Fluidity Local (Homogeneous Fluidity Model) — Handbook

Quick Reference

• Use when: Yield-stress fluids, thixotropic materials, aging systems with homogeneous (spatially uniform) flow

• Parameters: 9 (\(G\), \(\tau_y\), \(K\), \(n_{\text{flow}}\), \(f_{\text{eq}}\), \(f_\infty\), \(\theta\), \(a\), \(n_{\text{rejuv}}\))

• Key equation: \(\dot{\sigma} = G\dot{\gamma} - f(t)\sigma\)

• Test modes: Oscillation, relaxation, creep, steady shear, startup, LAOS

• Material examples: Mayonnaise, drilling muds, waxy crude oils, colloidal gels, greases, thixotropic paints

Notation Guide

Symbol	Meaning
\(f(t)\)	Fluidity (inverse relaxation time), \(f = 1/\tau\)
\(G\)	Elastic modulus
\(\sigma\)	Shear stress
\(\dot{\gamma}\)	Shear rate
\(f_{\rm eq}\)	Equilibrium (resting) fluidity
\(f_\infty\)	Infinite-shear fluidity
\(\tau_{\rm age}\)	Aging timescale
\(a, n\)	Rejuvenation parameters

Overview

The Local Fluidity Model provides a minimal yet powerful description of yield-stress fluids and thixotropic materials. Rather than prescribing a fixed viscosity, the model introduces a time-dependent fluidity field \(f(t) = 1/\tau(t)\), which represents the material’s inverse characteristic relaxation time.

The key physical insight is the competition between two opposing processes:

1. Aging (structural buildup): At rest, the material’s microstructure rebuilds, decreasing fluidity toward a solid-like equilibrium

2. Rejuvenation (shear-induced breakdown): Under flow, mechanical forcing breaks down structure, increasing fluidity toward a liquid-like state

This competition naturally produces:

• Yield stress behavior (solid-like at rest)

• Thixotropy (time-dependent viscosity)

• Stress overshoots in startup flows

• Power-law creep and relaxation

Historical Context

Fluidity-based models trace their origins to Bingham’s concept of a “coefficient of mobility” (1922) and were formalized by Coussot, Nguyen, and collaborators [1] [2]. The local (0D) model presented here is the spatially homogeneous limit, suitable when shear banding and spatial gradients are negligible.

The approach is closely related to:

• Lambda models for thixotropy (de Souza Mendes, Mujumdar)

• Inelastic/elastic thixotropic models (Mewis, Wagner)

• Kinetic theory approaches (Dullaert, Mewis)

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Physical Foundations

Maxwell-Like Constitutive Framework

The model adopts a Maxwell-type viscoelastic framework with a dynamically evolving relaxation rate:

\[\dot{\sigma}(t) = G \dot{\gamma}(t) - f(t) \sigma(t)\]

This equation describes:

• Elastic loading: The \(G\dot{\gamma}\) term represents elastic stress buildup

• Viscous relaxation: The \(-f\sigma\) term represents stress dissipation with rate proportional to current fluidity

When \(f\) is constant, this reduces to the standard Maxwell model with \(\tau = 1/f\). The novelty is allowing \(f\) to evolve dynamically based on deformation history.

Aging vs. Rejuvenation Dynamics

The fluidity evolution captures the microstructural kinetics:

\[\dot{f}(t) = \underbrace{\frac{f_{\rm eq} - f}{\tau_{\rm age}}}_{\text{aging}} + \underbrace{a |\dot{\gamma}(t)|^n (f_\infty - f)}_{\text{rejuvenation}}\]

Aging term: Drives fluidity toward \(f_{\rm eq}\) on timescale \(\tau_{\rm age}\). For yield-stress fluids, \(f_{\rm eq} \approx 0\) (solid-like at rest).

