A lattice Boltzmann method for shallow flows of generalized Newtonian fluids

This study develops a lattice Boltzmann method (LBM) for shallow flows of generalized Newtonian fluids (GNFs), providing a unified framework for modeling both Newtonian and non-Newtonian rheologies. The formulation avoids viscosity-dependent relaxation times and pressure-density coupling, improving both physical consistency and numerical robustness. A novel equilibrium distribution function is introduced, derived directly from the macroscopic governing equations, ensuring fidelity to the underlying physics. Stability is inherently maintained across subcritical and supercritical regimes by dynamically adjusting the mesoscopic velocity via a Froude number–dependent parameter, eliminating the need for complex lattice modifications. This approach extends to the boundaries, where physically consistent conditions are implemented based on the new equilibrium function, allowing accurate mass and momentum exchange regardless of flow regime or rheology. Comprehensive validation against benchmark problems confirms the method's capability to capture stress-dependent behaviors of GNFs, including power-law and viscoplastic fluids. The dam-break case study illustrates how fluid rheology influences wave evolution and demonstrates the model's suitability for real-world studies. The proposed LBM provides a robust and physically grounded tool for simulating environmental engineering scenarios, including flood propagation and debris flows involving non-Newtonian fluids. • Novel LBM solves viscous shallow water equations for complex fluids. • Captures Newtonian, power-law, and viscoplastic shallow flow behaviours. • Benchmarks confirm accuracy in complex shallow flow simulations. • Analyses how fluid rheology affects dam break with non-Newtonian flows. • Applicable to flood modelling and debris or mudflow hazard simulations.