) with activation energy \(E_a\).

For graduate students: The consistency index encodes both molecular weight and concentration effects. For entangled polymers, \(K \propto M_w^{3.4}\) following the reptation scaling. For suspensions, \(K\) scales as \(\eta_s(1 - \phi/\phi_m)^{-2}\) near the maximum packing fraction.

For practitioners: Use \(K\) for batch-to-batch QC. A 20% increase in \(K\) at fixed \(n\) suggests higher molecular weight or concentration. Temperature control is critical—a 10°C change can shift \(K\) by 50%.

Material Classification

Material Classification from Power-Law Parameters

Flow Index Range	Material Behavior	Typical Materials	Processing Implications
\(n > 1.2\)	Strong thickening	Dense cornstarch, silica in PEG	Mixing challenges, equipment damage risk
\(1.0 < n < 1.2\)	Mild thickening	Some particle suspensions	Careful rate control needed
\(n = 1.0 \pm 0.05\)	Newtonian	Simple fluids, dilute solutions	Standard process design
\(0.5 < n < 1.0\)	Mild thinning	Dilute polymer solutions	Good pumpability, moderate flow enhancement
\(0.2 < n < 0.5\)	Strong thinning	Melts, pastes, concentrated suspensions	Significant pressure reduction at high rates
\(n < 0.2\)	Extreme thinning	High-solid coatings, greases	Near-plug flow, yield-like behavior

Pipe Flow and Pumping Calculations

The Power-Law enables analytical solutions for pressure-driven flow:

Pressure Drop in Pipes:

\[\Delta P = \left(\frac{3n+1}{n}\right)^n \frac{2K L}{R} \left(\frac{Q}{\pi R^3}\right)^n\]

where \(Q\) is volumetric flow rate, \(L\) is pipe length, and \(R\) is pipe radius.

Velocity Profile:

\[v(r) = \frac{n}{n+1} \left(\frac{\Delta P}{2KL}\right)^{1/n} \left(R^{(n+1)/n} - r^{(n+1)/n}\right)\]

• For \(n < 1\): Blunted profile (approaches plug flow as \(n \to 0\))

• For \(n = 1\): Parabolic (Newtonian)

• For \(n > 1\): More peaked profile

For practitioners: Shear-thinning fluids (\(n < 1\)) require less pumping power than equivalent Newtonian fluids. The power saving scales as \((3n+1)/(4n)\) relative to Newtonian flow at the same flow rate.

Process Window Estimation

From fitted \(K\) and \(n\), estimate operating conditions:

Shear Rate from Viscosity Target:

\[\dot{\gamma}_{target} = \left(\frac{\eta_{target}}{K}\right)^{1/(n-1)}\]

Example: For a coating with \(K = 50\) Pa·sⁿ, \(n = 0.5\), requiring \(\eta = 0.5\) Pa·s for spray application:

\[\dot{\gamma} = \left(\frac{0.5}{50}\right)^{1/(0.5-1)} = (0.01)^{-2} = 10{,}000 \text{ s}^{-1}\]

Diagnostic Indicators

Warning signs in fitted parameters:

• n approaching 0: Model may be masking yield stress behavior. Consider Herschel-Bulkley if residuals are systematic at low rates.

• n > 1.5: Rare for true shear thickening. Check for inertial artifacts (Taylor vortices above Re ≈ 1000) or slip at high rates.

• K changes with shear rate range: Power-law region not isolated. Narrow the fitting range to exclude plateaus.

• Large confidence intervals on n: Insufficient data points or narrow shear rate range. Expand measurement range by at least one decade.

Application Examples

Quality Control:

Monitor \(K\) at fixed \(n\) for batch consistency. A control chart with ±10% limits on \(K\) catches molecular weight or concentration drift.

Process Optimization:

Use \(n\) to optimize mixing. Strongly thinning materials (\(n < 0.3\)) need high-shear impellers; mildly thinning materials (\(n > 0.7\)) work with standard designs.