Rejuvenation term: Flow at rate \(|\dot{\gamma}|\) drives fluidity toward \(f_\infty\) (fluid-like). The power \(n\) controls the sensitivity to shear rate:

• \(n = 1\): Linear shear-rate dependence

• \(n > 1\): Super-linear (stronger at high rates)

• \(n < 1\): Sub-linear (saturating at high rates)

Physical Interpretation

The fluidity \(f\) can be interpreted as:

1. Inverse viscosity: \(\eta_{\rm eff} = G/f\)

2. Relaxation rate: Characteristic time \(\tau = 1/f\)

3. Structural parameter: Higher \(f\) means more “broken down” structure

4. Mobility: Rate at which the material can flow under stress

Tip

Key Physical Intuition

Think of \(f\) as measuring “how liquid” the material currently is:

• Low \(f\) (near \(f_{\rm eq}\)): Solid-like, structured, high viscosity

• High \(f\) (near \(f_\infty\)): Liquid-like, broken down, low viscosity

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Mathematical Formulation

Core Equations

The Local Fluidity Model is defined by two coupled ordinary differential equations:

Constitutive stress equation:

(1)\[\dot{\sigma}(t) = G \dot{\gamma}(t) - f(t) \sigma(t)\]

Fluidity evolution equation:

(2)\[\dot{f}(t) = \frac{f_{\rm eq} - f}{\tau_{\rm age}} + a |\dot{\gamma}(t)|^n (f_\infty - f)\]

Initial conditions: \(\sigma(0) = \sigma_0\), \(f(0) = f_0\) (typically \(f_0 = f_{\rm eq}\) for a well-rested sample).

Steady-State Analysis

Under constant shear rate \(\dot{\gamma}\), the system reaches steady state with \(\dot{\sigma} = 0\) and \(\dot{f} = 0\):

Steady fluidity:

(3)\[f_{\rm ss}(\dot{\gamma}) = \frac{f_{\rm eq}/\tau_{\rm age} + a|\dot{\gamma}|^n f_\infty}{1/\tau_{\rm age} + a|\dot{\gamma}|^n}\]

Steady stress (flow curve):

(4)\[\sigma_{\rm ss}(\dot{\gamma}) = \frac{G \dot{\gamma}}{f_{\rm ss}(\dot{\gamma})}\]

This produces a flow curve with:

• Yield stress \(\sigma_y = G \cdot \lim_{\dot{\gamma}\to 0} \dot{\gamma}/f_{\rm ss}(\dot{\gamma})\) when \(f_{\rm eq} \approx 0\)

• Shear-thinning at high rates as \(f_{\rm ss} \to f_\infty\)

Effective Viscosity

The effective viscosity at steady state:

\[\eta_{\rm eff}(\dot{\gamma}) = \frac{\sigma_{\rm ss}}{\dot{\gamma}} = \frac{G}{f_{\rm ss}(\dot{\gamma})}\]

At low shear rates (with \(f_{\rm eq} \ll f_\infty\)):

\[\eta_{\rm eff} \sim \frac{G \tau_{\rm age}}{f_{\rm eq}} \to \infty \quad \text{as } f_{\rm eq} \to 0\]

At high shear rates:

\[\eta_{\rm eff} \to \frac{G}{f_\infty}\]

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Protocol-Specific Equations

Rotation (Steady-State Flow Curve)

Protocol: Constant global shear rate \(\dot{\gamma} = \text{const}\).

Equations:

\[\dot{\sigma} = G\dot{\gamma} - f\sigma, \qquad \dot{f} = (f_{\rm eq}-f)/\tau_{\rm age} + a|\dot{\gamma}|^n(f_\infty-f)\]

Steady state:

\[\sigma(\dot{\gamma}) = \frac{G\dot{\gamma}}{f(\dot{\gamma})}, \qquad f(\dot{\gamma}) = \frac{f_{\rm eq}/\tau_{\rm age} + a|\dot{\gamma}|^n f_\infty}{1/\tau_{\rm age} + a|\dot{\gamma}|^n}\]

