$\displaystyle\frac{\partial f}{\partial t}+\displaystyle\sum_iv_i\displaystyle\frac{\partial f}{\partial x_i}+\displaystyle\sum_ia_i\displaystyle\frac{\partial f}{\partial v_i}=\displaystyle\frac{df}{dt}\mid_c$

\displaystyle\frac{\partial f}{\partial t}+\displaystyle\sum_iv_i\displaystyle\frac{\partial f}{\partial x_i}+\displaystyle\sum_ia_i\displaystyle\frac{\partial f}{\partial v_i}=\displaystyle\frac{df}{dt}\mid_c

$m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3vf+m\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v v_if+m\displaystyle\sum_i\displaystyle\int d^3v a_i\displaystyle\frac{\partial f}{\partial v_i}=m\displaystyle\int d^3v \displaystyle\frac{df}{dt}\mid_c$

m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3vf+m\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v v_if+m\displaystyle\sum_i\displaystyle\int d^3v a_i\displaystyle\frac{\partial f}{\partial v_i}=m\displaystyle\int d^3v \displaystyle\frac{df}{dt}\mid_c

$m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v f=\displaystyle\frac{\partial\rho}{\partial t}$

m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v f=\displaystyle\frac{\partial\rho}{\partial t}

$m\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v v_if=\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}(\rho u_i)=\vec{\nabla}\cdot(\rho\vec{u})$

m\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v v_if=\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}(\rho u_i)=\vec{\nabla}\cdot(\rho\vec{u})

$m\displaystyle\int d^3v \displaystyle\sum_ia_i\displaystyle\frac{\partial f}{\partial v_i}=0$

m\displaystyle\int d^3v \displaystyle\sum_ia_i\displaystyle\frac{\partial f}{\partial v_i}=0

$m\displaystyle\int d^3v \displaystyle\frac{df}{dt}\mid_c=0$

m\displaystyle\int d^3v \displaystyle\frac{df}{dt}\mid_c=0

$\displaystyle\frac{\partial\rho}{\partial t}+\vec{\nabla}\cdot(\rho\vec{u})=0$

\displaystyle\frac{\partial\rho}{\partial t}+\vec{\nabla}\cdot(\rho\vec{u})=0

$m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v v_jf+m\displaystyle\sum_i \displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v v_jv_if+m\displaystyle\sum_i\displaystyle\int d^3v v_ja_i\displaystyle\frac{\partial f}{\partial v_i}=m\displaystyle\int d^3v v_j\displaystyle\frac{df}{dt}\mid_c$

m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v v_jf+m\displaystyle\sum_i \displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v v_jv_if+m\displaystyle\sum_i\displaystyle\int d^3v v_ja_i\displaystyle\frac{\partial f}{\partial v_i}=m\displaystyle\int d^3v v_j\displaystyle\frac{df}{dt}\mid_c

$m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v v_jf=\displaystyle\frac{\partial}{\partial t}(\rho u_j)$

m\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v v_jf=\displaystyle\frac{\partial}{\partial t}(\rho u_j)

$\rho\langle v_jv_i\rangle\equiv m\displaystyle\int d^3v v_jv_if$

\rho\langle v_jv_i\rangle\equiv m\displaystyle\int d^3v v_jv_if

$m\displaystyle\int d^3v \displaystyle\sum_i v_ja_i\displaystyle\frac{\partial f}{\partial v_i}=-\rho a_j$

m\displaystyle\int d^3v \displaystyle\sum_i v_ja_i\displaystyle\frac{\partial f}{\partial v_i}=-\rho a_j

$\vec{w}=\vec{v}-\vec{u}$

\vec{w}=\vec{v}-\vec{u}

$\langle\vec{w}\rangle=\langle\vec{v}\rangle-\langle\vec{u}\rangle=\langle\vec{v}\rangle-\vec{u}=0$

\langle\vec{w}\rangle=\langle\vec{v}\rangle-\langle\vec{u}\rangle=\langle\vec{v}\rangle-\vec{u}=0

