Development and validation of a phase-field lattice Boltzmann method for non-Newtonian Herschel-Bulkley fluids in three dimensions

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.