Material Development:

During formulation, track how additives affect \(n\). Thickeners typically decrease \(n\); plasticizers may increase it. Target \(n\) and \(K\) values for desired application performance

Experimental Design

The Steady State Flow Curve is the standard test:

1. Rate Sweep: Logarithmic sweep of \(\dot{\gamma}\) (e.g., 0.1 to 1000 s^-1).

2. Equilibration: Ensure steady state at each point (30-60s typical).

3. Visualization: Plot \(\eta\) vs \(\dot{\gamma}\) on log-log axes. * Check: Is it a straight line? If yes, Power-Law fits. If curved, use Carreau.

Fitting Guidance

Initialization

• Log-Log Regression: The best way to initialize. * \(n\) = slope of \(\log(\sigma)\) vs \(\log(\dot{\gamma})\). * \(K\) = exponent of intercept (\(e^{\text{intercept}}\)).

Optimization

• Bounds (recommended):

  • \(K\): [1e-6, 1e6] Pa·sⁿ

  • \(n\): (0.01, 2.0)

• Loss function: Standard least squares suitable for mid-range data

• Weighted fitting: Optional weights to emphasize process-relevant shear rate range

Troubleshooting

Problem	Cause	Solution
Fit deviates at low rate	Zero-shear plateau (\(\eta_0\)) reached	Truncate low-rate data or switch to Carreau Model model
Fit deviates at high rate	Infinite-shear plateau or instability	Truncate high-rate data or switch to Cross Model model
\(n > 1\) unexpectedly	Inertia or Taylor vortices at high shear	Check Reynolds number; valid thickening is rare in simple fluids
\(K\) varies with test time	Thixotropy or evaporation	Use solvent trap; ensure steady state (no thixotropy loop)
Large confidence intervals	Insufficient data range	Extend shear rate sweep by at least one decade
Systematic residuals	Power-law region not isolated	Narrow fitting range to exclude plateaus

Usage

Basic Fitting

from rheojax.core.jax_config import safe_import_jax
jax, jnp = safe_import_jax()
from rheojax.models import PowerLaw
from rheojax.core.data import RheoData

# Steady shear flow curve data
gamma_dot = jnp.array([0.1, 1, 10, 100, 1000])  # s^-1
eta = jnp.array([500, 150, 45, 14, 4.5])  # Pa·s

# Create model and fit
model = PowerLaw()
model.fit(gamma_dot, eta, test_mode='flow_curve')

# Extract parameters
K = model.parameters.get_value('K')  # Consistency index
n = model.parameters.get_value('n')  # Flow index
print(f"K = {K:.1f} Pa·s^n, n = {n:.3f}")

# Predict viscosity at new shear rates
gamma_dot_new = jnp.logspace(-1, 4, 50)
eta_pred = model.predict(gamma_dot_new, test_mode='flow_curve')

Using RheoData

from rheojax.core.data import RheoData

# Load data with automatic test mode detection
data = RheoData(x=gamma_dot, y=eta, test_mode='flow_curve')

model = PowerLaw()
model.fit(data)

# Access fit quality
print(f"R² = {model.r_squared:.4f}")

Bayesian Parameter Estimation

from rheojax.models import PowerLaw

model = PowerLaw()
model.fit(gamma_dot, eta, test_mode='flow_curve')  # NLSQ warm-start

# Bayesian inference with uncertainty quantification
result = model.fit_bayesian(
    gamma_dot, eta,
    test_mode='flow_curve',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4
)

# Get credible intervals
intervals = model.get_credible_intervals(result.posterior_samples)
print(f"K: {intervals['K']['mean']:.1f} [{intervals['K']['hdi_2.5%']:.1f}, {intervals['K']['hdi_97.5%']:.1f}]")
print(f"n: {intervals['n']['mean']:.3f} [{intervals['n']['hdi_2.5%']:.3f}, {intervals['n']['hdi_97.5%']:.3f}]")

Pipeline Workflow

from rheojax.pipeline import Pipeline

# Complete workflow from file to results
(Pipeline()
    .load('flow_curve.csv', x_col='shear_rate', y_col='viscosity')
    .fit('power_law', test_mode='flow_curve')
    .plot(log_scale=True, title='Power-Law Fit')
    .save('results.hdf5'))

