Bayesian Model Selection for Complex Flows of Yield Stress Fluids

TL;DR

The paper develops a Bayesian uncertainty quantification framework to calibrate and select constitutive models for yield-stress fluids in complex flows, explicitly modeling both observational noise and model bias. It applies the approach to Carbopol 980 rheology and squeeze-flow experiments, comparing five generalized Newtonian models via MCMC and model evidence to obtain plausibilities, and demonstrates how prior information can strongly influence predictions in non-elastic flows. Key findings show that simple rheological models may suffice in controlled rheometry, but translating those priors to squeeze-flow analysis can lead to significant biases; an expert-informed (broader) prior approach yields more accurate predictions, underscoring the value of Bayesian bias quantification. The work highlights practical implications for soft matter rheology and suggests methodological avenues, including computational accelerations and reduced-order modeling, to extend Bayesian model selection to industrially relevant complex flows.

Abstract

Modeling yield stress fluids in complex flow scenarios presents significant challenges, particularly because conventional rheological characterization methods often yield material parameters that are not fully representative of the intricate constitutive behavior observed in complex conditions. We propose a Bayesian uncertainty quantification framework for the calibration and selection of constitutive models for yield stress fluids, explicitly accounting for uncertainties in both modeling accuracy and experimental observations. The framework addresses the challenge of complex flow modeling by making discrepancies that emanate from rheological measurements explicit and quantifiable. We apply the Bayesian framework to rheological measurements and squeeze flow experiments on Carbopol 980. Our analysis demonstrates that Bayesian model selection yields robust probabilistic predictions and provides an objective assessment of model suitability through evaluated plausibilities. The framework naturally penalizes unnecessary complexity and shows that the optimal model choice depends on the incorporated physics, the prior information, and the availability of data. In rheological settings, the Herschel-Bulkley and biviscous power law models perform well. However, when these rheological outcomes are used as prior information for a rheo-informed squeeze flow analysis, a significant mismatch with the experimental data is observed. This is due to the yield stress inferred from rheological measurements not being representative of the complex squeeze flow case. In contrast, an expert-informed squeeze flow analysis, based on broader priors, yields accurate predictions. These findings highlight the limitations of translating rheological measurements to complex flows and underscore the value of Bayesian approaches in quantifying model bias and guiding model selection under uncertainty.

Figures (27)

• Figure 1: Example of the dependence of the evidence estimator on the truncation radius $R_s$. The uncertainty band shows the 95% confidence interval of the estimator. The fraction of samples included in the truncation is represented by $N_s/N$.
• Figure 2: Overview of the considered rheological models: (N) Newtonian; (B) Bingham; (BV) biviscous; (HB) Herschel-Bulkley; and (BVPL) biviscous power law. Note that parameters may have model-dependent interpretations, e.g., $\eta$ in the biviscous power law should not be interpreted as an effective Newtonian viscosity.
• Figure 3: Rheological data of Carbopol 980 from two batches, where batch 1 is obtained using 50mm rough plates and batch 2 is obtained using 50mm smooth plates measured at different gap heights. We combine these batches into a single data set of which the 95% confidence interval is visualized.
• Figure 4: Predictive posterior distribution per model when omitting model bias. The left column shows all models on a linear scale, whereas the right columns shows the same results on a logarithmic scale.
• Figure 5: Model plausibilities when omitting model bias for the Newtonian (N), Bingham (B), biviscous (BV), Herschel-Bulkley (HB) and biviscous power law (BVPL) models. For the first three models, the plausibility is zero up to machine precision.

Theorems & Definitions (5)

• Remark 1: Sampler implementation aspects
• Remark 2: Apparent viscosity in the biviscous model
• Remark 3: Regularization interpretation
• Remark 4: Model hierarchy
• Remark 5: Shear rates in the squeeze flow problem