Bingham Plastic

Quick Reference

• Use when: Rigid below yield stress, Newtonian flow after yielding

• Parameters: 2 (\(\sigma_y\), \(\eta_p\))

• Key equation: \(\tau = \tau_y + \eta_p \dot{\gamma}\) for \(|\tau| > \tau_y\)

• Test modes: Flow (steady shear)

• Material examples: Cement pastes, drilling muds, mayonnaise, ketchup, toothpaste

Overview

The Bingham Plastic model describes viscoplastic materials that behave as rigid bodies below a yield stress (\(\tau_y\)) and flow linearly with a plastic viscosity (\(\eta_p\)) once yielded. Named after Eugene Bingham (1922), this is the simplest model capturing yield-stress behavior—materials that require a minimum stress to initiate flow. The Bingham model is foundational for understanding cement pastes, drilling muds, slurries, toothpaste, mayonnaise, and suspensions whose post-yield flow curves are approximately Newtonian.

The model represents a critical transition in non-Newtonian fluid mechanics: below \(\tau_y\), the material acts as an elastic or rigid solid; above \(\tau_y\), it flows as a shear-thinning or Newtonian liquid. This behavior arises from microstructural networks (particle contacts, hydrogen bonds, electrostatic interactions) that must be broken before flow can occur.

Notation Guide

Symbol	Meaning
\(\tau\)	Shear stress (Pa)
\(\tau_y\)	Yield stress (Pa). Minimum stress for flow initiation.
\(\dot{\gamma}\)	Shear rate (s-1)
\(\eta_p\)	Plastic viscosity (Pa·s). Post-yield slope.
\(\eta_{app}\)	Apparent viscosity (Pa·s) = \(\tau/\dot{\gamma}\)
\(Bi\)	Bingham number = \(\tau_y L / (\eta_p U)\). Ratio of yield to viscous stress.

Physical Foundation

Microstructural Origin of Yield Stress:

The yield stress \(\tau_y\) arises from:

1. Particle Networks: Colloidal particles forming space-spanning structures (e.g., clay suspensions, cement)

2. Attractive Interactions: Van der Waals, electrostatic, or depletion forces creating particle bridges

3. Entangled Structures: Fiber networks, polymer chains, or droplet clusters

4. Jamming Transitions: Dense suspensions where particles cage each other

Post-Yield Flow:

Once \(\tau > \tau_y\), the network breaks and particles/droplets flow past each other with a constant viscosity \(\eta_p\) (plastic viscosity), analogous to Newtonian flow but offset by the yield stress.

Governing Equations

Constitutive Equation:

\[\begin{split}\tau = \begin{cases} \tau_y \operatorname{sgn}(\dot{\gamma}) & \text{if } |\tau| \le \tau_y \text{ (unyielded)}, \\ \tau_y + \eta_p \dot{\gamma} & \text{if } |\tau| > \tau_y \text{ (yielded)}, \end{cases}\end{split}\]

where:

• \(\tau\) = shear stress (Pa)

• \(\dot{\gamma}\) = shear rate (s^-1)

• \(\tau_y\) = yield stress (Pa), \(\tau_y \geq 0\)

• \(\eta_p\) = plastic viscosity (Pa·s), \(\eta_p > 0\)

Apparent Viscosity:

\[\eta(\dot{\gamma}) = \frac{\tau_y}{|\dot{\gamma}|} + \eta_p \qquad \text{for } |\dot{\gamma}| > 0\]

The apparent viscosity diverges as \(\dot{\gamma} \to 0\) (infinite viscosity at very low shear rates).

Flow Curve Interpretation:

Plot \(\tau\) vs \(\dot{\gamma}\):

• Intercept: \(\tau_y\) (extrapolation to \(\dot{\gamma} = 0\))

• Slope: \(\eta_p\) (linear post-yield region)

Parameters

Parameter summary

Name	Units	Description / Constraints
sigma_y	Pa	Yield stress; ≥ 0. Sets the plateau torque required to initiate motion. Typical range: 1-500 Pa.
eta_p	Pa·s	Plastic viscosity governing the linear post-yield segment; > 0. Typical range: 0.001-10 Pa·s.

