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.

翻译：我们求解了一个贝叶斯逆纳维-斯托克斯（N-S）问题，该问题通过同化速度测量数据来联合重建流场并学习未知的N-S参数。通过将Carreau剪切稀化黏度模型纳入N-S问题，我们设计了一种算法，该算法仅从速度测量数据即可学习剪切稀化流体最可能的Carreau参数，并估计其不确定性。随后，我们通过流动磁共振成像实验，获取了血液模拟流体在理想化医疗器械（FDA喷嘴）中轴对称层流射流的速度测量数据。研究表明，该算法能够通过学习最可能的Carreau参数成功重建流场，且学习所得参数与流变测量结果高度吻合。该算法可兼容任意代数形式的有效黏度模型（仅需满足模型可微条件），若在N-S问题中引入粘弹性模型，还可扩展应用于更复杂的非牛顿流体（如Oldroyd-B流体）。