A lattice Boltzmann‐BGK algorithm for a diffusion equation with Robin boundary condition—application to NMR relaxation

Abstract

SUMMARYWe present a lattice Boltzmann-BGK (LBGK) algorithm for a diffusion equation together with a Robin boundary condition, which we apply in the case of nuclear magnetic resonance relaxation. The boundary condition we employ is independent of the direction of the wall. This makes the algorithm very suitable for complicated geometries, such as porous media. We discuss the effect of lattice topology by using, respectively, an eight-speed and a four-speed lattice. The numerical algorithm is compared with analytical results for a square and an equilateral triangle. The eight-speed lattice performs well in both cases. The four-speed lattice performs well for the square, but fails in the case of an equilateral triangle. Comparison with a random walk algorithm is also included. The LBGK algorithm presented here can also be used for a convective diffusion problem if the speed of the fluid can be neglected close to the boundary.

A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions

“…It is based on the diffusive scaling ∆t ∼ (∆x) 2 , as this seems to be the appropriate scaling for the diffusion equation (although the Chapman-Enskog analysis presented elsewhere, see e.g. [6,10], also yields correct results). See [12] for a discussion of the differences between the two approaches.…”

“…[7,13,14,16]. Applications include solute transport in porous media [13], dissolution phenomena [17], dispersion [18,19] and comparisons to NMR experiments [10].…”

Influence of asperities on fluid and thermal flow in a fracture: A coupled lattice Boltzmann study

“…It is assumed that the heat is passively transported by the fluid: viscous heat dissipation is neglected, as well as the viscosity dependence with the temperature. Using square lattices and appropriate collision rules, it has been shown [Wolf-Gladrow, 2005;Hiorth et al, 2008] that the LBM solves advection-diffusion equation. The internal energy and its flux are conserved during the collision phase, which is done with a BGK scheme, exactly as in, e.g., Hiorth et al [2008].…”

“…The chosen lattices for the flow and temperature variables are respectively hypercubic and cubic, with a single lattice speed for each lattice. This choice is seldom used in literature, but it is suitable for three-dimensional (3-D) mass and heat transport modeling [d 'Humières et al, 1986;Wolf-Gladrow, 2005;Hiorth et al, 2008]. Despite the methods being implemented in 3-D, for simplicity reasons, the parameter exploration is done in two dimensions, translational invariance being assumed along the third dimension.…”

A volume-preserving sharpening approach for the propagation of sharp phase boundaries in multiphase lattice Boltzmann simulations

“…We now construct a lattice Boltzmann formulation for our model equation (6), following the usual approaches for advection-reaction-diffusion equations [12,13,14,15]. We express the phase field φ as the zeroth moment of a set of distribution functions f i (x, t),…”