Herschel-Bulkley Model

Quick Reference

• Use when: Yield stress fluids (pastes, gels, foams), shear-thinning after yielding

• Parameters: 3 (\(\sigma_y\), \(K\), \(n\))

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

• Test modes: Flow curve (Steady Shear), Stress Ramp

• Material examples: Toothpaste, mayonnaise, drilling muds, fresh concrete, paints

Overview

The Herschel-Bulkley (HB) model is the most generic and widely used constitutive equation for yield stress fluids that demonstrate non-Newtonian flow behavior after yielding. It generalizes the Bingham plastic model (which assumes linear post-yield flow) and the Power-law model (which assumes no yield stress), making it the standard choice for complex fluids like pastes, emulsions, foams, and slurries.

Key Characteristics:

• Yield Stress ( \(\sigma_y\) ): Material acts as a rigid solid below a critical stress.

• Consistency ( \(K\) ): Measures the viscous resistance to flow.

• Flow Index ( \(n\) ): Characterizes post-yield behavior (usually shear-thinning, \(n < 1\)).

The model was introduced by Herschel and Bulkley in 1926 while studying rubber-benzene solutions and has since become the workhorse model for yield stress fluid characterization in industries ranging from food processing to oil drilling.

Notation Guide

Symbol	Meaning
\(\sigma\)	Shear stress (Pa)
\(\dot{\gamma}\)	Shear rate (s-1)
\(\sigma_y\)	Yield stress (Pa) - stress required to initiate flow
\(K\)	Consistency index (Pa·sn) - viscosity magnitude
\(n\)	Flow index (dimensionless) - slope of log-log flow curve
\(\eta_{app}\)	Apparent viscosity, \(\sigma / \dot{\gamma}\) (Pa·s)

Physical Foundations

Microstructural Interpretation

The Herschel-Bulkley model describes materials with a jammed microstructure at rest that breaks down under flow:

1. Jammed State (at Rest): Particles, droplets, or bubbles form a volume-spanning network or glassy cage. Brownian motion is insufficient to break this structure. * Result: Material behaves as an elastic solid (\(G' > G''\)) for small stresses.

2. Yielding Transition (\(\sigma \approx \sigma_y\)): The applied stress exceeds the inter-particle attractive forces or cage strength. The structure “un-jams” or fractures. * Result: Onset of irreversible flow.

3. Flowing State (\(\sigma > \sigma_y\)): The microstructure flows but retains interactions. Forces between particles lead to viscous dissipation. * Shear-Thinning ( \(n < 1\) ): Most common. Structure aligns, organizes (e.g., lanes), or breaks down further as \(\dot{\gamma}\) increases, reducing resistance. * Shear-Thickening ( \(n > 1\) ): Rare for simple yield stress fluids (usually seen in dense suspensions at high rates).

Governing Equations

Stress-Strain Rate Relationship

\[\begin{split}\sigma = \begin{cases} \sigma_y + K \dot{\gamma}^n & \text{if } \sigma > \sigma_y \\ \dot{\gamma} = 0 & \text{if } \sigma \le \sigma_y \end{cases}\end{split}\]

Apparent Viscosity

The viscosity is not constant but depends on shear rate:

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

• Low shear limit: \(\eta \to \infty\) as \(\dot{\gamma} \to 0\) (infinite viscosity at rest).

• High shear limit: \(\eta \to K \dot{\gamma}^{n-1}\) (approaches power-law behavior).

Parameters

Parameters

Name	Symbol	Units	Description
sigma_y	\(\sigma_y\)	Pa	Yield Stress. Critical stress for flow. High \(\sigma_y\) means “stiff” paste.
K	\(K\)	Pa·sn	Consistency. Viscosity scale. Note units depend on \(n\).
n	\(n\)		Flow Index. \(n<1\) (thinning), \(n=1\) (Bingham), \(n>1\) (thickening).

