From 968326faf5b91656e037bf000ef955fc9aa08306 Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Fri, 15 Mar 2024 16:58:40 +0100
Subject: [PATCH] Added some details for the bgk / moments
---
.../content/2-hermite-expansion.md | 16 ++++++++++++++++
1 file changed, 16 insertions(+)
diff --git a/lattice-boltzmann.com/content/2-hermite-expansion.md b/lattice-boltzmann.com/content/2-hermite-expansion.md
index 6fb3f83..40201e7 100644
--- a/lattice-boltzmann.com/content/2-hermite-expansion.md
+++ b/lattice-boltzmann.com/content/2-hermite-expansion.md
@@ -28,6 +28,22 @@ $$
$$
where the density, $\rho$, and the momentum $\rho\bm{u}$ are the zeroth and first order moment of $f$.
+Before proceeding further we remind here that the non-dimensional BGK equation reads
+$$
+\begin{equation}
+\partial_t f+\bm{\xi}\cdot \bm{\nabla}f = -\frac{1}{\mathrm{Kn}}\left(f-f^{eq}\right).
+\end{equation}
+$$
+In order to make the link with the momentum and mass conservation equation in the Navier--Stokes framework it is therefore
+natural to take the zeroth and first order moments of the BGK equation, and one gets.
+$$
+\begin{align}
+&\partial_t\rho + \bm{\nabla}\cdot(\rho\bm{u})=0,\\\\
+&\partial_t(\rho\bm{u}) + \bm{\nabla}\cdot\left(\int_{-\infty}^\infty\bm{\xi}\bm{\xi}f\mathrm{d}\bm{\xi}\right)=0.
+\end{align}
+$$
+We must note that the rhs of these equations is zero which is actually not a surprise, since the BGK equation is an approximation of the collision operator which must conserve mass and momentum.
+
--
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