Fluidity Models¶

This section documents the Fluidity family of models for thixotropic and elastoviscoplastic materials.

Overview¶

The Fluidity family provides constitutive equations for complex fluids where the relaxation time (or its inverse, fluidity) evolves dynamically due to competing aging and shear-rejuvenation processes. These models capture:

Thixotropy: Time-dependent viscosity from structural buildup/breakdown
Yield stress behavior: Solid-like response at rest, liquid-like under flow
Stress overshoot: Transient peak during startup after aging
Shear banding: Spatial flow heterogeneity (nonlocal variants)
Normal stresses: First normal stress difference \(N_1\) (Saramito EVP)

These models are well-suited for:

Colloidal gels and pastes
Concentrated emulsions (mayonnaise, cosmetics)
Polymer gels (Carbopol, hydrogels)
Drilling fluids and muds
Waxy crude oils
Cement and concrete

Thixotropy Fundamentals

Thixotropy is the reversible, time-dependent decrease in viscosity under constant shear rate, with subsequent recovery at rest. It arises from competition between microstructural breakdown (shear) and buildup (aging).

Physical Mechanisms:

Breakdown: Shear disrupts network bonds, aggregates, or particle structures
Buildup (aging): Brownian motion, attractive forces, or reaction kinetics rebuild structure
Structure parameter (\(\lambda\)): Dimensionless variable tracking microstructural state (0-1)

Characteristic Experimental Signatures:

Hysteresis loops: Different stress-strain rate curves for increasing vs decreasing shear
Stress overshoot: Peak stress in startup flow before steady-state
Delayed yielding: Time-dependent creep response, viscosity bifurcation
Recovery kinetics: Gradual viscosity increase after shear cessation

Common Kinetic Equation:

\[\frac{d\lambda}{dt} = \underbrace{\frac{1-\lambda}{t_{eq}}}_{\text{aging}} - \underbrace{a\lambda|\dot{\gamma}|^c/t_{eq}}_{\text{rejuvenation}}\]

where \(t_{eq}\) is equilibration time, \(a\) is breakdown rate, and \(c\) is shear-rate exponent.

Model Selection Guide:

Model Family	Best For	Key Features
DMT Thixotropic Models	Industrial fluids	Simple kinetics, exponential/HB closures
Isotropic-Kinematic Hardening (IKH) Models	Metal plasticity	Hardening/softening, yield surface evolution
Fluidity Models	Yield stress fluids	Fluidity evolution, Saramito viscoelasticity

Experimental Protocols for Thixotropic Materials:

Three-interval test: Low rate → high rate → low rate to measure breakdown/recovery
Step-rate tests: Instantaneous rate changes to probe kinetics
Startup flow: Constant rate from rest to observe overshoot
Creep: Constant stress to observe delayed yielding

Model Hierarchy¶

Fluidity Family
│
├── FluidityLocal (0D Homogeneous)
│   └── Scalar stress σ
│   └── 9 parameters: G, tau_y, K, n_flow, f_eq, f_inf, theta, a, n_rejuv
│   └── Maxwell-like viscoelasticity
│
├── FluidityNonlocal (1D Spatial)
│   └── Adds cooperativity length ξ
│   └── Shear banding resolution
│   └── Couette/channel flow profiles
│
└── FluiditySaramito EVP (Tensorial)
    │
    ├── Minimal Coupling
    │   └── λ = 1/f only
    │   └── Fewer parameters, identifiable
    │
    └── Full Coupling
        └── λ + τ_y(f) aging yield
        └── Wait-time dependent yield stress

When to Use Which Model¶

Feature / Use Case	FluidityLocal	FluidityNonlocal	FluiditySaramito EVP
Homogeneous flow	✓ Use this	Overkill	✓ If \(N_1\) needed
Shear banding	Cannot capture	✓ Use this	✓ Nonlocal variant
Stress overshoot	✓ Scalar	✓ Scalar	✓ Tensorial
Normal stresses (\(N_1\))	✗	✗	✓ Use this
Von Mises yield	✗ (implicit)	✗ (implicit)	✓ Explicit
Aging yield stress	✗	✗	✓ Full coupling
Creep bifurcation	✓	✓	✓ (enhanced)
Parameters	9	10	10-12
Computational cost	1× (baseline)	2-5×	3-5×

Decision Guide:

Start with FluidityLocal for exploratory analysis and homogeneous flows
Use FluidityNonlocal when shear banding is observed or expected
Use FluiditySaramito when normal stresses, tensorial stress state, or aging-dependent yield stress are important

Quick Comparison¶

Model	Stress Type	Key Extension	Primary Use
FluidityLocal	Scalar \(\sigma\)	Base model	Thixotropic flow curves
FluidityNonlocal	Scalar \(\sigma(y)\)	Cooperativity \(\xi\)	Shear banding
FluiditySaramitoLocal	Tensor [\(\tau_{xx}\), \(\tau_{yy}\), \(\tau_{xy}\)]	UCM + Von Mises	EVP with \(N_1\)
FluiditySaramitoNonlocal	Tensor \(\tau(y)\)	Spatial + tensorial	Banding with \(N_1\)

Key Equations¶

Scalar fluidity evolution (all models):

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

Maxwell constitutive (Local/Nonlocal):

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

Upper-convected Maxwell with plasticity (Saramito EVP):

\[\lambda \overset{\nabla}{\boldsymbol{\tau}} + \alpha(\boldsymbol{\tau})\boldsymbol{\tau} = 2\eta_p \mathbf{D}\]

where \(\lambda = 1/f\) and \(\alpha = \max(0, 1 - \tau_y/|\boldsymbol{\tau}|)\).