Material Behavior Guide

Typical Parameter Ranges by Material Class

Material Class	\(\sigma_y\) (Pa)	K (Pa·sn)	n	Notes
Mayonnaise	50–200	5–30	0.3–0.5	Highly shear-thinning emulsion
Toothpaste	100–300	10–50	0.2–0.4	Stiff paste with strong thinning
Drilling Mud	5–50	0.5–5	0.4–0.7	Bentonite suspensions
Fresh Concrete	10–200	50–500	0.2–0.5	Self-compacting has lower \(\sigma_y\)
Ketchup	10–50	5–20	0.3–0.5	Lower yield than mayo
Cosmetic Cream	20–100	2–20	0.3–0.6	O/W or W/O emulsions
Food Purees	5–50	2–15	0.2–0.4	Fruit/vegetable pastes
Foam (Shaving)	20–100	1–10	0.2–0.4	Gas-liquid system
Waxy Crude Oil	1–100	0.1–10	0.5–0.9	Temperature-dependent wax network

Validity and Assumptions

When Herschel-Bulkley Applies

The HB model is appropriate when:

1. Clear yield stress exists: Material does not flow below a critical stress. The stress-strain rate curve shows a stress intercept at zero rate.

2. Post-yield power-law behavior: After yielding, the material follows \(\sigma - \sigma_y = K \dot{\gamma}^n\) over the measured range.

3. Steady-state flow: Material reaches equilibrium at each shear rate (no thixotropy or aging during measurement).

4. No slip at walls: The material shears uniformly without wall slip.

When to Use Alternatives

Model Selection

Observation	Issue	Better Model
Fitted n ≈ 1 (within ±0.1)	Newtonian post-yield	Bingham Plastic (simpler)
Fitted \(\sigma_y\) ≈ 0	No yield stress	Power-Law (Ostwald–de Waele) or Carreau Model
Thixotropic hysteresis	Time-dependent structure	Fluidity models, DMT
Stress overshoot in startup	Viscoelastic effects	Saramito EVP, SGR

What You Can Learn

This section explains how to translate fitted Herschel-Bulkley parameters into material insights and actionable knowledge.

Parameter Interpretation

Yield Stress ( \(\sigma_y\) ):

The yield stress reveals the strength of the material’s microstructural network:

• \(\sigma_y\) < 10 Pa: Weak network. Material will flow under its own weight or light handling. Common in dilute suspensions and thin gels.

• 10 < \(\sigma_y\) < 100 Pa: Moderate network. Material holds shape against gravity but yields to reasonable forces. Typical for most commercial pastes.

• \(\sigma_y\) > 100 Pa: Strong network. Requires significant force to initiate flow. Common in stiff pastes like toothpaste and cement.

For graduate students: The yield stress scales with microstructural parameters. For colloidal gels, \(\sigma_y \propto \phi^m G_0\) where \(\phi\) is volume fraction, \(m \approx 2-4\) depends on network structure, and \(G_0\) is the elastic modulus. For emulsions, \(\sigma_y \propto \gamma/R \cdot (\phi - \phi_c)^2\) where \(\gamma\) is interfacial tension and \(R\) is droplet radius.

For practitioners: Use \(\sigma_y\) for packaging and dispensing design. A mayonnaise with \(\sigma_y = 80\) Pa needs approximately 80 Pa of shear stress to flow from a squeeze bottle. For vertical surfaces, material thickness \(h\) should satisfy \(h < \sigma_y / (\rho g)\) to prevent sagging.

Consistency Index (K):

The consistency governs the viscous response after yielding:

• Low K (< 5 Pa·s^n): Thin flow once yielded. Good pumpability but may spray or splash.

• Moderate K (5–50 Pa·s^n): Balanced viscous resistance. Typical for controlled spreading.

• High K (> 50 Pa·s^n): Thick flow requiring sustained energy input. Common in stiff mortars and heavy pastes.

For graduate students: For concentrated suspensions above the yield stress, \(K\) reflects hydrodynamic interactions between particles. It scales approximately as \(K \propto \eta_s (1 - \phi/\phi_m)^{-2.5}\) where \(\eta_s\) is solvent viscosity.