Material Examples

Cement and Construction Materials (\(\tau_y \approx 10-200\) Pa, \(\eta_p \approx 0.1-5\) Pa·s):

• Cement pastes (water-cement ratio dependent)

• Concrete slurries (fresh concrete)

• Mortar and grouts

• 3D printing inks (cementitious)

Drilling and Mining Fluids (\(\tau_y \approx 5-50\) Pa, \(\eta_p \approx 0.01-0.5\) Pa·s):

• Bentonite drilling muds

• Barite-weighted fluids

• Oil well drilling fluids

Food Products (\(\tau_y \approx 10-100\) Pa, \(\eta_p \approx 0.1-5\) Pa·s):

• Mayonnaise (\(\tau_y \approx 80-150\) Pa)

• Ketchup (\(\tau_y \approx 20-50\) Pa)

• Mustard (\(\tau_y \approx 30-70\) Pa)

• Chocolate (molten, \(\tau_y \approx 5-20\) Pa)

Personal Care and Pharmaceuticals (\(\tau_y \approx 50-300\) Pa):

• Toothpaste (\(\tau_y \approx 100-200\) Pa)

• Lotions and creams

• Ointments and gels

Suspensions and Slurries (\(\tau_y \approx 1-100\) Pa):

• Clay suspensions (kaolin, montmorillonite)

• Mineral slurries (tailings, coal slurries)

• Activated sludge (wastewater treatment)

Experimental Design

Flow Curve (Controlled Shear Rate):

1. Shear rate sweep: 0.001-1000 s^-1 (log-spaced, 10 points/decade)

2. Pre-shear: High shear rate (100 s^-1, 60 s) to erase history

3. Rest period: 2-5 min to allow structure recovery

4. Ramp protocol: Low → high or bidirectional to check thixotropy

5. Geometry: Vane or serrated plates to minimize wall slip

Yield Stress Determination Methods:

1. Flow Curve Extrapolation: - Linear regression of \(\tau\) vs \(\dot{\gamma}\) in post-yield region - Intercept at \(\dot{\gamma} = 0\) → \(\tau_y\) - Caution: Sensitive to fitting range selection

2. Controlled Stress Ramp: - Apply increasing stress, monitor strain rate - \(\tau_y\) = stress where \(\dot{\gamma}\) jumps from ~0 to finite value - More reliable for materials with sharp yielding

3. Vane Method (ASTM D4648): - Insert vane into sample, rotate at constant speed - Peak torque → \(\tau_y\) (accounts for 3D geometry) - Minimizes wall slip artifacts

Avoiding Common Artifacts:

• Wall slip: Use roughened surfaces, vane geometry, or serrated plates

• Sedimentation: Short measurement times, homogenize before test

• Evaporation: Solvent trap, short test duration

• Thixotropy: Control rest time, use consistent pre-shear protocol

Physical Foundations

Microstructural Origin of Yield Stress

The yield stress \(\tau_y\) arises from:

1. Particle Networks: Colloidal particles forming space-spanning structures (e.g., clay suspensions, cement)

2. Attractive Interactions: Van der Waals, electrostatic, or depletion forces creating particle bridges

3. Entangled Structures: Fiber networks, polymer chains, or droplet clusters

4. Jamming Transitions: Dense suspensions where particles cage each other

Post-Yield Flow

Once \(\tau > \tau_y\), the network breaks and particles/droplets flow past each other with a constant viscosity \(\eta_p\) (plastic viscosity), analogous to Newtonian flow but offset by the yield stress.

Validity and Assumptions

When Bingham Model Applies

The Bingham 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. Newtonian post-yield behavior: After yielding, the material follows linear flow with constant plastic viscosity.