Behavior:

• For \(f_{\rm eq} \approx 0\): Apparent yield stress emerges

• Shear-thinning at all rates

• Flow index related to exponent \(n\)

Start-Up Shear

Protocol: Apply constant shear rate \(\dot{\gamma}_0\) at \(t = 0\) from rest.

\[\dot{\gamma}(t) = \dot{\gamma}_0 H(t)\]

Equations:

\[\dot{\sigma} = G\dot{\gamma}_0 - f\sigma, \qquad \dot{f} = (f_{\rm eq}-f)/\tau_{\rm age} + a|\dot{\gamma}_0|^n(f_\infty-f)\]

Initial conditions: \(\sigma(0) = 0\), \(f(0) = f_{\rm eq}\)

Behavior:

• Initial elastic response: \(\sigma \approx G \dot{\gamma}_0 t\) for \(t \ll 1/f_{\rm eq}\)

• Stress overshoot if \(\tau_{\rm age}\) is large (aging competes with rejuvenation)

• Approach to steady state: \(\sigma \to \sigma_{\rm ss}(\dot{\gamma}_0)\)

The stress overshoot reflects the lag between structural breakdown and stress relaxation.

Stress Relaxation (Step Strain)

Protocol: Apply step strain \(\gamma_0\) at \(t = 0\), then hold \(\dot{\gamma} = 0\).

Initial stress: \(\sigma(0^+) = G\gamma_0\)

Equations for \(t > 0\):

\[\dot{\sigma} = -f\sigma, \qquad \dot{f} = (f_{\rm eq}-f)/\tau_{\rm age}\]

Solution:

\[\sigma(t) = \sigma(0^+) \exp\left(-\int_0^t f(s) \, ds\right)\]

With \(f(t) = f_0 e^{-t/\tau_{\rm age}} + f_{\rm eq}(1 - e^{-t/\tau_{\rm age}})\):

Behavior:

• If \(f_{\rm eq} \approx 0\): Stress decays more slowly as \(f \to 0\) (aging)

• Non-exponential relaxation due to evolving fluidity

• For aged samples (\(f_0 = f_{\rm eq} \approx 0\)): \(\sigma(t) \approx \sigma(0^+)\) (solid-like plateau)

Creep (Step Stress)

Protocol: Apply constant stress \(\Sigma_0\) at \(t = 0\).

Equations:

\[\dot{\gamma}(t) = \frac{\Sigma_0}{G} f(t)\]

\[\dot{f} = (f_{\rm eq}-f)/\tau_{\rm age} + a\left|\frac{\Sigma_0}{G}f\right|^n (f_\infty-f)\]

Strain accumulation:

\[\gamma(t) = \frac{\Sigma_0}{G} \int_0^t f(s) \, ds\]

Behavior:

• Bifurcation at yield stress \(\sigma_y\):

  • \(\Sigma_0 < \sigma_y\): Fluidity decays, strain saturates (solid-like)

  • \(\Sigma_0 > \sigma_y\): Fluidity grows, steady flow (liquid-like)

• Delayed yielding near \(\sigma_y\): Long induction time before flow onset

• Power-law creep regime: \(\gamma(t) \sim t^\alpha\) with \(\alpha < 1\) during transient

Oscillatory Shear (SAOS and LAOS)

Protocol (strain-controlled):

\[\gamma(t) = \gamma_0 \sin(\omega t), \qquad \dot{\gamma}(t) = \gamma_0 \omega \cos(\omega t)\]

Equations:

\[\dot{\sigma} = G\dot{\gamma}(t) - f(t)\sigma, \qquad \dot{f} = (f_{\rm eq}-f)/\tau_{\rm age} + a|\dot{\gamma}(t)|^n(f_\infty-f)\]

SAOS (Small Amplitude):

• Fluidity nearly constant: \(f(t) \approx \bar{f}\)

