# Confusion About Boundary Condition Implementations

I was reading the explanations on LBM BCs from Succi’s book (pages 77 to 88). Certain details about the implementation don’t seem to make physical sense to me.

Before proceeding, I want to explain what I have understood of the whole process here:

[i]The particle distribution labels, for reference:
4 3 2
5 … 1
6 7 8

Also, this discussion pertains to on-grid boundaries[/i]

f[sub]4[/sub], f[sub]3[/sub], and f[sub]2[/sub] are the outgoing (from the fluid to the boundary) particle distributions. In the collision step at this boundary node, we compute the incoming (from the boundary into the liquid) distribution f[sub]6[/sub], f[sub]7[/sub], and f[sub]8[/sub] in terms of the outgoing ones. Refering to the equation 6.42 in the book, it is clear that we compute the x-momentum and y-momentum at the node using all of these values (outgoing and incoming distributions):

J[sub]x[/sub] = (f[sub]2[/sub] + f[sub]8[/sub]) - (f[sub]4[/sub] + f[sub]6[/sub]) = 2(1-p)(f[sub]2[/sub] - f[sub]4[/sub]).

I don’t understand why we consider f[sub]2[/sub] and f[sub]4[/sub] when calculating J[sub]x[/sub]. After the collision, these two get converted to f[sub]6[/sub] and f[sub]8[/sub], right? So why do we not set them to zero after the collision? It seems to me that this results in new particle populations f[sub]6[/sub] and f[sub]8[/sub] being created out of thin air…how does this not result in problems with conservation of mass/momentum?

Succi then goes on to say: “Contributions from f[sub]1[/sub], f[sub]5[/sub] are barred out since these populations do not contribute to the wall-fluid coupling”.

This also seems weird to me. When we want to calculate the value of the momentum at any node, we will have to consider the contributions of f[sub]1[/sub] and f[sub]5[/sub], so why neglect them when satisfying boundary conditions? My doubt about the falsity of this statement is further exacerbated by the fact that in the in the derivation of the Inamuro BCs (as found in the original paper by Inamuro et. al.), the contributions from f[sub]1[/sub] and f[sub]5[/sub] are considered.