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
Post-yield shear-thinning/thickening	Non-linear flow beyond yield	Herschel-Bulkley Model (\(n \neq 1\))
Fitted \(\tau_y\) ≈ 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

Fitting Guidance

Initialization

1. From flow curve: Linear fit of high shear rate region → slope = \(\eta_p\), intercept = \(\tau_y\)

2. From controlled stress: Stress at flow onset → \(\tau_y\)

3. Robust estimation: Median of multiple yield determinations

Optimization

• Use Huber loss to down-weight noisy pre-yield data

• Weighted least squares: Higher weights on post-yield region where \(\dot{\gamma}\) is reliable

• Constrain \(\tau_y \geq 0\) and \(\eta_p > 0\)

• Verify: Residuals should be random in post-yield region

Handling Pre-Yield Data

• Option 1: Exclude data below \(\dot{\gamma} < 0.01\) s^-1 (noisy, not truly rigid)

• Option 2: Fit only post-yield region (\(\tau > 1.1 \tau_y\))

• Option 3: Use robust loss (Huber, Tukey) to reduce influence of outliers

Troubleshooting

Problem	Cause	Solution
Negative \(\tau_y\) estimates	Noisy sub-yield data dominating fit	Use robust loss (Huber), exclude low \(\dot{\gamma}\) points, or constrain \(\tau_y \geq 0\)
Poor fit quality despite linear appearance	Thixotropic hysteresis (up-ramp ≠ down-ramp)	Use consistent pre-shear protocol or fit up/down separately
Apparent viscosity shows curvature	Material exhibits shear-thinning beyond yielding	Use Herschel-Bulkley Model (\(n < 1\)) or Casson model
Scatter at low shear rates	Instrument torque resolution, slip, or structural recovery	Use vane geometry, faster ramp rate, or exclude \(\dot{\gamma} < 0.01\) s-1

Usage

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

# Generate synthetic data (toothpaste)
gamma_dot = jnp.logspace(-2, 2, 80)  # 0.01 - 100 s⁻¹
tau_exp = 120.0 + 2.5 * gamma_dot + jnp.random.normal(0, 3, size=gamma_dot.shape)

# Initialize and fit
model = Bingham()
model.parameters.set_value('sigma_y', 120.0)
model.parameters.set_value('eta_p', 2.5)
model.fit(gamma_dot, tau_exp, loss="huber", ftol=1e-6)

# Inspect fitted parameters
print(f"Yield stress: {model.parameters.get_value('sigma_y'):.2f} Pa")
print(f"Plastic viscosity: {model.parameters.get_value('eta_p'):.3f} Pa·s")

# Predict and plot
tau_pred = model.predict(gamma_dot)

Model Comparison

Bingham vs Herschel-Bulkley:

• Bingham: Linear post-yield (\(n = 1\))

• Herschel-Bulkley: Power-law post-yield (\(\tau = \tau_y + K\dot{\gamma}^n\))

• Use Bingham when post-yield flow is Newtonian; HB for shear-thinning/thickening

Bingham vs Casson:

• Bingham: \(\tau = \tau_y + \eta_p \dot{\gamma}\)

• Casson: \(\sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\eta_{\infty} \dot{\gamma}}\)

• Casson better for blood and chocolate; Bingham for suspensions

Bingham vs Carreau:

• Bingham: Discontinuous yielding

• Carreau: Smooth shear-thinning without yield

• Combine for materials with both yield stress and gradual thinning

Limitations

1. Pre-yield behavior: Assumes rigid solid; real materials show viscoelastic creep

2. Sharp yielding: Real yield is gradual transition, not instantaneous

3. Newtonian post-yield: Cannot capture shear-thinning/thickening beyond yield

4. No thixotropy: Static model, ignores structural evolution

5. Wall slip: Requires careful geometry selection

What You Can Learn

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

Parameter Interpretation

Yield Stress ( \(\tau_y\) ):

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

• Low yield stress (< 10 Pa): Weak structure. Material flows easily under gravity or light handling. Examples: dilute suspensions, low-fat mayo.

• Moderate yield stress (10–100 Pa): Functional structure for most applications. Sufficient to prevent sedimentation and sagging, yet dispensable with reasonable force.

• High yield stress (> 100 Pa): Strong network requiring significant force to initiate flow. Examples: toothpaste, heavy-duty grease, cement paste.

For graduate students: The yield stress scales with microstructural parameters. For colloidal suspensions: \(\tau_y \propto \phi^2 G_p / a\) where \(\phi\) is volume fraction, \(G_p\) is particle modulus, and \(a\) is particle size. For attractive systems, \(\tau_y\) increases exponentially with interparticle attraction strength.