• Linear moduli from stress-strain relationship

• Storage modulus: \(G' \approx \frac{G \omega^2}{f^2 + \omega^2}\)

• Loss modulus: \(G'' \approx \frac{G \omega f}{f^2 + \omega^2}\)

LAOS (Large Amplitude):

• Strong \(f(t)\) modulation within each cycle

• Higher harmonics in stress response

• Intracycle softening and stiffening

• Lissajous-Bowditch curves show nonlinear features

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Governing Equations

Core Coupled ODEs

The Local Fluidity Model is governed by two coupled ordinary differential equations that describe the evolution of stress and fluidity under applied deformation:

Stress Evolution:

\[\frac{d\sigma}{dt} = G \frac{d\gamma}{dt} - f(t) \sigma(t)\]

Fluidity Evolution:

\[\frac{df}{dt} = \frac{f_{\rm eq} - f}{\tau_{\rm age}} + a \left|\frac{d\gamma}{dt}\right|^n (f_\infty - f)\]

Initial Conditions:

\[\sigma(0) = \sigma_0, \quad f(0) = f_0 \quad \text{(typically } f_0 = f_{\rm eq} \text{ for aged samples)}\]

These equations capture the key physics:

• Elastic loading via \(G\dot{\gamma}\) (strain rate drives stress buildup)

• Viscous relaxation via \(-f\sigma\) (fluidity controls stress dissipation)

• Structural aging via \((f_{\rm eq} - f)/\tau_{\rm age}\) (rebuilding at rest)

• Shear rejuvenation via \(a|\dot{\gamma}|^n(f_\infty - f)\) (breakdown under flow)

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What You Can Learn

From fitting Local Fluidity to experimental data, you can extract insights about yield stress emergence, thixotropic kinetics, and microstructural evolution in homogeneous flows.

Parameter Interpretation

f (Fluidity):

Time-dependent inverse relaxation time \(f = 1/\tau\), providing direct interpretation of effective viscosity \(\eta_{\text{eff}}(t) = G/f(t)\). For graduate students: \(f\) tracks microstructural state: low \(f\) (\(f \to f_{\text{eq}}\)) = highly structured solid-like, high \(f\) (\(f \to f_\infty\)) = broken-down liquid-like. Evolution: \(df/dt = (f_{\text{eq}} - f)/\tau_{\text{age}} + a|\dot{\gamma}|^n(f_\infty - f)\). At steady state: \(f_{ss} = [f_{\text{eq}}/\tau_{\text{age}} + a|\dot{\gamma}|^n \cdot f_\infty]/[1/\tau_{\text{age}} + a|\dot{\gamma}|^n]\). Connects to SGR effective temperature \(x\) via \(f \sim x\). For practitioners: Measure indirectly via \(\eta(t) = G/f(t)\) in startup tests. For yield-stress fluids, \(f_{\text{eq}} \approx 0\) (solid at rest), \(f_\infty = G/\eta_\infty\) (liquid at high shear).

f_eq (Equilibrium Fluidity):

Fluidity at complete rest, controlling yield stress behavior. For graduate students: For true yield-stress fluids, \(f_{\text{eq}} \to 0\), giving \(\sigma_y = G \cdot \lim(\dot{\gamma}/f_{ss})\) as \(\dot{\gamma} \to 0\). Nonzero \(f_{\text{eq}}\) produces viscoelastic liquid (no yield stress). Sets solid-like viscosity \(\eta_{\text{rest}} = G/f_{\text{eq}}\). For practitioners: \(f_{\text{eq}} \approx 10^{-6}\) to \(10^{-3}\) s^-1 for yield-stress fluids (mayonnaise, drilling muds). \(f_{\text{eq}} > 10^{-2}\) s^-1 indicates viscoelastic liquid without true yield stress.

f_∞ (Infinite-Shear Fluidity):

Fluidity limit at very high shear rates (fully broken-down structure). For graduate students: Sets minimum viscosity \(\eta_{\infty} = G/f_\infty\) at high shear. Difference \(f_\infty - f_{\text{eq}}\) quantifies maximum structural change. Shear-thinning ratio \(\eta_{\text{rest}}/\eta_{\infty} = f_\infty/f_{\text{eq}}\) (typically \(10^3\)–\(10^6\) for strong thixotropic materials). For practitioners: Extract from high-shear plateau in flow curves. Typical: \(f_\infty = 10^{-1}\) to \(10^2\) s^-1. Higher \(f_\infty\) = lower high-shear viscosity.