For practitioners: The pumping power in the post-yield regime scales with \(K\). Reducing particle concentration or adding dispersant lowers \(K\) and pumping costs.

Flow Index (n):

The flow index characterizes the degree of post-yield shear-thinning:

• n ≈ 1.0: Bingham-like. Post-yield viscosity is constant (linear flow curve above yield).

• 0.5 < n < 1.0: Mild thinning. Common in dilute systems or materials with weak interparticle attractions.

• 0.2 < n < 0.5: Strong thinning. Indicates significant microstructural breakdown with increasing shear. Common in concentrated emulsions and pastes.

• n < 0.2: Extreme thinning. May indicate a near-critical system (approaching glass or jamming transition). Check data quality.

For graduate students: For soft glassy materials, the SGR model predicts \(n = x - 1\) where \(x\) is the noise temperature. Materials near the glass transition (\(x \to 1\)) show \(n \to 0\). The flow index also connects to the Cole-Cole distribution width for polydisperse relaxation.

For practitioners: Lower \(n\) means the material “thins out” more dramatically at high shear rates. This aids mixing and pumping but may cause coating non-uniformity as the material levels differently at different rates.

Material Classification

HB Material Classification

Parameter Pattern	Material Behavior	Typical Materials	Process Implications
High \(\sigma_y\), low n	Stiff, thinning paste	Toothpaste, grease	Extrusion-based processing
Moderate \(\sigma_y\), n ≈ 0.5	Standard paste	Mayo, lotions	Conventional pumping/mixing
Low \(\sigma_y\), n close to 1	Near-Bingham	Thin suspensions	Consider Bingham model
High \(\sigma_y\), n close to 1	Stiff plastic	Cement, clay	High-pressure extrusion

Dimensional Analysis

The Oldroyd number (Od) characterizes flow regime:

\[Od = \frac{\sigma_y}{K} \left(\frac{L}{U}\right)^n\]

where \(L\) is length scale and \(U\) is velocity. Large Od means yield-stress-dominated flow (plug flow); small Od means power-law dominated.

For pipe flow, the fraction of unyielded material (plug) is:

\[\frac{r_{plug}}{R} = \frac{\sigma_y}{\sigma_{wall}} = \frac{\sigma_y}{\Delta P R / (2L)}\]

Diagnostic Indicators

Warning signs in fitted parameters:

• \(\sigma_y\) → 0 or negative: Data may not have a true yield stress. Consider Power-Law or Carreau model. Check that low-rate data is reliable.

• n > 1: Shear-thickening post-yield is rare. Check for inertial artifacts, Taylor vortices, or slip at high rates.

• K very small with high \(\sigma_y\): Unusual combination. Check units and data scaling.

• Strong parameter correlations: Especially \(\sigma_y\)– \(K\) correlation. Extend measurement range; ensure data spans transition region well.

• Systematic residuals at low rates: May indicate wall slip or viscoelastic creep below yield.

Application Examples

Food Product Design:

For spreadable products, target \(\sigma_y \approx 30-80\) Pa (easy spreading but no dripping). Track \(n\) to ensure consistent texture— lower \(n\) gives more “slip” on the knife.

Drilling Fluid Optimization:

Target \(\sigma_y\) high enough to suspend cuttings (typically 5–15 Pa) with \(n\) low enough for easy circulation. The API specifies 6 rpm and 300 rpm readings for HB parameter estimation.

Concrete Mix Design:

Self-compacting concrete requires \(\sigma_y < 60\) Pa for gravity-driven flow. Standard concrete has \(\sigma_y \approx 100-200\) Pa requiring vibration for compaction.

Cosmetic Formulation:

Body lotions need \(\sigma_y \approx 20-50\) Pa for good dispensing. Track \(n\) to ensure smooth spreading—values around 0.4-0.5 give pleasant sensory properties.

Experimental Design

Recommended Test Modes

1. Steady State Flow Curve (Step-Rate): * Protocol: Apply range of \(\dot{\gamma}\) (e.g., \(10^{-3}\) to \(10^2\) s^-1), measure \(\sigma\). * Duration: Allow steady state at each point (crucial for thixotropic materials). * Best for: Accurate parameter fitting over wide range.