Quick Start¶

Local (homogeneous) model:

from rheojax.models.fluidity import FluidityLocal

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

# Simulate startup with stress overshoot
t, stress, fluidity = model.simulate_startup(t, gamma_dot=1.0, t_wait=100)

Nonlocal (shear banding) model:

from rheojax.models.fluidity import FluidityNonlocal

model = FluidityNonlocal(N_y=51, H=1e-3, xi=1e-5)
result = model.simulate_startup(t, gamma_dot=0.1)

# Check for shear banding
is_banded, cv, ratio = model.detect_shear_bands()

Saramito EVP (tensorial) model:

from rheojax.models.fluidity import FluiditySaramitoLocal

# Minimal coupling (most identifiable)
model = FluiditySaramitoLocal(coupling="minimal")
model.fit(gamma_dot, sigma, test_mode='flow_curve')

# Note: tensorial stress (τ_xx, τ_yy, τ_xy) is tracked internally;
# access N1 via transient simulations (simulate_startup, simulate_laos)

Protocol-Specific Recommendations¶

Different experimental protocols and material types are best served by different models in the Fluidity family. Use this guide to select the appropriate variant.

By Experimental Protocol:

Protocol	Recommended Model	Rationale
Standard rheometry (cone-plate, parallel plate)	FluidityLocal or FluiditySaramitoLocal	Homogeneous flow assumption valid; no spatial resolution needed
LAOS with \(N_1\) extraction	FluiditySaramitoLocal	Tensorial stress required for first normal stress difference
Microfluidic confinement (\(H \sim \xi\))	FluidityNonlocal	Gap-dependent flow curves; spatial fluidity profiles
Wide-gap Couette \((R_o - R_i)/R_i > 0.1\)	FluidityNonlocal (with curvature)	Stress gradient matters; velocity profiles accessible
Startup with velocity profiles (PIV, USV)	FluiditySaramitoNonlocal	Validates spatial predictions; extracts \(\xi\)
Creep bifurcation tests	FluidityLocal or FluiditySaramitoLocal	Homogeneous; bifurcation point identifies \(\tau_y\)
Extensional flow (CaBER, filament stretching)	FluiditySaramitoLocal	Tensorial formulation handles uniaxial extension

By Material Type:

Material	Recommended Model	Notes
Carbopol gel	FluiditySaramitoLocal (minimal)	Well-characterized simple yield stress fluid; weak thixotropy; minimal coupling sufficient
Concentrated emulsion (mayonnaise, cosmetics)	FluiditySaramitoLocal (minimal)	Moderate \(N_1\); clear yield; standard thixotropy
Emulsion in microchannel	FluidityNonlocal	Strong confinement effects; \(\xi \sim\) 10-50 \(\mu\text{m}\) typically
Waxy crude oil	FluiditySaramitoLocal (full)	Strong aging-yield coupling; \(\tau_y\) increases significantly with rest
Drilling mud	FluiditySaramitoLocal (full) or DMT	Complex thixotropy; may need aging yield coupling
Cement/concrete	FluiditySaramitoLocal (full)	Hydration-dependent aging; \(\tau_y\) evolves with time
Colloidal glass near jamming	FluidityNonlocal or HL Trap	Cooperativity important; may need statistical mechanics model

Quick Decision Flowchart:

Start
  │
  ├── Need tensorial stress or N_1? ──Yes──► FluiditySaramito*
  │                                              │
  No                                             ├── Spatial profiles? ──Yes──► Nonlocal
  │                                              │
  │                                              └── Homogeneous ──► Local
  │
  ├── Shear banding or confinement? ──Yes──► FluidityNonlocal
  │
  No
  │
  └── Homogeneous thixotropy ──► FluidityLocal

Model Documentation¶

References¶

Coussot, P., Nguyen, Q. D., Huynh, H. T., and Bonn, D. (2002). “Viscosity bifurcation in thixotropic, yielding fluids.” J. Rheol., 46(3), 573-589. https://doi.org/10.1122/1.1459447
Bocquet, L., Colin, A., and Ajdari, A. (2009). “Kinetic theory of plastic flow in soft glassy materials.” Phys. Rev. Lett., 103, 036001. https://doi.org/10.1103/PhysRevLett.103.036001
Saramito, P. (2007). “A new constitutive equation for elastoviscoplastic fluid flows.” J. Non-Newtonian Fluid Mech., 145, 1-14. https://doi.org/10.1016/j.jnnfm.2007.04.004
de Souza Mendes, P. R. & Thompson, R. L. (2012). “A critical overview of elasto-viscoplastic thixotropic modeling.” J. Non-Newtonian Fluid Mech., 187-188, 8-15. https://doi.org/10.1016/j.jnnfm.2012.08.006