Lattice Boltzmann methods for simulating non-Newtonian fluids: A comprehensive review

TL;DR

The paper surveys lattice Boltzmann method (LBM) approaches for non-Newtonian fluids, focusing on generalized Newtonian fluids (GNFs) and viscoelastic flows. It outlines core LBM formulations for simple hydrodynamics, force coupling, and multiphase extensions, then details GNFs—covering models such as Power-law, Carreau–Yasuda, and Herschel–Bulkley—and how their shear-rate dependent viscosity is implemented in LBM. For viscoelastic flows, the review comprehensively covers MRT-based, linear Maxwell forcing, lattice Fokker–Planck, advection–diffusion, and hybrid LB–continuum strategies, highlighting stability, memory effects, and high Weissenberg number challenges, along with recent stabilization techniques like log-conformation representations. The authors discuss validation benchmarks, practical applications in porous media and complex geometries, and offer perspectives on future directions, including advanced stabilization, GPU-accelerated HPC, and comprehensive cross-method benchmarks to assess accuracy and efficiency. Overall, the work positions LBM as a versatile, scalable framework capable of bridging microstructural polymer dynamics with macroscopic flow in complex non-Newtonian systems, while identifying key numerical and modeling gaps to be addressed for industrial-scale applications.

Abstract

Non-Newtonian fluids encompass a large family of fluids with additional nonlinear material properties, contributing to non-trivial flow behaviour that cannot be captured through a single constant viscosity term. Common non-Newtonian characteristics include shear-thinning, shear-thickening, viscoplasticity, and viscoelasticity, commonly encountered in everyday fluids, such as ketchup, blood, toothpaste, mud, etc., as well as practical applications involving porous media, cosmetics, food processing, and pharmaceuticals. Due to the complex nature of these fluids, accurate computational fluid dynamics simulations are essential for predicting their behaviour under various flow conditions. Recent advancements have highlighted the growing trend of using the lattice Boltzmann method to solve such complex flows, owing to its ability to handle intricate boundary conditions, ease of including additional multiphysics, and providing computationally efficient parallel simulations. Since the initial review over a decade ago [Phillips & Roberts, IMA J. Appl. Math. 76, 790-816 (2011)], significant advancements have been made to the lattice Boltzmann method to simulate non-Newtonian fluids. Here, we present a comprehensive review of different lattice Boltzmann techniques used to solve non-Newtonian fluid systems, specifically dealing with shear-dependent viscosity, viscoplasticity, and viscoelasticity. In addition, we discuss various benchmark cases that validate these approaches and highlight their growing application to realistic and challenging complex flow problems. We further address outstanding issues in current lattice Boltzmann models, as well as future directions for numerical advancement and application.

Figures (8)

• Figure 1: Illustration of common non-Newtonian fluid examples, which encompass various types of anomalous flow behaviour. These involve shear-thinning in (a) ketchup and (b) yoghurt, as well as thixotropy in (c) paints, while shear-thickening is observed in (d) a suspension of cornstarch and water (commonly referred to as Oobleck). Viscoplasticity is observed in materials that exhibit a yield-stress transition, such as (e) squeezing toothpaste and (f) mud. An additional elastic material response can be naturally observed in certain biofluids, such as (g) blood and (h) mucus. Note, certain examples outlined above, such as ketchup, can indeed experience multiple anomalous flow behaviours, such as shear-thinning and viscoplasticity (i.e., shaking the ketchup bottle causes it to flow from rest and reduce its viscosity).
• Figure 2: Example of computation fluid dynamics (CFD) techniques applied across different scales: (from left to right) microscopic, mesoscopic, and macroscopic. Each resolution highlights the distinct approaches used to capture fluid behaviour, ranging from detailed molecular interactions at the microscopic scale [i.e., molecular dynamics (MD)], to particle-based models at the mesoscopic scale [i.e., lattice Boltzmann method (LBM), smoothed-particle hydrodynamics (SPH), dissipative particle dynamics (DPD), etc.], and continuum-based equations at the macroscopic scale (i.e., NSE). The transitions emphasize the trade-offs in computational complexity and the level of physical detail represented at each scale. Although microscopic simulations provide unparalleled detail, they are computationally expensive, while macroscopic models prioritise efficiency at the expense of fine-grained details.
• Figure 3: The two-dimensional nine-velocity lattice structure set $\{ w_{\alpha}, \bm{\xi}_{\alpha} : \alpha=0,\dots, 8 \}$, known as the D2Q9, where $c_s^2=1/3$.
• Figure 4: Qualitative flow curves illustrating the rheological behavior of various generalized Newtonian fluids (GNFs): a) shear stress ($\sigma_s$) as a function of shear rate ($\dot{\gamma}$), and b) apparent viscosity ($\mu$) as a function of shear rate ($\dot{\gamma}$). These curves demonstrate typical shear-thinning, shear-thickening, and viscoplastic responses characteristic of GNFs, highlighting the nonlinear dependence of stress and viscosity on the applied shear rate.
• Figure 5: Illustration of a) the molecular build-up of long-chain polymers comprised of binding monomers, which can b) be simplified towards a coarse-grained treatment, known as the dumbbell model. The dumbbell model consists of two beads connected by a spring of distance $\bm{r}$, representing the entropic elasticity of the polymer chain. The beads account for the hydrodynamic drag experienced by the polymer, while the spring captures the force dipole $\bm{F}_s$ arising from the entropic restoring force, providing a mesoscopic description of polymer dynamics in a solvent