For practitioners: Use \(\tau_y\) to assess shelf stability. A mayonnaise needs \(\tau_y > 50\) Pa to prevent oil separation; a paint needs \(\tau_y > 5\) Pa to avoid sagging on vertical surfaces.

Plastic Viscosity ( \(\eta_p\) ):

The plastic viscosity governs post-yield energy dissipation:

• Low \(\eta_p\) (< 0.1 Pa·s): Thin flow once yielded. Good for easy pumping but may cause splashing or poor coating uniformity.

• Moderate \(\eta_p\) (0.1–5 Pa·s): Balanced flow. Typical for most applications requiring controlled spreading or mixing.

• High \(\eta_p\) (> 5 Pa·s): Viscous flow requiring sustained energy input. Common in heavy pastes and slurries.

For graduate students: The plastic viscosity includes contributions from the continuous phase viscosity, hydrodynamic interactions between particles, and the rate of network breakdown. For concentrated suspensions: \(\eta_p \approx \eta_s (1 - \phi/\phi_m)^{-2}\) where \(\phi_m\) is the maximum packing fraction.

For practitioners: The pumping power scales with \(\eta_p\). Reducing particle size or concentration lowers \(\eta_p\) and pumping costs, but may also reduce \(\tau_y\) and shelf stability.

Material Classification

Bingham Fluid Classification

\(\tau_y / \eta_p\) Ratio	Behavior	Typical Materials	Process Implications
Low ratio (< 10 s\(^{-1}\))	“Thick and easy”	Light sauces, thin lotions	Easy pumping, gravity flow possible
Moderate (10–100 s\(^{-1}\))	Balanced plasticity	Mayonnaise, drilling mud	Standard processing equipment
High ratio (> 100 s\(^{-1}\))	“Stiff paste”	Toothpaste, cement	High pressure extrusion needed

The ratio \(\tau_y / \eta_p\) has units of shear rate and indicates the characteristic rate where yield stress and viscous stress are comparable.

Engineering Applications

Pipe Flow Design:

The Buckingham-Reiner equation predicts pressure drop:

\[\frac{\Delta P}{L} = \frac{8 \eta_p Q}{\pi R^4} \left[1 + \frac{1}{3} Bi - \frac{4}{3} Bi^{-3} \right]^{-1}\]

where \(Bi = \tau_y R / (\eta_p \bar{v})\) is the Bingham number. For \(Bi > 3\), plug flow dominates and pressure scales with \(\tau_y/R\).

Coating and Spreading:

For gravity-driven leveling on an inclined surface:

• Material will not flow if \(\tau_y > \rho g h \sin\theta\)

• Use this to size layer thickness \(h\) for sag prevention

Mixing Power:

Anchor or helical impellers are preferred. Power requirement scales as:

\[P \propto \tau_y V + \eta_p (\dot{\gamma}_{avg}) V\]

where \(V\) is vessel volume and \(\dot{\gamma}_{avg}\) is average shear rate in the mixer.

Diagnostic Indicators

Warning signs in fitted parameters:

• \(\tau_y\) → 0: Material is Newtonian or nearly so. Check if yield stress model is appropriate; consider using Carreau Model instead.

• \(\tau_y\) negative: Fitting artifact from noisy low-rate data. Constrain to \(\tau_y \geq 0\) or use robust fitting.

• \(\eta_p\) unexpectedly low: Check for wall slip or instrument calibration issues.

• Strong correlation between \(\tau_y\) and \(\eta_p\): Insufficient data range. Extend measurements to higher shear rates for better separation.

• Systematic residuals: If residuals curve, the material shows shear-thinning post-yield. Use Herschel-Bulkley Model instead.

Application Examples

Quality Control for Food Products:

Monitor \(\tau_y\) as primary QC metric. A 20% drop in \(\tau_y\) indicates batch problems (wrong emulsifier ratio, insufficient homogenization).

Drilling Mud Formulation:

Target \(\tau_y = 5-15\) Pa for cuttings suspension with \(\eta_p\) < 0.1 Pa·s for easy circulation. The API recommends reporting both 6 rpm and 300 rpm readings for Bingham analysis.

Cement Mix Design:

Fresh concrete workability correlates with Bingham parameters. Self-compacting concrete requires \(\tau_y < 60\) Pa and \(\eta_p < 50\) Pa·s.

