Carreau–Yasuda Model

• The Carreau–Yasuda model is a rheological law that quantitatively describes shear-dependent viscosity using minimal, physically interpretable parameters.
• It interpolates between zero-shear and infinite-shear viscosities, capturing power-law behavior in the mid-regime of non-Newtonian fluids.
• The model underpins computational and experimental applications in fields like polymer solutions, blood rheology, and nanoconfined fluids.

The Carreau–Yasuda (CY) model is a rheological law that provides a quantitatively flexible, algebraic description of the steady shear viscosity of generalized Newtonian fluids exhibiting shear-thinning or shear-thickening behavior. Its widespread adoption in both computational and experimental studies stems from its ability to interpolate between two Newtonian plateaux (zero-shear and infinite-shear viscosities) via a broad, power-law-like thinning regime, using a minimal set of physically interpretable parameters. The model is foundational in the analysis and simulation of polymer solutions, biological fluids such as blood, nanoconfined fluids, and a wide range of industrial suspensions and process fluids.

1. Mathematical Formulation and Parameter Definition

The constitutive relation for a Carreau–Yasuda fluid expresses the shear viscosity $\eta(\dot\gamma)$ as a function of the local scalar shear rate $\dot\gamma$: $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$ where:

• $\eta_0$ is the zero-shear viscosity (Pa·s), the plateau at vanishingly slow flows,
• $\eta_\infty$ is the infinite-shear viscosity (Pa·s), the high-shear limiting value,
• $\lambda$ is a characteristic time constant (s), setting the scale for the onset of non-Newtonian behavior,
• $a$ is the Yasuda transition exponent (dimensionless), which controls the breadth and sharpness of the shear-thinning regime,
• $n$ is the power-law index: $n<1$ for shear-thinning, $n>1$ for shear-thickening, $\dot\gamma$0 recovers Newtonian behavior.

This formulation admits three limiting regimes:

1. $\dot\gamma$1,
2. $\dot\gamma$2 and $\dot\gamma$3 finite $\dot\gamma$4,
3. $\dot\gamma$5 but not asymptotically large $\dot\gamma$6, i.e., effective power-law behavior.

The deviatoric stress in a generalized Newtonian fluid is then $\dot\gamma$7, with $\dot\gamma$8 the rate-of-strain tensor (Ghosh et al., 2022).

2. Physical Interpretation, Parameter Fitting, and Model Calibration

Each parameter has a concrete physical meaning established by steady shear rheometry:

• $\dot\gamma$9: Viscosity in the Newtonian plateau at low shear rates, typically reflecting equilibrium or near-equilibrium polymer coil dimensions, particle microstructure, or molecular entanglement (Chun et al., 2022).
• $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$0: Viscosity at extreme shear, generally set by the solvent viscosity (for polymer solutions) or a "solvent plus aligned/fully stretched" microstate.
• $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$1: Relates to the fastest relaxation process controlling the transition from Newtonian to power-law, often comparable to the dominant (longest) relaxation time in a polymeric fluid (Ghosh et al., 2022).
• $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$2: The Yasuda exponent, experimentally found to vary, controls how rapidly the viscosity transitions. Choices $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$3 (Carreau), $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$4 (frequent in fits), $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$5 (broader transitions).
• $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$6: Dictates the degree of shear-thinning: $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$7 closer to zero implies stronger thinning (Sontti et al., 21 Oct 2025).

Parameter estimation is typically performed via nonlinear regression to steady-shear experimental data (viscometry). Many studies fix certain parameters based on physical reasoning or external calibration (e.g., set $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$8 to solvent viscosity), then fit $$\eta(\dot\gamma) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot\gamma)^a\right]^{\frac{n-1}{a}}$$9 (Chun et al., 2022, Sontti et al., 21 Oct 2025, Bernabeu et al., 2012).

3. Analytical Structure and Theoretical Properties

The Carreau–Yasuda model supports analytic treatment for a range of canonical flows, including Poiseuille pipe flow, creeping planar flow, and flows in perforated or tapered domains. In theoretical analysis, existence and uniqueness of solutions depend on the monotonicity of $\eta_0$0; for certain ($\eta_0$1) parameter sets and sufficiently strong driving, generalization beyond classical $\eta_0$2 solutions (e.g., weak or pluglike flows) is required (Kutev et al., 2022, Kutev et al., 2023).

In unsteady or spatially complex flows, the model is algebraically tractable—no evolution equations or internal-state variables beyond the local $\eta_0$3 are needed. As a result, CY-based fluids are compatible with a wide array of numerical discretizations (VEM, HHO, Lattice-Boltzmann, finite element, finite volume), which exploit the monotonicity and coercivity afforded by the model for robust solution existence proofs and error estimates (Antonietti et al., 2024, Botti et al., 2020).

A further analytical refinement is the identification of critical cut-offs in instability analysis: for pipe flow, wall and core instability modes are found only if the power-law index $\eta_0$4 falls below critical cut-offs ($\eta_0$5, $\eta_0$6), fundamentally linked to the local structure of $\eta_0$7 in the high-shear (power-law) regime (He et al., 2024).