Temperature Dependence

import numpy as np

# Fit at multiple temperatures
temperatures = [25, 40, 60, 80]  # °C
K_values = []

for T, data in zip(temperatures, datasets):
    model = PowerLaw()
    model.fit(data)
    K_values.append(model.parameters.get_value('K'))

# Arrhenius analysis: ln(K) vs 1/T
T_kelvin = np.array(temperatures) + 273.15
ln_K = np.log(K_values)

# Fit for activation energy
from scipy.stats import linregress
slope, intercept, _, _, _ = linregress(1/T_kelvin, ln_K)
E_a = -slope * 8.314  # J/mol
print(f"Activation energy: {E_a/1000:.1f} kJ/mol")

Computational Implementation

JAX Vectorization

The Power-Law model is fully JIT-compiled for optimal performance:

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

@partial(jax.jit, static_argnums=(2,))
def power_law_viscosity(gamma_dot, params, n_points):
    K, n = params
    return K * gamma_dot ** (n - 1)

# Vectorized over multiple datasets
batched_predict = jax.vmap(power_law_viscosity, in_axes=(0, None, None))

Numerical Considerations

1. Log-space fitting: For numerical stability, the model internally works in log-space: \(\log(\eta) = \log(K) + (n-1)\log(\dot{\gamma})\).

2. Bounds: Default bounds are \(K \in [10^{-6}, 10^{6}]\) and \(n \in [0.01, 3.0]\) to ensure physical results.

3. Initialization: Smart initialization uses linear regression on log-log data, providing excellent starting points for optimization.

See Also

Related Flow Models

• Carreau Model — Adds zero-shear plateau; 4 parameters

• Cross Model — Alternative transition function; 4 parameters

• Carreau–Yasuda Model — Extra shape parameter for transition sharpness; 5 parameters

• Herschel-Bulkley Model — Power-law with yield stress; 3 parameters

• Bingham Plastic — Linear plastic with yield stress; 2 parameters

Transforms

• Mastercurve (Time-Temperature Superposition) — Time-temperature superposition

• Strain-Rate Frequency Superposition (SRFS) — Strain-rate frequency superposition for flow curves

API Reference

• rheojax.models.PowerLaw

• rheojax.core.data.RheoData

References

[1]

Ostwald, W. “Über die Geschwindigkeitsfunktion der Viskosität disperser Systeme.” Kolloid-Zeitschrift, 36, 99–117 (1925). https://doi.org/10.1007/BF01431449

[2]

de Waele, A. “Viscometry and plastometry.” Journal of the Oil and Colour Chemists’ Association, 6, 33–69 (1923).

[3]

Macosko, C. W. Rheology: Principles, Measurements, and Applications. Wiley-VCH, New York (1994). ISBN: 978-0471185758

[4]

Bird, R. B., Armstrong, R. C., and Hassager, O. Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics. 2nd ed., Wiley, New York (1987). ISBN: 978-0471802457

[5]

Barnes, H. A., Hutton, J. F., and Walters, K. An Introduction to Rheology. Elsevier, Amsterdam (1989). ISBN: 978-0444871404

[6]

Wilkinson, W. L. Non-Newtonian Fluids: Fluid Mechanics, Mixing and Heat Transfer. Pergamon Press, Oxford (1960).

[7]

Skelland, A. H. P. Non-Newtonian Flow and Heat Transfer. Wiley, New York (1967).

[8]

Chhabra, R. P., and Richardson, J. F. Non-Newtonian Flow and Applied Rheology: Engineering Applications. 2nd ed., Butterworth-Heinemann (2008). https://doi.org/10.1016/B978-0-7506-8532-0.X0001-7

[9]

Steffe, J. F. Rheological Methods in Food Process Engineering. 2nd ed., Freeman Press, East Lansing (1996). ISBN: 978-0963203618

[10]

Rao, M. A. Rheology of Fluid, Semisolid, and Solid Foods: Principles and Applications. 3rd ed., Springer (2014). https://doi.org/10.1007/978-1-4614-9230-6