From 9ca5f7614f76b00db03cd26e23106475d0803a64 Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Thu, 14 Mar 2024 17:48:20 +0100
Subject: [PATCH] completed post
---
.../content/1-dimensionless-lbm.md | 26 ++++++++++++++++---
1 file changed, 22 insertions(+), 4 deletions(-)
diff --git a/lattice-boltzmann.com/content/1-dimensionless-lbm.md b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
index 1aab1ae..50df1a6 100644
--- a/lattice-boltzmann.com/content/1-dimensionless-lbm.md
+++ b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
@@ -97,6 +97,11 @@ $$
\left[M \cdot L^{-2D}\cdot T^D\right]
$$
By defining the characteristic velocity of our particles by $\xi_0$ (due to thermal agitation)
+$$
+\begin{equation}
+\xi_0^2=\frac{k_BT}{m},
+\end{equation}
+$$
the non-dimensional quantities of interest become
$$
\begin{equation}
@@ -120,11 +125,24 @@ $$
\mathrm{Kn}=\frac{\tau\xi_0}{L},
\end{equation}
$$
-is the Knudsen number.
-
-As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized
-with an unique transport coefficient which is the Knudsen number. In the next episodes of this series
+is the Knudsen number. As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized
+with an unique transport coefficient which is the Knudsen number in this case. In the next episodes of this series
we will make the link between the BGK and the Navier--Stokes equations and show how we discretize
the BGK equation in order to simulate weakly compressible fluid flows.
+Finally we want to express the non-dimensional Maxwell-Boltzmann distribution as a function of only non-dimensional quantities and it reads
+$$
+\begin{equation}
+{f^{eq}}^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast) = \frac{\rho^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)}{(2\pi\theta^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast))^{D/2}}\exp\left(-\frac{\left(\bm{\xi}^\ast-\bm{u}^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)\right)^2}{2\theta^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)}\right),
+\end{equation}
+$$
+where $\theta^\ast$ is given by
+$$
+\begin{equation}
+\theta^\ast = \frac{k_B T}{m\cdot \xi_0^2}
+\end{equation}
+$$
+In the next episode we will discuss in more details, the link between the description of a fluid in the Boltzmann framework
+and in the Navier--Stokes framework.
+
--
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