a few aditions

lattice-boltzmann.com/content/1-dimensionless-lbm.md

@@ -8,9 +8,9 @@ slug = "dimensionless"
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-# Dimensionless Navier-Stokes
+## Thed imensionless Navier-Stokes
 
-In order to write the dimensionless lattice Boltzmann equation, we start by reminding the basis of dimensionless formulation by writing the dimensionless incompressible Navier--Stokes which are more common in the litterature.
+In order to write the dimensionless lattice Boltzmann equation, we start by reminding the basis of dimensional analysis for fluid flows by writing the dimensionless incompressible Navier--Stokes which are more common in the litterature.
 
 The incompressible Navier--Stokes equations reads
 
@@ -36,7 +36,66 @@ Replacing these equations into the Navier--Stokes equations one gets
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 \begin{aligned}
 &\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,
+&\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,
 \end{aligned}
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-where $\mathrm{Re}=U\cdot L/\nu$ is the _famous_ Reynolds number which represents the ratio of the .
+where $\mathrm{Re}=U\cdot L/\nu$ is the _famous_ Reynolds number which represents the ratio of the inertial over the viscous forces.
+
+It must me noted that this non-dimensionnalization procedure heavily relies on the choice of the characteristic timescale of the flow 
+which is this case can be constructed through the characteristic velocity of the flow, $T=L/U$. By chosing the kinematic viscosity instead of the
+velocity, the characteristic time of the flow becomes, $T=L^2/\nu$. With this new characteristic time the dimensionless quantities become
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+\begin{aligned}
+&\bm{u}^\ast=\frac{L}{\nu}\bm{u}, &p^\ast=\frac{L^2}{\nu^2\rho}p,\\\\
+&t^\ast=\frac{\nu}{L^2}t, &\bm{x}^\ast=\frac{\bm{x}}{L},\\\\
+&\frac{\partial}{\partial t^\ast}=\frac{L^2}{\nu}\frac{\partial}{\partial t}, &\bm{\nabla}^\ast=L\bm{\nabla}
+\end{aligned}
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+which in turn give the following dimensionless Navier--Stokes equations
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+\begin{aligned}
+&\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+\bm{\nabla}^{\ast 2}\bm{u}^\ast,
+\end{aligned}
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+where we see that no dimensionless number is present anymore. This kind of non-dimensionalization is only valid for creeping flows that move only very slowly and will not interest us in this series of tutorials.
+
+## The dimensionless Boltzmann equation
+
+The quantity of interest in the Boltzmann equation is the desity probability distribution function
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+f(\bm{x},\bm{\xi}, t)
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+which represents the probability of finding a particle at position $\bm{x}$ with velocity $\bm{\xi}$ at time $t$.
+
+In this tutorial we will only be interested in the Boltzmann-BGK equation[^1]
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+\partial_t f(\bm{x}, \bm{\xi}, t)+\bm{\xi}\cdot \bm{\nabla}f(\bm{x},\bm{\xi}, t)=-\frac{1}{\tau}\left(f(\bm{x}, \bm{\xi}, t)-f^{eq}(\bm{x}, \bm{\xi}, t)\right),
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+where $\tau$ is the relaxation time of the fluid, and $f^{eq}$ the equilibrium distribution function which is typically the Maxwell-Boltzmann distribution (more on this later).
+
+The dimensionless density distribution function is then given by
+
+
+[^1]: See reference TODO
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