Asymptotic Analysis of the Lattice Boltzmann Method for Generalized Newtonian Fluid Flows

Abstract

In this paper, we present a detailed asymptotic analysis of the lattice Boltzmann method with two different collision mechanisms of BGK-type on the D2Q9-lattice for generalized Newtonian fluids. Unlike that based on the Chapman-Enskog expansion leading to the compressible Navier-Stokes equations, our analysis gives the incompressible ones directly and exposes certain important features of the lattice Boltzmann solutions. Moreover, our analysis provides a theoretical basis for using the iteration to compute the rate-of-strain tensor, which makes sense especially for generalized Newtonian fluids. As a by-product, a seemingly new structural condition on the generalized Newtonian fluids is singled out. This condition reads as "the magnitude of the stress tensor increases with increasing the shear rate." We verify this condition for all the existing constitutive relations which are known to us. In addition, it is straightforward to extend our analysis to multiple-relaxation-time models or to three-dimensional lattices. Introduction.During the last two decades, the lattice Boltzmann method (LBM) has been developed into an effective and viable tool for simulating various fluid flow problems. It has been proved to be quite successful in simulating complex Newtonian fluid flows such as turbulent flows, microflows, multiphase and multicomponent flows, particulate suspensions, and interfacial dynamics. We refer the reader to [1, 2, 3] for a comprehensive account of the method and its applications. Moreover, the potential of the LBM in simulating flows of generalized Newtonian fluids, for which the dynamic shear viscosity depends on the shear rate [4], was shown by Aharonov and Rothman [5] as early as 1993. In recent years, this potential has attracted much attention [6,7,8,9,10,11,12,13,14,15,16].There are at least two kinds of LB-BGK models for generalized Newtonian fluids. In the first model, the relaxation time is not a constant anymore but depends on the shear rate. The non-Newtonian effects are embedded in the LBM through a dynamic change of the relaxation time. The second model has a constant relaxation time, but its equilibrium distribution contains the shear rate.The original goal of this paper is to extend the asymptotic analysis of LBM [17, 18] for classical Newtonian fluids to generalized ones. The analysis is based on the diffusive, instead of convective, scaling developed in [19] for the Boltzmann equation and in [17] for LB equations. It turns out that the diffusive scaling is the most natural choice if the LBE is viewed as a numerical solver of the incompressible Navier-Stokes