From e10b4cfc29a51363ceeb75ceacaad5c957f0e525 Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Thu, 14 Mar 2024 10:32:51 +0100
Subject: [PATCH] removed toucan
---
.gitmodules | 3 ---
lattice-boltzmann.com/content/1-dimensionless-lbm.md | 8 +++++---
2 files changed, 5 insertions(+), 6 deletions(-)
diff --git a/.gitmodules b/.gitmodules
index 36c2ce1..744bd3b 100644
--- a/.gitmodules
+++ b/.gitmodules
@@ -1,6 +1,3 @@
-[submodule "toucan"]
- path = lattice-boltzmann.com/themes/toucan
- url = https://git.42l.fr/HugoTrentesaux/toucan.git
[submodule "lattice-boltzmann.com/themes/abridge"]
path = lattice-boltzmann.com/themes/abridge
url = https://github.com/jieiku/abridge.git
diff --git a/lattice-boltzmann.com/content/1-dimensionless-lbm.md b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
index b2cb286..c893f99 100644
--- a/lattice-boltzmann.com/content/1-dimensionless-lbm.md
+++ b/lattice-boltzmann.com/content/1-dimensionless-lbm.md
@@ -21,7 +21,7 @@ The incompressible Navier--Stokes equations reads
$$
\begin{align}
&\bm{\nabla}\cdot\bm{u}=0,\\\\
-&\frac{\partial}{\partial t}\bm{u}+\bm{u}\cdot\bm{\nabla}\bm{u}=-\frac{1}{\rho}\bm{\nabla}p+\nu\bm{\nabla}^2\bm{u},
+&\partial_t\bm{u}+\bm{u}\cdot\bm{\nabla}\bm{u}=-\frac{1}{\rho}\bm{\nabla}p+\nu\bm{\nabla}^2\bm{u},
\end{align}
$$
where $p$, $\rho$, $\nu$, and $\bm{u}$ are respectively the pressure, density, kinematic viscosity, and velocity of the flow.
@@ -34,7 +34,7 @@ $$
\begin{aligned}
&\bm{u}^\ast=\frac{\bm{u}}{U}, &p^\ast=\frac{p}{\rho U^2},\\\\
&t^\ast=\frac{U}{L}t, &\bm{x}^\ast=\frac{\bm{x}}{L},\\\\
-&\frac{\partial}{\partial t^\ast}=\frac{L}{U}\frac{\partial}{\partial t}, &\bm{\nabla}^\ast=L\bm{\nabla}
+&\partial_t^\ast=\frac{L}{U}\partial_t, &\bm{\nabla}^\ast=L\bm{\nabla}
\end{aligned}
\end{equation}
$$
@@ -42,7 +42,7 @@ Replacing these equations into the Navier--Stokes equations one gets
$$
\begin{align}
&\bm{\nabla}^\ast\cdot\bm{u}^\ast=0,\\\\
-&\frac{\partial}{\partial t^\ast}\bm{u}^\ast+\bm{u}^\ast\cdot\bm{\nabla}^\ast\bm{u}^\ast=-\bm{\nabla}^\ast p^\ast+\frac{1}{\mathrm{Re}}\bm{\nabla}^{\ast 2}\bm{u}^\ast,
+&\partial_t^\ast\bm{u}^\ast+\bm{u}^\ast\cdot\bm{\nabla}^\ast\bm{u}^\ast=-\bm{\nabla}^\ast p^\ast+\frac{1}{\mathrm{Re}}\bm{\nabla}^{\ast 2}\bm{u}^\ast,
\end{align}
$$
where $\mathrm{Re}=U\cdot L/\nu$ is the _famous_ Reynolds number which represents the ratio of the inertial over the viscous forces.
@@ -115,7 +115,9 @@ $$
$$
where the space, time, and microscopic velocity dependence is omitted and where
$$
+\begin{equation}
\mathrm{Kn}=\frac{\tau\xi_0}{L},
+\end{equation}
$$
is the Knudsen number.
--
GitLab