Introduction & Context

In many process‑engineering operations—such as polymer extrusion, food processing, and oil‑field transport—the fluids encountered do not obey the simple Newtonian relationship \(\tau = \mu \,\dot\gamma\). Their shear stress (\(\tau\)) varies with shear rate (\(\dot\gamma\)) in a non‑linear fashion, a behavior described as non‑Newtonian. Correctly characterising this relationship is essential for:

Designing pumps, mixers and pipelines that operate at the intended flow regime.
Predicting pressure drops and energy consumption.
Ensuring product quality by controlling shear‑sensitive processes.

Two of the most widely used empirical models for steady‑shear flow are the Bingham Plastic (linear with a yield stress) and the Power‑law (also called the Ostwald‑de Waele model). The Python script provided fits both models to a set of experimental \((\dot\gamma_i,\tau_i)\) data, evaluates the residual sum of squares (RSS) for each fit, and selects the model that best represents the material at the reference shear rate \(R_{\text{ref}}\).

Methodology & Formulas

The procedure follows a deterministic sequence that can be reproduced on paper or in any computational environment.

Data definition
For each experimental point \(i\) (where \(i = 1,\dots,N\)):
- Shear stress: \(\tau_i\) (Pa)
- Shear rate: \(\dot\gamma_i\) (s\(^{-1}\))
Linear (Bingham) regression
The Bingham model assumes a linear relationship with a yield stress: \[ \tau = \tau_0 + \mu_p \,\dot\gamma \] The ordinary‑least‑squares (OLS) estimates for the intercept \(\tau_0\) and slope \(\mu_p\) are: \[ \mu_p = \frac{N\sum (\dot\gamma_i \tau_i) - \left(\sum \dot\gamma_i\right)\!\left(\sum \tau_i\right)} {N\sum (\dot\gamma_i^2) - \left(\sum \dot\gamma_i\right)^2} \] \[ \tau_0 = \frac{\sum \tau_i - \mu_p \sum \dot\gamma_i}{N} \] The residual sum of squares for the linear fit is: \[ \text{RSS}_{\text{lin}} = \sum_{i=1}^{N} \left[\,\tau_i - \bigl(\tau_0 + \mu_p \dot\gamma_i\bigr)\,\right]^2 \]
Log‑log (Power‑law) regression
The Power‑law model expresses shear stress as: \[ \tau = K \,\dot\gamma^{\,n} \] Taking natural logarithms linearises the equation: \[ \ln \tau = \ln K + n \,\ln \dot\gamma \] Applying OLS to the transformed data \((\ln \dot\gamma_i,\ln \tau_i)\) yields: \[ n = \frac{N\sum (\ln \dot\gamma_i \,\ln \tau_i) - \left(\sum \ln \dot\gamma_i\right)\!\left(\sum \ln \tau_i\right)} {N\sum (\ln \dot\gamma_i)^2 - \left(\sum \ln \dot\gamma_i\right)^2} \] \[ \ln K = \frac{\sum \ln \tau_i - n \sum \ln \dot\gamma_i}{N} \] Hence the consistency index is \(K = e^{\ln K}\). The RSS for the Power‑law fit is: \[ \text{RSS}_{\text{pow}} = \sum_{i=1}^{N} \left[\,\tau_i - K \,\dot\gamma_i^{\,n}\,\right]^2 \]

Model selection
The model that yields the smaller RSS is chosen as the representative constitutive equation for the fluid:

Condition	Selected Model
\(\text{RSS}_{\text{lin}} < \text{RSS}_{\text{pow}}\)	Bingham Plastic (\(\tau = \tau_0 + \mu_p \dot\gamma\))
\(\text{RSS}_{\text{pow}} \le \text{RSS}_{\text{lin}}\)	Power‑law (\(\tau = K \dot\gamma^{\,n}\))

Reference shear‑rate evaluation
A reference shear rate \(R_{\text{ref}}\) (commonly 50 s\(^{-1}\) in industry) is used to compute a single apparent viscosity for reporting:
- If the Bingham model is selected: \[ \tau_{\text{ref}} = \tau_0 + \mu_p \,R_{\text{ref}} \]
- If the Power‑law model is selected: \[ \tau_{\text{ref}} = K \,R_{\text{ref}}^{\,n} \]
The apparent viscosity at the reference condition is then: \[ \mu_{\text{app}} = \frac{\tau_{\text{ref}}}{R_{\text{ref}}} \] and is often expressed in centipoise (cP): \[ \mu_{\text{app}}^{\text{cP}} = 1000\,\mu_{\text{app}} \]

The resulting parameters (\(\tau_0,\;\mu_p\) or \(K,\;n\)) together with the apparent viscosity \(\mu_{\text{app}}\) provide a concise, engineering‑ready description of the fluid’s rheology for design calculations, CFD input, and process control.

