Giesekus Nonlinear Viscoelastic Models¶

This section documents the Giesekus family of models for polymer melts and solutions exhibiting shear-thinning, normal stress differences, and stress overshoot behavior.

Overview¶

The Giesekus model (1982) extends the Upper-Convected Maxwell (UCM) framework with a quadratic stress term representing anisotropic molecular mobility:

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

Where:

\(\boldsymbol{\tau}\) is the polymer extra stress tensor
\(\lambda\) is the relaxation time
\(\alpha\) is the mobility factor (\(0 \leq \alpha \leq 0.5\))
\(\eta_p\) is the polymer viscosity
\(\overset{\nabla}{\boldsymbol{\tau}}\) is the upper-convected derivative
\(\mathbf{D}\) is the rate-of-deformation tensor

The mobility factor \(\alpha\) controls shear-thinning:

\(\alpha\) = 0: Recovers UCM (no shear-thinning)
\(\alpha\) = 0.5: Maximum anisotropy/shear-thinning

Model Variants¶

Model	Description
`GiesekusSingleMode`	Single relaxation time with all 6 protocols
`GiesekusMultiMode`	N parallel modes for broad relaxation spectra

Key Features¶

Shear-Thinning Viscosity:

\[\eta(\dot{\gamma}) = \eta_s + \eta_p \cdot f(\text{Wi})\]

where f(Wi) decreases with Weissenberg number Wi = \(\lambda\dot{\gamma}\).

Normal Stress Differences:

\[ \begin{align}\begin{aligned}N_1 = \tau_{xx} - \tau_{yy} > 0 \quad \text{(first normal stress)}\\N_2 = \tau_{yy} - \tau_{zz} < 0 \quad \text{(second normal stress)}\end{aligned}\end{align} \]

Diagnostic Ratio:

\[\frac{N_2}{N_1} = -\frac{\alpha}{2}\]

This provides a direct experimental route to determine \(\alpha\).

Multi-Mode Superposition:

For polydisperse systems with broad relaxation spectra, the multi-mode Giesekus model sums N independent mode contributions:

\[G'(\omega) = \sum_{k=1}^{N} \frac{\eta_{p,k} \lambda_k \omega^2}{1 + (\lambda_k \omega)^2}, \qquad G''(\omega) = \eta_s \omega + \sum_{k=1}^{N} \frac{\eta_{p,k} \omega}{1 + (\lambda_k \omega)^2}\]

See the Handbook for multi-mode fitting strategies and protocol-specific equations.

Supported Protocols¶

Protocol	Method	Notes
FLOW_CURVE	Analytical	Steady shear \(\sigma(\dot{\gamma})\), \(\eta(\dot{\gamma})\)
OSCILLATION	Analytical	SAOS \(G'(\omega)\), \(G''(\omega)\) (\(\alpha\)-independent)
STARTUP	ODE (diffrax)	Stress overshoot at constant rate
RELAXATION	ODE (diffrax)	Faster-than-exponential decay
CREEP	ODE (diffrax)	Strain evolution under constant stress
LAOS	ODE + FFT	Nonlinear harmonics \(I_3\), \(I_5\), …

Quick Start¶

Basic Usage:

from rheojax.models.giesekus import GiesekusSingleMode
import numpy as np

# Create model
model = GiesekusSingleMode()
model.parameters.set_value("eta_p", 100.0)
model.parameters.set_value("lambda_1", 1.0)
model.parameters.set_value("alpha", 0.3)

# Predict flow curve
gamma_dot = np.logspace(-2, 2, 50)
sigma = model.predict(gamma_dot, test_mode='flow_curve')

# Predict SAOS
omega = np.logspace(-1, 3, 50)
G_prime, G_double_prime = model.predict_saos(omega)

# Simulate startup with overshoot
t = np.linspace(0, 10, 500)
sigma_t = model.simulate_startup(t, gamma_dot=10.0)

Multi-Mode:

from rheojax.models.giesekus import GiesekusMultiMode

# Create 3-mode model
model = GiesekusMultiMode(n_modes=3)

# Set mode parameters
model.set_mode_params(0, eta_p=100.0, lambda_1=10.0, alpha=0.3)
model.set_mode_params(1, eta_p=50.0, lambda_1=1.0, alpha=0.2)
model.set_mode_params(2, eta_p=20.0, lambda_1=0.1, alpha=0.1)

# Predict SAOS
G_prime, G_double_prime = model.predict_saos(omega)

Note

For comprehensive theory including analytical steady-state solutions, dimensionless formulations, protocol-specific ODE systems, LAOS analysis, and multi-mode fitting strategies, see the Giesekus Handbook.

When to Use Giesekus¶

Use Giesekus when you observe:

Shear-thinning viscosity
Non-zero first and second normal stress differences
Stress overshoot in startup flow
Linear SAOS that can be fit by Maxwell modes

Decision Tree:

Is N_2 measurable (negative)?
├── YES → Giesekus captures N_2/N_1 = -α/2
│
└── NO → Is only shear-thinning needed?
    ├── YES → Consider simpler Carreau/Cross
    └── NO → Consider PTT or FENE-P for extensional

Material-Specific Recommendations:

Material	Typical \(\alpha\)	n_modes	Key Protocol
Polymer melts	0.1-0.3	3-5	Flow curve + SAOS
Polymer solutions	0.2-0.4	1-3	Startup + SAOS
Wormlike micelles	0.3-0.5	1	Startup overshoot

References¶

Giesekus, H. (1982). “A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility.” J. Non-Newtonian Fluid Mech., 11, 69-109.
Bird, R.B., Armstrong, R.C., & Hassager, O. (1987). Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics. 2nd ed. Wiley.
Larson, R.G. (1988). Constitutive Equations for Polymer Melts and Solutions. Butterworths.

Detailed Documentation

Giesekus Model — Handbook