2. Stress Ramp: * Protocol: Linear ramp of \(\sigma\) from 0 to \(>\sigma_y\). * Best for: Precise determination of \(\sigma_y\) (observe sudden strain rate jump).

Experimental Considerations

• Wall Slip: Common in pastes/gels. Material slips at geometry wall instead of flowing. * Symptom: Apparent “kink” in flow curve, or lower viscosity than expected. * Fix: Use sandpaper/serrated plates or vane geometry.

• Thixotropy: Time-dependent breakdown. * Check: Perform hysteresis loop (ramp up, then ramp down). If curves differ, material is thixotropic. Use steady-state averaging to fit equilibrium HB model.

• Geometry: * Cone-Plate: Constant shear rate (preferred). * Parallel Plate: Shear rate gradient (requires correction, but better for varying gaps/slip). * Vane: Best for preventing slip in yield stress fluids.

Fitting Guidance

Initialization

1. Estimate Yield Stress ( \(\sigma_y\) ): Extrapolate the low-shear stress plateau to \(\dot{\gamma} = 0\), or take the stress at the lowest measured rate.

2. Estimate Power-Law Parameters ( \(K, n\) ): Plot \((\sigma - \sigma_y)\) vs \(\dot{\gamma}\) on log-log scale. * Slope = \(n\) * Intercept (at \(\dot{\gamma}=1\)) = \(K\)

Troubleshooting Fitting Issues

Fitting Diagnostics

Symptom	Possible Cause	Solution
Negative Yield Stress	Data shows Newtonian plateau at low shear (no yield)	Switch to Cross or Carreau model (pseudoplastic with zero-shear viscosity).
Fit passes below data at high shear	Shear thickening onset or Taylor vortices	Restrict fit range to laminar region (remove high \(\dot{\gamma}\) points).
Poor fit at low shear	Wall slip or incomplete yielding	Check for slip (serrated plates). Down-weight low-shear points if noisy.
n close to 1	Material is Bingham Plastic	Simplify to Bingham model (\(n=1\)) for robustness.

Model Comparison

• Bingham: HB with \(n=1\). Simpler, assumes constant post-yield viscosity.

• Power Law: HB with \(\sigma_y = 0\). No yield stress.

• Casson: Alternative yield stress model (\(\sqrt{\sigma} = \sqrt{\sigma_y} + \sqrt{\eta \dot{\gamma}}\)), mainly for blood/chocolate.

• Carreau: No yield stress, but finite zero-shear viscosity. Better for polymer melts/solutions.

Usage

Basic Fitting

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

# Flow curve data (stress vs shear rate)
gamma_dot = jnp.logspace(-2, 2, 50)  # s^-1
# Example data for toothpaste
sigma = jnp.array([120.5, 125.3, 135.2, 148.7, 165.3, 185.2])  # Pa (measured stress)

# Fit the model
model = HerschelBulkley()
model.fit(gamma_dot[:6], sigma, test_mode='flow_curve')

# Extract parameters
sigma_y = model.parameters.get_value('sigma_y')
K = model.parameters.get_value('K')
n = model.parameters.get_value('n')
print(f"σ_y = {sigma_y:.1f} Pa, K = {K:.2f} Pa·s^n, n = {n:.3f}")

With Custom Initialization

from rheojax.models import HerschelBulkley

# Initialize with estimates from data inspection
model = HerschelBulkley()
model.parameters.set_value('sigma_y', 50.0)  # From low-rate plateau
model.parameters.set_value('K', 10.0)
model.parameters.set_value('n', 0.4)

# Constrain n for shear-thinning only
model.parameters.set_bounds('n', lower=0.1, upper=1.0)

model.fit(gamma_dot, sigma, test_mode='flow_curve')