\(\tau_{age}\) (Aging Timescale):

Characteristic time for structure rebuilding at rest. For graduate students: First-order aging kinetics: \(f \to f_{\text{eq}}\) with time constant \(\tau_{\text{age}}\). Sets width of thixotropic hysteresis loops and stress overshoot position in startup. Competes with rejuvenation time \(\tau_{\text{rej}} \sim 1/(a|\dot{\gamma}|^n)\). For thermally-activated processes, \(\tau_{\text{age}} \sim \tau_0 \exp(\Delta E_{\text{build}}/k_B T)\). For practitioners: Measure via rest-time dependent startup tests or creep recovery. Fast aging (\(\tau_{\text{age}} = 1\text{--}10\) s) vs slow aging (\(\tau_{\text{age}} = 10^2\text{--}10^4\) s). Critical for pumping restart protocols.

a, n (Rejuvenation Parameters):

Control shear-induced breakdown: \(df/dt|_{rej} = a|\dot{\gamma}|^n(f_{\infty} - f)\). For graduate students: \(a\) is breakdown amplitude, \(n\) is rate sensitivity (\(n = 1\) linear, \(n > 1\) superlinear). Characteristic shear rate: \(\dot{\gamma}_c \sim (1/(a\tau_{\text{age}}))^{1/n}\) where structure is half-broken. Connects to Herschel-Bulkley exponent via steady-state analysis. For practitioners: Extract from flow curve curvature. Typical: \(a \sim 0.1\text{--}10\), \(n \sim 0.5\text{--}1.5\). Higher \(a\) or \(n\) = more rapid breakdown under flow.

Material Classification

Material Classification from Local Fluidity Parameters

Parameter Range	Material Behavior	Typical Materials	Processing Implications
\(f_{\text{eq}} < 10^{-4}\) s-1, \(\tau_{\text{age}} > 100\) s	Strong yield stress, slow aging	Waxy crude oils, cement pastes	High yield stress, long memory, pumping challenges
\(f_{\text{eq}} = 10^{-4}\) to \(10^{-2}\) s-1, \(\tau_{\text{age}} = 10\text{--}100\) s	Moderate yield stress, intermediate aging	Mayonnaise, drilling muds, paints	Pronounced thixotropy, restart protocols needed
\(f_{\text{eq}} > 10^{-2}\) s-1, \(\tau_{\text{age}} < 10\) s	Weak/no yield stress, fast recovery	Soft gels, cosmetics, dilute emulsions	Minimal thixotropy, easy flow
\(n \approx 1\)	Linear breakdown	Simple thixotropic fluids	Predictable shear-thinning
\(n > 1.5\)	Superlinear breakdown	Complex soft solids with abrupt yielding	Strong rate-dependence, flow instabilities

