Theory of the lattice Boltzmann method: three-dimensional model for linear viscoelastic fluids

A three-dimensional lattice Boltzmann model with thirty two discrete velocity distribution functions for viscoelastic fluid is presented in this work. The model is based upon the generalized lattice Boltzmann equation constructed in moment space. The nonlinear equilibria of the model have a number of coupling constants that are free parameters. The dispersion equation of the model is analyzed under various conditions to obtain the constraints on the free parameters such that the model satisfies isotropy and Galilean invariance. The macroscopic equations are also derived from the lattice Boltzmann model through the dispersion equation analysis and the Chapman-Enskog analysis. We demonstrate that the dispersion equation analysis can be used as a general and effective means to derive hydrodynamic equations, excluding some nonlinear source terms, from the lattice Boltzmann model, to obtain conditions for its isotropy and Galilean invariance, and to optimize its stability. We show that the hydrodynamic behavior of the lattice Boltzmann model has memory effects, and that in the linear regime, it behaves as a viscoelastic fluid described by the Jeffreys model. Some numerical results to verify the theoretical analysis of the model are also presented.