Rheo-Informed Squeeze Flow Analysis

Updated 22 January 2026

The paper demonstrates that expert-informed Bayesian frameworks yield effective yield stress values (≈8.8 Pa) significantly lower than those from canonical shear tests.
The methodology combines lubrication theory and regularized constitutive models (e.g., BVPL) to resolve the squeeze-flow paradox and ensure invertible pressure–flux relations.
Key practical insights include enhanced uncertainty quantification and model selection for reliably predicting non-Newtonian, yield-stress, and thixotropic fluid behaviors.

Rheo-informed squeeze flow analysis refers to the quantitative study of squeeze flow configurations using constitutive rheological models and parameter sets tailored to the actual flow environment, rather than relying directly on parameters from canonical rheometry (such as simple shear). This approach integrates detailed rheological characterization, uncertainty quantification, and model selection frameworks—most notably Bayesian techniques—to optimally describe, calibrate, and predict the complex flow behavior encountered in squeeze geometries. Recent advances in this framework highlight fundamental differences between parameters inferred from classic rheological experiments and those genuinely governing squeeze flows, especially for yield stress, viscoplastic, and thixotropic fluids.

1. Theoretical Basis and Lubrication-Scale Governing Equations

The rheo-informed squeeze flow scenario typically involves axisymmetric or planar flow between two plates or in Hele–Shaw geometries under prescribed force, displacement, or pressure gradient, with the gap height $H(t)$ much smaller than lateral dimensions ( $R$ , $L$ ). Under the lubrication approximation, mass conservation and momentum balance reduce to

$\frac{1}{r} \frac{\partial}{\partial r}(r Q) = -r\,\frac{\partial H}{\partial t}$

$Q(dp/dr) = -\frac{r}{2}\,\frac{dH}{dt}$

The pressure gradient–flux relation $Q(dp/dr)$ is closed by integrating the local momentum equation in the gap:

$\tau(\partial u/\partial z) = \frac{dp}{dr}\,z$

with $u(z=\pm H/2)=0$ and a suitable rheological closure (e.g., Newtonian, Bingham, Herschel–Bulkley, biviscous power law, thixotropic viscoplastic).

Boundary conditions are typically:

zero pressure at the spread radius ( $p(R)=0$ ),
initial gap $H(0)=H_0$ ,
either prescribed force or gap velocity depending on experiment.

Once the constitutive law $\tau(\dot \gamma)$ is specified, the above can be integrated to yield volumentric flux $Q$ as a function of the local pressure gradient, which, combined with the force balance, determines the time evolution of $H(t)$ or the pressure and velocity fields (Rinkens et al., 15 Jan 2026).

2. Rheo-informed Constitutive Models and the Squeeze-Flow Paradox

For Newtonian fluids, classical lubrication provides

$Q = -\frac{\pi R^4}{8\eta H} \frac{dH}{dt}$

yielding standard scaling laws for force-gap relations.

For yield stress and shear-thinning materials, direct use of classic rheometer (e.g., Herschel–Bulkley, Bingham) parameters in squeeze flow leads to fundamental issues:

The Herschel–Bulkley and ideal Bingham models produce a non-invertible $Q$ – $dp/dr$ relation under the symmetry and boundary constraints of squeeze geometry (the “squeeze-flow paradox”).
Realistic modeling requires regularization or adoption of alternative forms such as the biviscous or biviscous power law models:
- Biviscous (BV):
$\dot \gamma = \frac{\tau}{\eta_0} (\lvert \tau \rvert < \tau_y), \qquad \tau = \tau_y + \eta \dot \gamma (\lvert \tau \rvert \geq \tau_y)$ - Biviscous power law (BVPL): as above but with

$\tau = \tau_0 + K \dot \gamma^n \quad (\lvert \tau \rvert \geq \tau_y), \quad \tau_0 = (1-\varepsilon)\tau_y, \quad \varepsilon = K \tau_y^{n-1}/\eta_0^n$

This regularization is necessary for stable mathematical and computational treatment in squeeze flows (Rinkens et al., 15 Jan 2026).