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Parameters

Parameters

Name	Symbol	Units	Bounds	Notes
G	\(G\)	Pa	\(G > 0\)	Elastic modulus; sets stress scale
tau_y	\(\tau_y\)	Pa	\(\tau_y \geq 0\)	Yield stress
K	\(K\)	Pa·sn	\(K > 0\)	Flow consistency (Herschel-Bulkley K parameter)
n_flow	\(n_{\rm flow}\)	—	\(0.1 \leq n_{\rm flow} \leq 2\)	Flow exponent (Herschel-Bulkley n parameter)
f_eq	\(f_{\rm eq}\)	1/(Pa·s)	\(f_{\rm eq} \geq 0\)	Equilibrium fluidity; \(\approx 0\) for yield-stress fluids
f_inf	\(f_\infty\)	1/(Pa·s)	\(f_\infty > f_{\rm eq}\)	Infinite-shear fluidity; sets minimum viscosity
theta	\(\theta\)	s	\(\theta > 0\)	Aging timescale; controls buildup rate at rest
a	\(a\)	—	\(a \geq 0\)	Rejuvenation amplitude
n_rejuv	\(n_{\rm rejuv}\)	—	\(0 \leq n_{\rm rejuv} \leq 2\)	Rejuvenation exponent; typically \(0.5 \leq n \leq 2\)

Parameter Interpretation

G (Elastic Modulus):

• Physical meaning: Stiffness of the elastic network

• Typical ranges:

  • Colloidal gels: \(10^0 - 10^2\) Pa

  • Emulsions: \(10^1 - 10^3\) Pa

  • Drilling muds: \(10^1 - 10^2\) Pa

f_eq (Equilibrium Fluidity):

• Physical meaning: Fluidity at complete rest

• For yield-stress fluids: \(f_{\rm eq} \approx 10^{-6}\) to \(10^{-3}\) s^-1

• For viscoelastic fluids without yield stress: \(f_{\rm eq} > 0\)

f_inf (Infinite-Shear Fluidity):

• Physical meaning: Fluidity at very high shear (fully broken-down structure)

• Typical ranges: \(10^{-1}\) to \(10^2\) s^-1

• High-shear viscosity: \(\eta_\infty = G / f_\infty\)

tau_age (Aging Timescale):

• Physical meaning: Time for structure to rebuild at rest

• Typical ranges:

  • Fast aging: \(1 - 10\) s

  • Slow aging: \(10^2 - 10^4\) s

• Thixotropic time: Related to loop hysteresis in flow curves

a, n (Rejuvenation Parameters):

• Physical meaning: Sensitivity of breakdown to shear rate

• Typical values: \(a \sim 0.1 - 10\), \(n \sim 0.5 - 1.5\)

• Relationship to flow index: Connected to Herschel-Bulkley exponent

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Validity and Assumptions

Model Assumptions

1. Homogeneous flow: No spatial gradients in fluidity or velocity (shear banding excluded)

2. Affine deformation: Microstructure deforms with macroscopic strain

3. First-order kinetics: Simple exponential approach to equilibrium

4. Isothermal: Temperature effects not explicitly modeled

5. Scalar fluidity: Single internal variable (no tensorial microstructure)

Data Requirements

• Flow curves: Steady shear \(\sigma(\dot{\gamma})\) over 2+ decades

• Transient data: Start-up, step stress, or hysteresis loops

• Optional: Oscillatory sweeps, relaxation tests

Limitations

No shear banding:

The homogeneous model cannot capture spatial heterogeneity. For shear-banded flows, use the Fluidity Nonlocal (Coussot-Ovarlez Cooperative Model) — Handbook model.

Simple kinetics:

Real materials may have multiple structural timescales. Consider multi-lambda extensions for complex thixotropic behavior.

No normal stresses:

The scalar model does not predict \(N_1, N_2\). Use tensorial extensions for extensional flows or normal stress measurements.

Pre-yielding elasticity:

The linear \(G\dot{\gamma}\) term may not capture complex pre-yield behavior (e.g., fatigue, microslip).