Bayesian Inference

from rheojax.models import HerschelBulkley

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

# Bayesian inference with uncertainty quantification
result = model.fit_bayesian(
    gamma_dot, sigma,
    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"σ_y: {intervals['sigma_y']['mean']:.1f} "
      f"[{intervals['sigma_y']['hdi_2.5%']:.1f}, {intervals['sigma_y']['hdi_97.5%']:.1f}] Pa")

Comparing with Bingham

from rheojax.models import HerschelBulkley, Bingham

# Fit both models
hb = HerschelBulkley()
hb.fit(gamma_dot, sigma, test_mode='flow_curve')

bingham = Bingham()
bingham.fit(gamma_dot, sigma, test_mode='flow_curve')

# Compare fit quality
print(f"HB R² = {hb.r_squared:.5f}")
print(f"Bingham R² = {bingham.r_squared:.5f}")
print(f"HB n = {hb.parameters.get_value('n'):.3f}")

# If n ≈ 1 and R² similar, use Bingham for parsimony
if abs(hb.parameters.get_value('n') - 1.0) < 0.1:
    print("Consider using simpler Bingham model")

Pipeline Workflow

from rheojax.pipeline import Pipeline

(Pipeline()
    .load('flow_curve.csv', x_col='shear_rate', y_col='stress')
    .fit('herschel_bulkley', test_mode='flow_curve')
    .plot(log_scale=True, title='Herschel-Bulkley Fit')
    .save('results.hdf5'))

Computational Implementation

JAX Vectorization

The model uses JIT-compiled evaluation:

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

@partial(jax.jit, static_argnums=())
def herschel_bulkley_stress(gamma_dot, tau_y, K, n):
    """Compute stress for yielded flow (γ̇ > 0)."""
    return tau_y + K * jnp.power(gamma_dot, n)

Regularization for Numerics

The true HB model has a discontinuity at yield. For numerical stability, RheoJAX uses a regularized form:

\[\sigma = \sigma_y \left(1 - e^{-m \dot{\gamma}}\right) + K \dot{\gamma}^n\]

where \(m\) is a large regularization parameter (default \(m = 10^6\)). This produces smooth gradients while matching the true HB to high precision for \(\dot{\gamma} > 10^{-4}\) s^-1.

See Also

Related Flow Models

• Bingham Plastic — Special case with n = 1 (linear post-yield)

• Power-Law (Ostwald–de Waele) — Special case with \(\sigma_y = 0\) (no yield stress)

• Carreau Model — For materials without yield stress

• Cross Model — Alternative generalized Newtonian model

Advanced Yield Stress Models

• SGR Conventional (Soft Glassy Rheology) — Handbook — Soft glassy rheology for pastes near glass transition

• Fluidity-Saramito EVP Model — Elastoviscoplastic with viscoelastic pre-yield

• Shear Transformation Zone (STZ) — Shear transformation zone model

Transforms

• Mastercurve (Time-Temperature Superposition) — Temperature superposition for HB materials

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

API Reference

• rheojax.models.HerschelBulkley

• rheojax.models.Bingham

References

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[2]

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[3]

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Larson, R. G. The Structure and Rheology of Complex Fluids. Oxford University Press, New York (1999). ISBN: 978-0195121971

[8]

Coussot, P. Rheometry of Pastes, Suspensions, and Granular Materials: Applications in Industry and Environment. Wiley (2005). https://doi.org/10.1002/0471720577

[9]

Mewis, J., and Wagner, N. J. Colloidal Suspension Rheology. Cambridge University Press (2012). ISBN: 978-0521515993

[10]

Møller, P. C. F., Mewis, J., and Bonn, D. “Yield stress and thixotropy: On the difficulty of measuring yield stresses in practice.” Soft Matter, 2, 274–283 (2006). https://doi.org/10.1039/b517840a

[11]

Ovarlez, G., Cohen-Addad, S., Krishan, K., Goyon, J., and Coussot, P. “On the existence of a simple yield stress fluid behavior.” Journal of Non-Newtonian Fluid Mechanics, 193, 68–79 (2013). https://doi.org/10.1016/j.jnnfm.2012.06.009

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