Learning rheological parameters of non-Newtonian fluids from velocimetry data

TL;DR

The paper tackles inferring non-Newtonian rheology from velocimetry data by casting a Bayesian inverse Navier–Stokes problem that jointly reconstructs the velocity field and learns Carreau model parameters with quantified uncertainties. It defines a forward map from N–S parameters to data, uses a Gaussian likelihood, and derives MAP estimates via a Taylor-expanded Jacobian and adjoint computations, aided by a damped BFGS update for stability. Applying the method to flow-MRI data of a blood-analogue jet in an FDA nozzle, the authors recover a MAP velocity field that aligns well with measurements and obtain Carreau parameters that are in good agreement with independent rheometry, validating the noninvasive approach. The approach is general to any differentiable generalized Newtonian model and can be extended to viscoelastic fluids, enabling patient-specific or device-specific rheology-informed CFD with uncertainty quantification. The work demonstrates a viable pathway for learning rheology from velocimetry alone and highlights practical considerations for experimental design to improve parameter identifiability.

Abstract

We solve a Bayesian inverse Navier-Stokes (N-S) problem that assimilates velocimetry data in order to jointly reconstruct the flow field and learn the unknown N-S parameters. By incorporating a Carreau shear-thinning viscosity model into the N-S problem, we devise an algorithm that learns the most likely Carreau parameters of a shear-thinning fluid, and estimates their uncertainties, from velocimetry data alone. We then conduct a flow-MRI experiment to obtain velocimetry data of an axisymmetric laminar jet through an idealised medical device (FDA nozzle) for a blood analogue fluid. We show that the algorithm can successfully reconstruct the flow field by learning the most likely Carreau parameters, and that the learned parameters are in very good agreement with rheometry measurements. The algorithm accepts any algebraic effective viscosity model, as long as the model is differentiable, and it can be extended to more complicated non-Newtonian fluids (e.g. Oldroyd-B fluid) if a viscoelastic model is incorporated into the N-S problem.

Figures (5)

• Figure 1: Overall flow system and setup around the MRI scanner with detail of the FDA flow nozzle geometry implemented. ID: Inner diameter, OD: outer diameter.
• Figure 2: Images and slices of reconstructed (MAP) flow, $\bm{u}^\circ$, vs. velocimetry data, $\bm{u}^\star$. In figures \ref{['fig:u_z_slices']} and \ref{['fig:u_r_slices']}, velocity is normalised by $U=20$ cm/s, and length is normalised by $L=5$ mm. We separate the transverse slices in the plot by applying a vertical offset of $0.1n$ to the $n$-th slice (the horizontal offset value is immaterial).
• Figure 3: Inferred (MAP) vs. prior strain-rate magnitude, $\dot{\gamma}$, and effective viscosity, $\mu_e$.
• Figure 4: Optimisation log (figure \ref{['fig:optim_log']}), and posterior p.d.f. evolution of the effective viscosity (figure \ref{['fig:mu_e_history']}) and the Carreau parameters (figure \ref{['fig:mu_e_param_history']}). In figure \ref{['fig:mu_e_param_history']} the axes are such that $d_\sigma x \coloneqq (x-\bar{x})/\sigma_{\bar{x}}$, where $\bar{x}$ is the prior mean, and $\sigma_{\bar{x}}$ is the prior standard deviation.
• Figure 5: Learned Carreau fit to rheometry data, learned model parameters (MAP estimates), and assumed priors. Uncertainties in the figures correspond to $3\sigma$ intervals.