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Fitting Guidance

Parameter Initialization

Method 1: From flow curve

Step 1: Identify yield stress \(\sigma_y\) from \(\sigma(\dot{\gamma})\) plot

Step 2: High-shear viscosity: \(\eta_\infty = \lim_{\dot{\gamma}\to\infty} \sigma/\dot{\gamma}\)

Step 3: Estimate \(G \approx 10 \sigma_y\) (typical for yield-stress fluids)

Step 4: \(f_\infty = G / \eta_\infty\)

Step 5: \(f_{\rm eq} \approx 10^{-4} f_\infty\) (small but nonzero for numerical stability)

Method 2: From transient response

Step 1: Fit exponential decay in step-strain relaxation to get \(\tau_{\rm age}\)

Step 2: Measure stress overshoot magnitude and peak time in startup

Step 3: Use overshoot characteristics to estimate \(a, n\)

Optimization Algorithm Selection

RheoJAX default: NLSQ (GPU-accelerated)

• Recommended for quick fits (6 parameters)

• Use good initial guesses from flow curve analysis

Bayesian inference (NUTS)

• Highly recommended for uncertainty quantification

• Critical for distinguishing yield stress vs. very high viscosity

• Use informative priors from physical constraints

Bounds:

• \(G\): [1e-1, 1e6] Pa

• \(f_{\rm eq}\): [1e-8, 1e-1] s^-1

• \(f_\infty\): [1e-2, 1e3] s^-1

• \(\tau_{\rm age}\): [1e-1, 1e5] s

• \(a\): [1e-3, 1e2]

• \(n\): [0.3, 2.5]

Troubleshooting

Fitting diagnostics

Problem	Diagnostic	Solution
\(f_{\rm eq}\) hits lower bound	True yield stress material	Fix at small value (e.g., \(10^{-6}\))
Poor transient fits	Wrong \(\tau_{\rm age}\) or \(a, n\)	Fit transient data separately first
Oscillations in prediction	Stiff ODE system	Reduce time step; use implicit integrator
Flow curve mismatch at low \(\dot{\gamma}\)	Wall slip or banding	Consider Fluidity Nonlocal (Coussot-Ovarlez Cooperative Model) — Handbook model
Fitted \(\tau_{\rm age}\) unrealistic	Data insufficient for aging dynamics	Include step-stress or hysteresis data

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Usage

Basic Example

import numpy as np
from rheojax.models import FluidityLocal

# Shear rate data
gamma_dot = np.logspace(-3, 2, 50)

# Create and fit model
model = FluidityLocal()
model.fit(gamma_dot, sigma_data, test_mode='steady_shear')

# Extract parameters
G = model.parameters.get_value('G')
f_eq = model.parameters.get_value('f_eq')
tau_age = model.parameters.get_value('tau_age')

print(f"Elastic modulus G = {G:.1f} Pa")
print(f"Aging timescale = {tau_age:.1f} s")

Bayesian Inference

from rheojax.models import FluidityLocal

model = FluidityLocal()
model.fit(gamma_dot, sigma_data, test_mode='steady_shear')

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

# Get credible intervals for yield stress
intervals = model.get_credible_intervals(result.posterior_samples)
print(f"tau_age 95% CI: [{intervals['tau_age'][0]:.1f}, {intervals['tau_age'][1]:.1f}] s")

Transient Start-Up

from rheojax.models import FluidityLocal
import numpy as np

model = FluidityLocal()

# Fit to flow curve first
model.fit(gamma_dot, sigma_ss, test_mode='steady_shear')

# Predict startup transient
t = np.linspace(0, 100, 1000)
gamma_dot_0 = 1.0  # Applied shear rate
sigma_t = model.predict_startup(t, gamma_dot_0)

Creep Analysis

from rheojax.models import FluidityLocal
import numpy as np

model = FluidityLocal()
model.fit(gamma_dot, sigma_data, test_mode='steady_shear')

# Predict creep response
t = np.logspace(-2, 4, 100)
sigma_0 = 50.0  # Applied stress (Pa)
gamma_t = model.predict_creep(t, sigma_0)

# Check for delayed yielding
yield_time = t[np.argmax(np.gradient(gamma_t) > threshold)]

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See Also

• Fluidity Nonlocal (Coussot-Ovarlez Cooperative Model) — Handbook — Nonlocal model for shear banding and cooperative flows

• SGR Conventional (Soft Glassy Rheology) — Handbook — Soft Glassy Rheology model with trap distributions

