The behaviour of non-Newtonian fluids, and their interaction with other fluid phases and components, is of interest in a diverse range of scientific and engineering problems. In the context of the lattice Boltzmann method (LBM), both non-Newtonian rheology and multiphase flows have received significant attention in the literature. This study builds on that work by presenting the development and validation of a phase-field LBM which combines these features in three-dimensional flows. Specifically, the model presented herein combines the simulation of Herschel-Bulkley fluids, which exhibit both a yield stress and power-law dependence on shear rate, interacting with a Newtonian fluid. The developed model is verified and validated using a diverse set of rheological properties and flow conditions, which in their totality represent an additional contribution of this work. Comparison with steady-state layered Poiseuille flow, where one fluid is Newtonian and the other is non-Newtonian, showed excellent correlation with the corresponding analytic solution. Validation against analytic solutions for the rise of a power-law fluid in a capillary tube also showed good correlation, but highlighted some sensitivity to initial conditions and high velocities occurring early in the simulation. A demonstration of the model in a microfluidic junction highlighted how non-Newtonian rheology can alter behaviour from cases where only Newtonian fluids are present. It also showed that significant changes in behaviour can occur when making small and smooth changes in non-Newtonian parameters. To summarise, this work broadens the range of physical phenomena that can be captured in computational analysis of complex fluid flows using the LBM.

中文翻译：

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开发和验证用于三维非牛顿 Herschel-Bulkley 流体的相场晶格玻尔兹曼方法


非牛顿流体的行为以及它们与其他流体相和成分的相互作用，在各种科学和工程问题中都引起了人们的兴趣。在晶格玻尔兹曼方法 （LBM） 的背景下，非牛顿流变学和多相流在文献中都受到了极大的关注。本研究以这项工作为基础，介绍了相场 LBM 的开发和验证，该 LBM 将这些特征结合在三维流中。具体来说，本文介绍的模型结合了 Herschel-Bulkley 流体的仿真，该流体与牛顿流体相互作用，表现出屈服应力和对剪切速率的幂律依赖性。开发的模型使用各种流变特性和流动条件进行了验证和确认，它们总体上代表了这项工作的额外贡献。与稳态分层泊肃叶流（其中一种流体是牛顿流体，另一种流体是非牛顿流体）的比较显示出与相应解析解的极好相关性。针对毛细管中幂律流体上升的解析解的验证也显示出良好的相关性，但突出了对初始条件和仿真早期发生的高速的一些敏感性。该模型在微流体结中的演示强调了非牛顿流变学如何改变仅存在牛顿流体的情况的行为。它还表明，当对非牛顿参数进行微小而平滑的更改时，行为会发生显着变化。总而言之，这项工作拓宽了使用 LBM 对复杂流体流动进行计算分析时可以捕获的物理现象的范围。