Fitting Guidance

Initialization

1. From flow curve: Linear fit of high shear rate region → slope = \(\eta_p\), intercept = \(\tau_y\)

2. From controlled stress: Stress at flow onset → \(\tau_y\)

3. Robust estimation: Median of multiple yield determinations

Optimization

• Use Huber loss to down-weight noisy pre-yield data

• Weighted least squares: Higher weights on post-yield region where \(\dot{\gamma}\) is reliable

• Constrain \(\tau_y \geq 0\) and \(\eta_p > 0\)

• Verify: Residuals should be random in post-yield region

Handling Pre-Yield Data

• Option 1: Exclude data below \(\dot{\gamma} < 0.01\) s^-1 (noisy, not truly rigid)

• Option 2: Fit only post-yield region (\(\tau > 1.1 \tau_y\))

• Option 3: Use robust loss (Huber, Tukey) to reduce influence of outliers

Troubleshooting

Problem	Cause	Solution
Negative \(\tau_y\) estimates	Noisy sub-yield data dominating fit	Use robust loss (Huber), exclude low \(\dot{\gamma}\) points, or constrain \(\tau_y \geq 0\)
Poor fit quality despite linear appearance	Thixotropic hysteresis (up-ramp ≠ down-ramp)	Use consistent pre-shear protocol or fit up/down separately
Apparent viscosity shows curvature	Material exhibits shear-thinning beyond yielding	Use Herschel-Bulkley Model (\(n < 1\)) or Casson model
Scatter at low shear rates	Instrument torque resolution, slip, or structural recovery	Use vane geometry, faster ramp rate, or exclude \(\dot{\gamma} < 0.01\) s-1

Tips & Best Practices

1. Pre-shear consistently: Erase mechanical history before each measurement

2. Use vane geometry: Minimizes wall slip for yield stress materials

3. Bidirectional sweeps: Check for thixotropic hysteresis

4. Robust fitting: Huber or Tukey loss to handle pre-yield noise

5. Validate yield stress: Compare flow curve, stress ramp, and vane methods

6. Temperature control: \(\tau_y\) and \(\eta_p\) are temperature-sensitive (±0.1°C)

7. Avoid evaporation: Use solvent trap for aqueous systems

References

[1]

Bingham, E. C. Fluidity and Plasticity. McGraw-Hill, New York (1922). The original description of the Bingham plastic model.

[2]

Barnes, H. A. “The yield stress—a review or ‘παντα ρει’—everything flows?” Journal of Non-Newtonian Fluid Mechanics, 81, 133–178 (1999). https://doi.org/10.1016/S0377-0257(98)00094-9

[3]

Coussot, P., and Ancey, C. “Rheophysical classification of concentrated suspensions and granular pastes.” Physical Review E, 59, 4445–4457 (1999). https://doi.org/10.1103/PhysRevE.59.4445

[4]

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

[5]

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

[6]

Balmforth, N. J., Frigaard, I. A., and Ovarlez, G. “Yielding to stress: Recent developments in viscoplastic fluid mechanics.” Annual Review of Fluid Mechanics, 46, 121–146 (2014). https://doi.org/10.1146/annurev-fluid-010313-141424

[7]

Cheng, D. C.-H. “Yield stress: A time-dependent property and how to measure it.” Rheologica Acta, 25, 542–554 (1986). https://doi.org/10.1007/BF01774406

[8]

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

[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

See Also

Related Flow Models

• Herschel-Bulkley Model — generalizes Bingham with power-law post-yield slope (\(n \neq 1\))

• Power-Law (Ostwald–de Waele) — zero-yield limit for simple shear-thinning/thickening fits

• Carreau Model — smooth transition between Newtonian plateaus without yield

• Fractional Zener Solid-Liquid (Fractional) — combines fractional relaxation with elastic plateau

Transforms

• SmoothDerivative — differentiate torque signals for stress calculation

• MutationNumber — monitor structural breakdown during yielding

Examples

• ../../examples/flow/01-bingham-fitting — step-by-step Bingham parameter estimation

• ../../examples/advanced/02-yield-stress-comparison — comparing Bingham, HB, and Casson