4. Applications in Fluid Mechanics and Rheological Modeling

The Carreau–Yasuda model underpins a broad spectrum of applications:

• Polymer solutions and FENE-P surrogate models: By matching shear-viscosity curves of molecular models (e.g., FENE-P), CY models serve as numerically efficient surrogates in flows dominated by shear, e.g., pipe or Couette geometry. However, they fail to capture elastic or extensional phenomena (normal stresses, chain stretching effects) in complex flow topologies (Ghosh et al., 2022).
• Blood rheology and hemodynamics: CY models accurately reproduce low-shear/thickening in blood, yielding results in large-scale vascular simulations that differ from Newtonian models only in pathologically relevant shear regimes (Bernabeu et al., 2012, Samavaki et al., 2023).
• Bubble and droplet motion in non-Newtonian media: Quantitative collapse of film-thickness and capillary-number data for Taylor bubbles in CMC/Carbopol solutions is achieved only when the full CY model is used. Piecewise or naive power-law approximations fail to capture the correct scalings or transitions (Chun et al., 2022, Sontti et al., 21 Oct 2025).
• Nanoconfined fluids: CY models account for observed flow enhancement or reduction due to confinement-induced shear-thinning in water and related systems at the nanoscale, and can be extended by inclusion of slip boundary conditions (Sekhon et al., 2017).
• Flow in porous and perforated media: Homogenization analyses in perforated domains (e.g., filtration, Darcy limit) confirm that Carreau–Yasuda fluids reduce to a Darcy law with effective infinite-shear viscosity and permeability tensor in the dilute hole limit (Lu et al., 2024).
• Process physics (extrusion, bioprinting, microfluidics): CY models integrate easily into quasi-analytical and numerical frameworks for engineering flows—e.g., bioprinter nozzles, microchannel design—enabling direct calculation of velocity, pressure, and stress fields with minimal computational overhead (Santesarti et al., 20 Feb 2025).

5. Computational Aspects and Algorithmic Integration

The algebraic, monotonic structure of the Carreau–Yasuda model simplifies its incorporation into modern numerical schemes:

• Virtual Element and Hybrid High-Order Methods: Both exploit the strong monotonicity and boundedness of the viscosity map to establish stability and predict convergence in $\eta_0$8 spaces, with precise error rates derived for the general Carreau–Yasuda structure (Antonietti et al., 2024, Botti et al., 2020).
• Lattice-Boltzmann solvers: CY viscosity is evaluated locally, used to set the relaxation time, and enables stable computation for both Newtonian and non-Newtonian regimes, with the only significant computational overhead being the evaluation of $\eta_0$9 at each step (Bernabeu et al., 2012).
• CFD, finite element, and VOF approaches: The model can be directly coded into solvers such as Ansys Fluent, without regularization or artificial cutoffs, provided sufficient care is taken near singular points (e.g., $\eta_\infty$0) (Sontti et al., 21 Oct 2025).
• Homogenization and upscaling: In perforated or hierarchical domain structures, the CY nonlinearity is preserved through compactness and two-scale expansions, allowing for quantitative convergence rates to be established for the averaged (Darcy) response (Lu et al., 2024).

6. Limitations, Identifiability, and Recent Advances

Despite its practical flexibility, the standard CY model displays structural identifiability issues when fitted to experimental rheometry:

• Parameter nonuniqueness: The $\eta_\infty$1 parameter space may exhibit flat cost-function valleys, allowing compensating changes in $\eta_\infty$2 and $\eta_\infty$3 that yield nearly indistinguishable fits, but physically unreasonable extrapolations. The estimated power-law slope from $\eta_\infty$4 need not coincide with the actual asymptotic slope observed in data (Santesarti et al., 20 Feb 2025).
• Slope-mismatch and physical interpretation: Direct identification of the mid-region power-law index from $\eta_\infty$5 is unreliable; analytical calculations reveal O(10–20%) errors can occur if this is assumed.
• Calibration strategies: Recent works propose "shear-rate-based" (SRB) formulas that directly parameterize the mid-range slope, transition scales, and endpoints, enabling robust, one-step direct fitting and eliminating the ill-posedness of the classical CY calibration (Santesarti et al., 20 Feb 2025). In the SRB formulation, the mid-range slope is explicit and the infinite-shear plateau is algebraically linked to the observed power-law and transition points.

These limitations do not diminish the value of CY in simulating shear-thinning flows, but require awareness when using fitted parameters for interpretation or design.

7. Comparative Table of Representative CY Model Parameters

The following table compiles typical ranges for the CY parameters in several fluid classes, as reported in recent studies:

Fluid/System	$\eta_\infty$6 (Pa·s)	$\eta_\infty$7 (Pa·s)	$\eta_\infty$8 (s)	$\eta_\infty$9	$\lambda$0
Aqueous CMC 1.0 wt% (Chun et al., 2022)	2.4	0.001	0.20	1.1	0.42
Human blood (Bernabeu et al., 2012)	0.16	0.0035	8.2	0.64	0.21
Blood (arterial) (Samavaki et al., 2023)	0.056	0.00345	1.902	1.25	0.22
Carbopol 0.2 wt% (Chun et al., 2022)	263.7	0.02	530	2.5	0.20
Nanoconfined Water (Mica) (Sekhon et al., 2017)	(fixed) 0.001	~0	2.8×10$\lambda$1	2.3	0.77
FENE-P equivalent (Ghosh et al., 2022)	matched	matched	fit	2	fit

Numerical values demonstrate the wide range of physical contexts and CY behaviors the model can represent, with $\lambda$2 ranging as low as 0.14 in strongly shear-thinning Carbopol solutions to $\lambda$3 in weakly thinning systems.

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The Carreau–Yasuda model thus provides an essential bridge between microscale rheological characterization and macroscopic flow predictions in non-Newtonian fluid mechanics. While purely viscous, it is foundational in both laboratory and computational studies, but its limitations in representing elasticity, extensional effects, and parameter identifiability must be explicitly recognized in advanced research and engineering practice.