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Modeling of fluids with complex rheology in the lattice Boltzmann method (LBM) is typically realized through the introduction of an effective viscosity. For fluids with a yield stress behavior, such as so-called Bingham fluids, the effective viscosity has a singularity for low shear rates and may become negative. This is typically avoided by regularization such as Papanastasiou’s method. Here we argue that the effective viscosity model can be re-interpreted as a generalized equilibrium in which no violation of the stability constraint is observed. We implement a Bingham fluid model in a three-dimensional cumulant lattice Boltzmann framework and compare the direct analytic effective viscosity/generalized equilibrium method to the iterative approach first introduced by Vikhansky which avoids the singularity in viscosity that can arise in the analytic method. We find that both methods obtain similar results at coarse resolutions. However, at higher resolutions the accuracy of the regularized method levels off while the accuracy of the direct method continuously improves. We find that the accuracy of the proposed direct method is not limited by the singularity in viscosity indicating that a regularization is not strictly necessary.

A Direct Effective Viscosity Approach for Modeling and Simulating Bingham Fluids with the Cumulant Lattice Boltzmann Method