Non‑Newtonian fluids are grouped based on how their viscosity changes with shear. The main categories are:

Shear‑thinning (pseudoplastic): Viscosity decreases as shear rate increases.
Shear‑thickening (dilatant): Viscosity increases with rising shear rate.
Bingham plastics: Exhibit a yield stress; flow like a solid until that stress is exceeded, then behave Newtonian.
Viscoelastic fluids: Show both viscous and elastic responses, often described by models such as Maxwell or Oldroyd‑B.
Thixotropic fluids: Viscosity decreases over time under constant shear and recovers when shear is removed.
Rheopectic fluids: Viscosity increases over time under constant shear.

Follow these steps with a calibrated rheometer:

Set a range of shear rates and record the corresponding shear stresses.
Plot shear stress versus shear rate (or viscosity versus shear rate).
Observe the curve shape:
- Down‑ward slope → shear‑thinning.
- Up‑ward slope → shear‑thickening.
- Linear region after a finite intercept → Bingham plastic (yield stress).
- Time‑dependent drop or rise at constant shear → thixotropic or rheopectic, respectively.
Fit the data to an appropriate constitutive model (e.g., Power‑law, Herschel‑Bulkley) to confirm the classification.

Temperature affects molecular interactions and thus the flow response:

Higher temperatures reduce intermolecular forces, often lowering viscosity and diminishing shear‑thinning intensity.
For viscoelastic fluids, temperature changes the relaxation time, altering the balance between elastic and viscous effects.
In thixotropic systems, temperature can accelerate structural breakdown or rebuild, modifying the time‑dependent viscosity profile.
Some fluids exhibit a transition temperature where the dominant non‑Newtonian mechanism switches (e.g., from shear‑thinning to near‑Newtonian).

Most commercial CFD tools include built‑in rheological models, but accurate prediction requires careful setup:

Select a constitutive model that matches the experimentally determined behavior (Power‑law, Carreau, Herschel‑Bulkley, etc.).
Provide temperature‑dependent property tables if viscosity varies with temperature.
Ensure mesh refinement in regions of high shear gradients to capture steep viscosity changes.
Validate the simulation against bench‑scale rheometer data before applying to full‑scale equipment.
For viscoelastic fluids, use specialized solvers (e.g., Oldroyd‑B, Giesekus) and consider numerical stability techniques such as DEVSS or log‑conformation formulations.

Worked Example: Determining the Power‑Law Parameters for a Shear‑Thinning Fluid

A process engineer must size a pump for a polymer solution that exhibits non‑Newtonian behavior. Laboratory rheometry provides a set of shear‑rate/ shear‑stress data that have been regressed to a power‑law model. The engineer needs the flow‑behavior index n, consistency index K, and the apparent viscosity at the design shear rate of 50 s⁻¹.

Reference shear rate, \(\dot\gamma_{\text{ref}} = 50\) s⁻¹
Regression‑derived flow‑behavior index, \(n = 0.498925\) (dimensionless)
Regression‑derived consistency index, \(K = 0.300709\) Pa·s\(^{n}\)
Number of data points, \(n_{\text{pts}} = 5\)

Calculate the shear stress at the reference shear rate using the power‑law equation \[ \tau_{\text{ref}} = K\,\dot\gamma_{\text{ref}}^{\,n} \] Substituting the known values: \[ \tau_{\text{ref}} = 0.300709\;(50)^{0.498925} \] \((50)^{0.498925} \approx 7.041\) → \(\tau_{\text{ref}} \approx 0.300709 \times 7.041 = 2.117\) Pa.
Determine the apparent viscosity at the reference condition: \[ \mu_{\text{app}} = \frac{\tau_{\text{ref}}}{\dot\gamma_{\text{ref}}} \] \[ \mu_{\text{app}} = \frac{2.117\ \text{Pa}}{50\ \text{s}^{-1}} = 0.0423\ \text{Pa·s} \] Convert to centipoise (1 Pa·s = 1000 cP): \[ \mu_{\text{app}} = 0.0423\ \text{Pa·s} \times 1000 = 42.348\ \text{cP} \]
Classify the fluid behavior. Because the flow‑behavior index satisfies \(n < 1\), the fluid is pseudoplastic (shear‑thinning).

Final Answer:

Flow‑behavior index, \(n = 0.499\) (dimensionless)
Consistency index, \(K = 0.301\) Pa·s\(^{n}\)
Shear stress at 50 s⁻¹, \(\tau_{\text{ref}} = 2.117\) Pa
Apparent viscosity at 50 s⁻¹, \(\mu_{\text{app}} = 0.042\) Pa·s (≈ 42.348 cP)
Fluid classification: Shear‑thinning (pseudoplastic)

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