• Hébraud–Lequeux (HL) Model — Handbook — Hébraud-Lequeux model for pasty materials

• Herschel-Bulkley Model — Simpler yield stress model for steady shear only

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API References

• Module: rheojax.models

• Class: rheojax.models.FluidityLocal

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References

[1]

Coussot, P., Nguyen, Q. D., Huynh, H. T., & Bonn, D. “Avalanche behavior in yield stress fluids.” Physical Review Letters, 88, 175501 (2002). https://doi.org/10.1103/PhysRevLett.88.175501

[2]

Coussot, P., Nguyen, Q. D., Huynh, H. T., & Bonn, D. “Viscosity bifurcation in thixotropic, yielding fluids.” Journal of Rheology, 46(3), 573-589 (2002). https://doi.org/10.1122/1.1459447

[3]

de Souza Mendes, P. R. “Modeling the thixotropic behavior of structured fluids.” Journal of Non-Newtonian Fluid Mechanics, 164(1-3), 66-75 (2009). https://doi.org/10.1016/j.jnnfm.2009.08.005

[4]

Mewis, J. & Wagner, N. J. “Thixotropy.” Advances in Colloid and Interface Science, 147-148, 214-227 (2009). https://doi.org/10.1016/j.cis.2008.09.005

[5]

Bocquet, L., Colin, A., & Ajdari, A. “Kinetic theory of plastic flow in soft glassy materials.” Physical Review Letters, 103, 036001 (2009). https://doi.org/10.1103/PhysRevLett.103.036001

[6]

Picard, G., Ajdari, A., Lequeux, F., & Bocquet, L. “Slow flows of yield stress fluids: Complex spatiotemporal behavior within a simple elastoplastic model.” Physical Review E, 71, 010501(R) (2005). https://doi.org/10.1103/PhysRevE.71.010501

[7]

Mansard, V., Colin, A., Chauduri, P., & Bocquet, L. “A molecular dynamics study of non-local effects in the flow of soft jammed particles.” Soft Matter, 9, 7489-7500 (2013). https://doi.org/10.1039/c3sm50847a

[8]

Shaukat, A., Sharma, A., & Joshi, Y. M. “Squeeze flow behavior of (soft glassy) thixotropic material.” Journal of Non-Newtonian Fluid Mechanics, 167-168, 9-17 (2012). https://doi.org/10.1016/j.jnnfm.2011.09.006

[9]

Blackwell, B. C. & Ewoldt, R. H. “A simple thixotropic-viscoelastic constitutive model produces unique signatures in large-amplitude oscillatory shear (LAOS).” Journal of Non-Newtonian Fluid Mechanics, 208-209, 27-41 (2014). https://doi.org/10.1016/j.jnnfm.2014.03.006

[10]

Divoux, T., Barentin, C., & Manneville, S. “Stress overshoot in a simple yield stress fluid: An extensive study combining rheology and velocimetry.” Soft Matter, 7, 9335-9349 (2011). https://doi.org/10.1039/c1sm05740e

Further Reading

• Mujumdar, A., Beris, A. N., & Metzner, A. B. “Transient phenomena in thixotropic systems.” Journal of Non-Newtonian Fluid Mechanics, 102(2), 157-178 (2002).

• Dullaert, K. & Mewis, J. “A structural kinetics model for thixotropy.” Journal of Non-Newtonian Fluid Mechanics, 139(1-2), 21-30 (2006).

• Moller, P., Fall, A., Chikkadi, V., Derks, D., & Bonn, D. “An attempt to categorize yield stress fluid behaviour.” Philosophical Transactions of the Royal Society A, 367(1909), 5139-5155 (2009).

• Ovarlez, G., Rodts, S., Chateau, X., & Coussot, P. “Phenomenology and physical origin of shear localization and shear banding in complex fluids.” Rheologica Acta, 48(8), 831-844 (2009).