A lattice-Boltzmann model for visco-elasticity

The classical lattice-Boltzmann scheme is extended in an attempt to represent viscoelastic fluids in two dimensions. At each lattice site, two new quantities are added. A suitable coupling of these quantities with the viscous stress tensor leads to a nonzero shear modulus and visco-elastic effects. A Chapman-Enskog expansion gives us the equilibrium populations and conditions for isotropy of the model. A finite wave vector analysis is needed to study the relaxation of sound waves and to determine the dependence of the transport coefficients upon the frequency.