diff --git a/lattice-boltzmann.com/content/2-hermite-expansion.md b/lattice-boltzmann.com/content/2-hermite-expansion.md index 6fb3f834e1f2481d6f66956dbf166389f2adef61..40201e7fdd02c657aa32ca3a7aafefaf21cd851e 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. +