$m\displaystyle\int d^3v v_jv_if=\rho\langle v_jv_i\rangle=\rho u_ju_i + \rho\langle w_jw_i\rangle$

m\displaystyle\int d^3v v_jv_if=\rho\langle v_jv_i\rangle=\rho u_ju_i + \rho\langle w_jw_i\rangle

$m\displaystyle\int d^3v v_j\displaystyle\frac{df}{dt}\mid_c=0$

m\displaystyle\int d^3v v_j\displaystyle\frac{df}{dt}\mid_c=0

$\displaystyle\frac{\partial}{\partial t}(\rho u_j)+\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}(\rho u_ju_i + \rho\langle w_jw_i\rangle)-\rho a_j=0$

\displaystyle\frac{\partial}{\partial t}(\rho u_j)+\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}(\rho u_ju_i + \rho\langle w_jw_i\rangle)-\rho a_j=0

$\rho\displaystyle\frac{\partial}{\partial t}\vec{u}+\vec{u}\cdot\nabla\vec{u}+\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}(\rho\langle w_jw_i\rangle)=\rho\vec{a}$

\rho\displaystyle\frac{\partial}{\partial t}\vec{u}+\vec{u}\cdot\nabla\vec{u}+\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}(\rho\langle w_jw_i\rangle)=\rho\vec{a}

$p=\displaystyle\frac{1}{3}\rho\displaystyle\sum_i\langle w_i^2\rangle$

p=\displaystyle\frac{1}{3}\rho\displaystyle\sum_i\langle w_i^2\rangle

$\pi_{ij}=p\delta_{ij}-\rho\langle w_iw_j\rangle$

\pi_{ij}=p\delta_{ij}-\rho\langle w_iw_j\rangle

$\rho\displaystyle\frac{\partial}{\partial t}\vec{u}+\vec{u}\cdot\nabla\vec{u}+\vec{\nabla} p-\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}\pi_{ij}=\rho\vec{a}$

\rho\displaystyle\frac{\partial}{\partial t}\vec{u}+\vec{u}\cdot\nabla\vec{u}+\vec{\nabla} p-\displaystyle\sum_i\displaystyle\frac{\partial}{\partial x_i}\pi_{ij}=\rho\vec{a}

$\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2f+\displaystyle\sum_i \displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2v_if+\displaystyle\sum_i\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2a_i\displaystyle\frac{\partial f}{\partial v_i}=\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2\displaystyle\frac{df}{dt}\mid_c$

\displaystyle\frac{\partial}{\partial t}\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2f+\displaystyle\sum_i \displaystyle\frac{\partial}{\partial x_i}\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2v_if+\displaystyle\sum_i\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2a_i\displaystyle\frac{\partial f}{\partial v_i}=\displaystyle\int d^3v \displaystyle\frac{1}{2}mv^2\displaystyle\frac{df}{dt}\mid_c

$\epsilon=\displaystyle\frac{1}{2}\langle\vec{w}\cdot\vec{w}\rangle$

\epsilon=\displaystyle\frac{1}{2}\langle\vec{w}\cdot\vec{w}\rangle

$\vec{F}=\displaystyle\frac{1}{2}\langle\vec{w}(\vec{w}\cdot\vec{w})\rangle$

\vec{F}=\displaystyle\frac{1}{2}\langle\vec{w}(\vec{w}\cdot\vec{w})\rangle

$\Psi=\displaystyle\sum_{i,j}\pi_{ij}\displaystyle\frac{\partial u_i}{\partial x_j}$

\Psi=\displaystyle\sum_{i,j}\pi_{ij}\displaystyle\frac{\partial u_i}{\partial x_j}

$\rho\displaystyle\frac{\partial\epsilon}{\partial t}+\rho\vec{u}\cdot\vec{\nabla}\epsilon=-p\vec{\nabla}\cdot\vec{u}-\vec{\nabla}\cdot\vec{F}+\Psi$

\rho\displaystyle\frac{\partial\epsilon}{\partial t}+\rho\vec{u}\cdot\vec{\nabla}\epsilon=-p\vec{\nabla}\cdot\vec{u}-\vec{\nabla}\cdot\vec{F}+\Psi