3. Bayesian Calibration and Model Selection in Squeeze Flow

Modern rheo-informed analysis applies Bayesian frameworks to rigorously quantify parameter and model uncertainty, drawing a sharp distinction between “rheo-informed” (parameters transferred directly from canonical rheometry) and “expert-informed” (broad, minimally informative priors, allowing complete exploration of parameter space) paradigms. The Bayesian workflow comprises:

Defining a parameter vector $\theta$ for all relevant rheological, experimental, and hyperparameters (e.g., $\tau_y$ , $K$ , $n$ , $\eta_0$ ).
Assigning priors:
- Rheo-informed: tightly constrained Gaussian priors from prior rheometry.
- Expert-informed: wide uniform or weakly-informative priors (e.g., $\tau_y \sim U(1,200)$ Pa).
Constructing the likelihood as a Gaussian error model for observable quantities (e.g., plate radius vs. time, with both noise and model bias components).
Drawing posterior samples $p(\theta|D)$ (typically via MCMC).
Using the model evidence $z_M$ to assign posterior model plausibilities.

Critically, Bayesian calibration reveals that squeeze flow experiments select for much lower effective yield stresses than those inferred from simple shear, and select for different preferred constitutive models (BVPL over BV or Newtonian) (Rinkens et al., 15 Jan 2026).

4. Quantitative and Predictive Outcomes

Analysis of squeeze flow data under expert-informed priors yields:

Posterior probability overwhelmingly in favor of the BVPL model (plausibility $\approx 1$ ).
Rheo-informed priors (from shear) result in poor fits, high model bias ( $\sigma_\text{bias} \gg 1$ mm), and effectively zero plausibility.
Inferred rheological parameters for the BVPL under squeeze are $\tau_y \approx 8.8$ Pa (vs. 75–130 Pa from rheometry), $n \approx 0.45$ , $\log_{10} \eta_0 \approx 2.53$ , $K \approx 44$ Pa·s $^n$ .
Predictive credible bands generated via Bayesian model averaging accurately envelop experimental squeeze spread curves only when using expert-informed frameworks.

Bayes factors for model (BVPL vs. BV) are $\gg 10^3$ , and for prior approach (expert-informed vs. rheo-informed for BVPL) exceed $10^6$ , demonstrating the statistical and predictive superiority of the former (Rinkens et al., 15 Jan 2026).

5. Broader Implications, Limitations, and Best Practices

Failure to recognize the non-universality of rheological parameters between rheological protocols and squeeze flow induces significant model bias and hinders accurate uncertainty quantification. Key recommendations include:

Always perform in situ squeeze-flow experiments when the target physical scenario diverges from canonical shear flows, especially for complex fluids.
Use expert-informed (broad) priors in Bayesian model calibration to allow the observed data to "speak" regarding parameter ranges.
For non-Newtonian, yield-stress, or thixotropic materials, employ regularized models (BVPL, Cross-type, Papanastasiou-Herschel–Bulkley for thixotropy), ensuring invertibility in the squeeze-flow $Q$ – $dp/dr$ relation (Florides et al., 2023).

The rheo-informed squeeze flow paradigm is thus characterized by an integrated experimental–statistical–theoretical approach:

Carefully tuned, model-aware experiments;
Bayesian frameworks for both parameter and model uncertainty;
Explicit quantification of the limitations and model bias incurred by parameter transfer from classical rheology to complex geometries (Rinkens et al., 15 Jan 2026).

This framework contrasts with classical squeeze flow literature for Newtonian or simply parameterized non-Newtonian fluids, where direct transfer of constitutive parameters suffices (Lorenz et al., 2010). Modern rheo-informed analysis is particularly essential for yield-stress and thixotropic systems, and for flows with substantial spatial or temporal variations in structure or flow regime. The approach is also differentiable from uncertainty quantification in generalized Newtonian squeeze flows, where model choice is not ambiguous but parameter recalibration is necessary when confronted with strong geometric or kinematic constraints (Rinkens et al., 2023).

In summary, rheo-informed squeeze flow analysis represents a high-fidelity, uncertainty-quantified methodology for interpreting and predicting squeeze flow in complex materials and geometries, with Bayesian model selection central to distinguishing suitable rheological descriptions and parameter sets in the context of real process conditions (Rinkens et al., 15 Jan 2026).