diff --git a/lattice-boltzmann.com/content/2-hermite-expansion.md b/lattice-boltzmann.com/content/2-hermite-expansion.md
index 40201e7fdd02c657aa32ca3a7aafefaf21cd851e..19496a21505671142a3e2443afe2951f68b480c1 100644
--- a/lattice-boltzmann.com/content/2-hermite-expansion.md
+++ b/lattice-boltzmann.com/content/2-hermite-expansion.md
@@ -39,13 +39,81 @@ natural to take the zeroth and first order moments of the BGK equation, and one
 $$
 \begin{align}
 &\partial_t\rho + \bm{\nabla}\cdot(\rho\bm{u})=0,\\\\
-&\partial_t(\rho\bm{u}) + \bm{\nabla}\cdot\left(\int_{-\infty}^\infty\bm{\xi}\bm{\xi}f\mathrm{d}\bm{\xi}\right)=0.
+&\partial_t(\rho\bm{u}) + \bm{\nabla}\cdot\left(\int_{-\infty}^\infty\bm{\xi}\bm{\xi}f\mathrm{d}\bm{\xi}\right)=0,
 \end{align}
 $$
-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.
+where $\\{\bm{\xi}\bm{\xi}\\}_{ab}=\xi_a\xi_b$.
 
+We must note that the rhs of these equations is zero which is actually not a surprise, since the rhs of BGK equation is an approximation of the collision operator which must conserve mass and momentum.
+Equation (4) is the well known mass conservation equation but equation (5) must be reworked
+a bit to remind us something similar to the momentum conservation of the
+Navier--Stokes equations. To do so we replace $\bm{\xi}$ by $\bm{\xi}-\bm{u}+\bm{u}$ and we get
+$$
+\begin{align}
+&\partial_t(\rho\bm{u}) + \bm{\nabla}\cdot\left(\int_{-\infty}^\infty(\bm{\xi}-\bm{u}+\bm{u})(\bm{\xi}-\bm{u}+\bm{u})f\mathrm{d}\bm{\xi}\right)=0,\nonumber\\\\
+&\partial_t(\rho\bm{u}) + \bm{\nabla}\cdot(\rho\bm{u}\bm{u}) + \bm{\nabla}\cdot \bm{P}=0,\\\\
+\end{align}
+$$
+where $\bm{P}$ is the stress tensor
+$$
+\begin{equation}
+\bm{P}=\int_{-\infty}^\infty (\bm{\xi}-\bm{u})(\bm{\xi}-\bm{u})f\mathrm{d}\bm{\xi}
+\end{equation}
+$$
+which is the second order moment of $f$ in the co-moving frame. 
 
+In the Navier--Stokes framework, for a Newtonian fluid, the stress tensor is is given by
+$$
+\begin{equation}
+\bm{P}=\bm{I}p-\frac{2}{\mathrm{Re}}\bm{S},
+\end{equation}
+$$
+where $\bm{I}$ is the identity matrix, and $\bm{S}$ is the strain rate tensor defined by
+$$
+\begin{equation}
+\bm{S}=\frac{1}{2}\left(\bm{\nabla}\bm{u}+(\bm{\nabla}\bm{u})^T\right),
+\end{equation}
+$$
+the superscript $T$ standing for the transposition operation.
 
+In the Boltzmann BGK framework we still need to derive a closure for $\bm{P}$.
 
+## Chapmann--Enskog expansion
 
+In order to obtain the constitutive equation for $\bm{P}$ in the BGK framework, we will perform the Chapmann--Enskog expansion
+where we assume $f$ is given by a perturbation around the equilibrium
+$$
+\begin{equation}
+f=f^{eq}+\mathrm{Kn}f^{neq},
+\end{equation}
+$$
+with $\mathrm{Kn}\ll 1$. With this assumption, equation (6) becomes
+$$
+\begin{equation}
+\partial_t(\rho\bm{u}) + \bm{\nabla}\cdot(\rho\bm{u}\bm{u}) + \bm{\nabla}\cdot \left(\bm{P}^{eq}+\mathrm{Kn}\bm{P}^{neq}\right)=0,
+\end{equation}
+$$
+with 
+$$
+\begin{align}
+\bm{P}^{eq}&=\int_{-\infty}^\infty (\bm{\xi}-\bm{u})(\bm{\xi}-\bm{u})f^{eq}\mathrm{d}\bm{\xi}=\rho\theta\bm{I},\\\\
+\bm{P}^{neq}&=\int_{-\infty}^\infty (\bm{\xi}-\bm{u})(\bm{\xi}-\bm{u})f^{neq}\mathrm{d}\bm{\xi}=\int_{-\infty}^\infty \bm{\xi}\bm{\xi}f^{neq}\mathrm{d}\bm{\xi},
+\end{align}
+$$
+where equation (12) is the perfect gaz law, and where we still need to find an explicit expression for $\bm{P}^{neq}$.
+In the second equality of equation (13), we used the mass and momentum conservation laws that imply that the zeroth and 
+first order moments of $f^{neq}$ are null since (see equations (4)-(5))
+$$
+\begin{align}
+&0=\int_{-\infty}^\infty (f - f^{eq})\mathrm{d}\bm{\xi}=\mathrm{Kn}\int_{-\infty}^\infty f^{neq}\mathrm{d}\bm{\xi},\\\\
+&0=\int_{-\infty}^\infty \bm{\xi}(f - f^{eq})\mathrm{d}\bm{\xi}=\mathrm{Kn}\int_{-\infty}^\infty \bm{\xi}f^{neq}\mathrm{d}\bm{\xi}.
+\end{align}
+$$
+In order to find a closure relation for $\bm{P}^{neq}$,
+we will replace the Chapmann--Enskog expansion in the BGK equation, keep only the lower order terms, and we obtain
+$$
+\begin{equation}
+\partial_t f^{eq}+\bm{\xi}\cdot \bm{\nabla}f^{eq} = -f^{neq}.
+\end{equation}
+$$