Flow Curve Models ================= This section documents models for steady-state shear flow behavior—the relationship between shear stress and shear rate under continuous deformation. Quick Reference --------------- .. list-table:: :widths: 25 15 60 :header-rows: 1 * - Model - Parameters - Use Case * - :doc:`power_law` - 2 (K, n) - Simple shear-thinning/thickening fluids * - :doc:`carreau` - 4 (:math:`\eta_0`, :math:`\eta_\infty`, :math:`\lambda`, n) - Full flow curve with Newtonian plateaus * - :doc:`carreau_yasuda` - 5 (+a) - Sharper transition region control * - :doc:`cross` - 4 (:math:`\eta_0, \eta_{\infty}`, K, n) - Alternative to Carreau, different transition * - :doc:`bingham` - 2 (:math:`\sigma_y`, :math:`\eta_p`) - Simple yield stress fluids * - :doc:`herschel_bulkley` - 3 (:math:`\sigma_y`, K, n) - Yield stress + power-law flow Overview -------- Flow curve models describe the steady-state relationship :math:`\sigma = f(\dot{\gamma})` between shear stress and shear rate. These models are essential for: - **Process design**: Pump selection, pipe flow calculations - **Material characterization**: Viscosity classification (ASTM, ISO) - **Quality control**: Batch-to-batch consistency - **Formulation development**: Additive effects on flow behavior **Key phenomena captured:** - **Shear thinning**: Decreasing viscosity with increasing shear rate (most polymeric fluids) - **Shear thickening**: Increasing viscosity (concentrated suspensions, some pastes) - **Yield stress**: Finite stress required to initiate flow - **Newtonian plateaus**: Constant viscosity at extreme shear rates Model Hierarchy --------------- :: Flow Curve Models │ ├── Newtonian Region Models (no yield stress) │ ├── Power Law (Ostwald-de Waele) │ │ └── σ = K · γ̇^n │ │ └── Simple, 2 parameters │ │ └── No plateaus │ │ │ ├── Carreau │ │ └── η = η∞ + (η_0-η∞)[1+(λγ̇)^2]^((n-1)/2) │ │ └── Both plateaus, smooth transition │ │ │ ├── Carreau-Yasuda │ │ └── η = η∞ + (η_0-η∞)[1+(λγ̇)^a]^((n-1)/a) │ │ └── Adjustable transition sharpness │ │ │ └── Cross │ └── η = η∞ + (η_0-η∞)/[1+(Kγ̇)^n] │ └── Different transition shape │ └── Yield Stress Models ├── Bingham │ └── σ = σ_y + η_p · γ̇ (if σ > σ_y) │ └── Linear above yield │ └── Herschel-Bulkley └── σ = σ_y + K · γ̇^n (if σ > σ_y) └── Power-law above yield When to Use Which Model ----------------------- .. list-table:: :widths: 25 12 12 12 12 12 15 :header-rows: 1 * - Feature - Power Law - Carreau - C-Y - Cross - Bingham - H-B * - Shear thinning - ✓ - ✓ - ✓ - ✓ - ✗ - ✓ * - Yield stress - ✗ - ✗ - ✗ - ✗ - ✓ - ✓ * - Zero-shear plateau - ✗ - ✓ - ✓ - ✓ - N/A - N/A * - High-shear plateau - ✗ - ✓ - ✓ - ✓ - ✗ - ✗ * - Transition control - ✗ - Fixed - ✓ - Fixed - N/A - N/A * - Simple fitting - ✓✓ - ✓ - ~ - ✓ - ✓✓ - ✓ **Decision Flowchart:** 1. Does the material have a yield stress? - **Yes** → Bingham (linear) or Herschel-Bulkley (power-law) - **No** → Continue 2. Do you observe Newtonian plateaus at low and/or high shear rates? - **Yes** → Carreau, Carreau-Yasuda, or Cross - **No** → Power Law (limited range) 3. Is the transition between plateaus sharp or gradual? - **Sharp** → Carreau-Yasuda (tune parameter a) - **Gradual** → Carreau or Cross Material Examples ----------------- .. list-table:: :widths: 25 20 25 30 :header-rows: 1 * - Material - Typical Model - Key Parameters - Industry * - Polymer solutions - Carreau - :math:`\eta_0` = 1-100 Pa·s, n = 0.3-0.7 - Plastics, coatings * - Polymer melts - Carreau-Yasuda - :math:`\eta_0` = :math:`10^3-10^5` Pa·s, a = 2 - Extrusion, injection * - Blood - Carreau - :math:`\eta_0` ≈ 50 mPa·s, n ≈ 0.4 - Biomedical * - Paints - Cross or H-B - :math:`\sigma_y` = 0.5-10 Pa - Coatings * - Toothpaste - Herschel-Bulkley - :math:`\sigma_y` = 10-100 Pa - Personal care * - Drilling mud - Herschel-Bulkley - :math:`\sigma_y` = 5-50 Pa, n = 0.5-0.8 - Oil & gas * - Ketchup - Herschel-Bulkley - :math:`\sigma_y` ≈ 15 Pa - Food * - Concrete - Bingham - :math:`\sigma_y` = 10-100 Pa - Construction Key Parameters -------------- .. list-table:: :widths: 15 10 15 60 :header-rows: 1 * - Parameter - Symbol - Units - Physical Meaning * - Zero-shear viscosity - :math:`\eta_0` - Pa·s - Viscosity at rest (Newtonian plateau) * - Infinite-shear viscosity - :math:`\eta_\infty` - Pa·s - High-rate limit (often ≈ 0) * - Consistency index - K - Pa·s^n - Power-law prefactor (magnitude) * - Flow index - n - — - n < 1: thinning, n > 1: thickening * - Relaxation time - :math:`\lambda` - s - Onset of shear thinning (1/:math:`\lambda`) * - Yield stress - :math:`\sigma_y` - Pa - Stress to initiate flow * - Yasuda parameter - a - — - Transition sharpness (a = 2 gives Carreau) Quick Start ----------- **Herschel-Bulkley (yield stress fluid):** .. code-block:: python from rheojax.models import HerschelBulkley import numpy as np model = HerschelBulkley() gamma_dot = np.logspace(-2, 2, 50) # Fit to flow curve data model.fit(gamma_dot, stress_data, test_mode='flow_curve') # Extract yield stress sigma_y = model.parameters.get_value('sigma_y') print(f"Yield stress: {sigma_y:.1f} Pa") **Carreau model (full flow curve):** .. code-block:: python from rheojax.models import Carreau model = Carreau() model.fit(gamma_dot, viscosity_data, test_mode='flow_curve') # Get zero-shear viscosity and critical shear rate eta_0 = model.parameters.get_value('eta_0') lambda_param = model.parameters.get_value('lambda') gamma_dot_c = 1 / lambda_param # Critical shear rate **Bayesian parameter estimation:** .. code-block:: python # Bayesian inference with NLSQ warm-start result = model.fit_bayesian( gamma_dot, data, test_mode='flow_curve', num_warmup=1000, num_samples=2000, num_chains=4, seed=42 ) # Yield stress uncertainty intervals = model.get_credible_intervals(result.posterior_samples) print(f"σ_y: [{intervals['sigma_y'][0]:.1f}, {intervals['sigma_y'][1]:.1f}] Pa") Model Documentation ------------------- **Simple Models:** .. toctree:: :maxdepth: 1 power_law bingham **Generalized Newtonian Models:** .. toctree:: :maxdepth: 1 carreau carreau_yasuda cross **Yield Stress Models:** .. toctree:: :maxdepth: 1 herschel_bulkley See Also -------- - :doc:`/models/dmt/index` — Thixotropic models (time-dependent flow curves) - :doc:`/models/ikh/index` — Kinematic hardening for complex yield behavior - :doc:`/models/fluidity/index` — Fluidity-based yield stress models - :doc:`/models/sgr/index` — Soft glassy rheology (power-law flow) - :doc:`/transforms/srfs` — Strain-rate frequency superposition References ---------- 1. Bird, R.B., Armstrong, R.C., & Hassager, O. (1987). *Dynamics of Polymeric Liquids*, Vol. 1, 2nd ed. Wiley. ISBN: 978-0471802457. 2. Carreau, P.J. (1972). "Rheological equations from molecular network theories." *Trans. Soc. Rheol.*, 16, 99-127. https://doi.org/10.1122/1.549276 3. Cross, M.M. (1965). "Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems." *J. Colloid Sci.*, 20, 417-437. 4. Herschel, W.H. & Bulkley, R. (1926). "Konsistenzmessungen von Gummi-Benzollösungen." *Kolloid-Z.*, 39, 291-300. 5. Yasuda, K., Armstrong, R.C., & Cohen, R.E. (1981). "Shear flow properties of concentrated solutions of linear and star branched polystyrenes." *Rheol. Acta*, 20, 163-178. https://doi.org/10.1007/BF01513059 6. Barnes, H.A., Hutton, J.F., & Walters, K. (1989). *An Introduction to Rheology*. Elsevier. ISBN: 978-0444871404. 7. Macosko, C.W. (1994). *Rheology: Principles, Measurements, and Applications*. Wiley-VCH. ISBN: 978-0471185758.