Lattice Boltzmann Formulation for Linear Viscoelastic Fluids Using an Abstract Second Stress

The kinetic theory of gases implies an independent evolution equation for the momentum flux tensor that closely resembles an evolution equation for the elastic stress in continuum descriptions of viscoelastic liquids. However, kinetic theory leads to a nonobjective convected derivative for the evolution of the deviatoric stress, and a fixed relation between the stress relaxation rate and the viscosity. We show that simulations of freely decaying shear flow using the standard two-dimensional lattice Boltzmann kinetic model develop a tangential stress consistent with this nonobjective convected derivative, and this fixed relation between parameters. By contrast, viscoelastic liquids are typically modeled by an upper convected derivative, and with two independent parameters for the viscosity and stress relaxation rate. Although we are unable to obtain an upper convected derivative from kinetic theory with a scalar distribution function, we show that introducing a general linear coupling to a second stress tensor yields the linear Jeffreys viscoelastic model with three independent parameters in the incompressible limit. Unlike previous work, we do not attempt to represent the additional stress through moments of additional distribution functions, but treat it only as an abstract tensor that couples to the corresponding tensorial moment of the hydrodynamic distribution functions. This greatly simplifies the derivation, and the implementation of flows driven by body forces. The utility of the approach is demonstrated through simulations of Stokes' second problem for an oscillating boundary, of the four-roller mill, and of three-dimensional Arnold--Beltrami--Childress and Taylor--Green flows.