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

Abstract

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.

An extensible lattice Boltzmann method for viscoelastic flows: complex and moving boundaries in Oldroyd-B fluids

“…Lattice Boltzmann [23][24][25] (LB) methods hold particular promise to achieve this goal due to their computational efficiency [26] and facile boundary [27] and particle coupling [28][29][30][31], as has been demonstrated in Newtonian media. A wide variety of viscoelastic LB schemes have been conceived over the years [32][33][34][35][36][37][38][39][40][41][42][43][44][45]. However, despite this long history, which we will summarize in Sect.…”

Study on Force Schemes in Pseudopotential Lattice Boltzmann Model for Two-Phase Flows

“…Yu and Fan [17] combined adaptive mesh refinement method and lattice Boltzmann method to improve the two-phase flow simulation. It should be noted that the EDM scheme is Galilean invariant and the results obtained by EDM scheme do not depend on relaxation time [11,18]. Kupershtokh et al [13] compared three kinds of approximation of the gradient of special potential ("local," "mean-value," and "general") and showed that the "general" approximation was most precise and stable.…”

On the Lattice Boltzmann Deviatoric Stress: Analysis, Boundary Conditions, and Optimal Relaxation Times

“…Interestingly, Yudisiawan [49] observed some kinetic effects in the stress field of the D2Q9 discrete Boltzmann model: solutions of the truncated PDE moment system suggested a non zero tangential stress in planar (force-driven) Poiseuille flow; something which is characteristic of Burnett and non-Newtonian behaviour. More recently, Dellar [9] took inspiration from [27] and [45] and showed the constitutive equation for stress embedded within the moments of the D2Q9 discrete Boltzmann equations is not that of the Navier-Stokes equations and resembles the upper convected Maxwell model for viscoelasticity. Reis [36] also noticed that that the LBE predicted non-Newtonian behaviour in the stress field and used the work of Dellar [9] to develo suitable boundary conditions for the tangential component of stress.…”

“…More recently, Dellar [9] took inspiration from [27] and [45] and showed the constitutive equation for stress embedded within the moments of the D2Q9 discrete Boltzmann equations is not that of the Navier-Stokes equations and resembles the upper convected Maxwell model for viscoelasticity. Reis [36] also noticed that that the LBE predicted non-Newtonian behaviour in the stress field and used the work of Dellar [9] to develo suitable boundary conditions for the tangential component of stress. Although Yong and Luo [47] showed that the D2Q9 lattice Boltzmann equation predicts the Newtonian viscous stress tensor with second order accuracy, their calculations assume a diffusive scaling where the timestep is proportional to the square of the grid spacing, ∆t ∝ ∆x 2 .…”

“…We hypothesise that standard local lattice Boltzmann boundary conditions that prescribe hydrodynamic conditions but do not fully account for the stress embedded within the moments are inconsistent with the underlying PDE and produce spurious oscillations in the numerical solutions to simple flows. We build on the work of He et al [25],D'Humieres and Ginzburg [10], Dellar [9], and Reis [36], and solve the BGK and TRT lattice Boltzmann equation analytically for the three components of the stress tensor in force driven Poiseuille flow in an infinitely long channel (no inflow/outflow conditions). This allows us to see precisely how the LBE is behaving and illuminates some physical and numerical aspects of the algorithm.…”