Navier-Stokes
$(\rho)_{,t} = -(\rho v_j)_{,j}$
$(\rho v_i)_{,t} = -(\rho v_i v_j + \delta_{ij} P - \tau_{ji})_{,j} + \rho g_i$
$(E_{total})_{,t} = -(v_j H_{total} - v_i \tau_{ij} + q_j)_{,j}$
$\tau_{ij} = 2 \mu S_{ij} $
$S_{ij} = \frac{1}{2} ( v_{i,j} + v_{j,i} ) - \frac{1}{3} \delta_{ij} v_{k,k} $
$q_j = -\lambda T_{,j} = -C_p \frac{\mu}{Pr} T_{,j}$
$(\rho v_i)_{,t}
= -(
\rho v_i v_j
+ \delta_{ij} P
- \mu (
v_{i,j}
+ v_{j,i}
- \frac{2}{3} \delta_{ij} v_{k,k}
)
)_{,j}
+ \rho g_i$
$(\rho v_i)_{,t} =
- \rho_{,j} v_i v_j
- \rho v_{i,j} v_j
- \rho v_i v_{j,j}
- \delta_{ij} P_{,j}
+ \mu (
v_{i,jj}
+ v_{j,ij}
- \frac{2}{3} \delta_{ij} v_{k,kj}
)
+ \rho g_i$
Let $a_{ij} = v_{i,j}$
$(\rho v_i)_{,t} =
- \rho_{,j} v_i v_j
- \rho a_{ij} v_j
- \rho v_i a_{jj}
- \delta_{ij} P_{,j}
+ \mu (
a_{ij,j}
+ a_{ji,j}
- \frac{2}{3} \delta_{ij} a_{kk,j}
)
+ \rho g_i$
$a_{ij,t} = v_{i,jt}$
$(\rho v_i)_{,t} = \rho_{,t} v_i + \rho v_{i,t} = -(\rho v_j)_{,j} v_i + \rho v_{i,t} = -\rho_{,j} v_j v_i - \rho v_{j,j} v_i + \rho v_{i,t}$
Combine:
$v_{i,t}
+ v_{i,j} v_j
+ \frac{1}{\rho} \delta_{ij} P_{,j}
- \frac{\mu}{\rho} (
v_{i,jj}
+ v_{j,ij}
- \frac{2}{3} \delta_{ij} v_{k,kj}
)
= g_i$
$v_{i,jt}
+ (
v_{i,k} v_k
+ \frac{1}{\rho} \delta_{ik} P_{,k}
- \frac{\mu}{\rho} (
v_{i,kk}
+ v_{k,ik}
- \frac{2}{3} \delta_{ik} v_{l,lk}
)
)_{,j}
= g_{i,j}$
$v_{i,jt}
+ v_{i,kj} v_k
+ v_{i,k} v_{k,j}
- \frac{1}{\rho^2} \rho_{,j} \delta_{ik} P_{,k}
+ \frac{1}{\rho} \delta_{ik} P_{,kj}
+ \frac{\mu}{\rho^2} \rho_{,j} (
v_{i,kk}
+ v_{k,ik}
- \frac{2}{3} \delta_{ik} v_{l,lk}
)
- \frac{\mu}{\rho} (
v_{i,kkj}
+ v_{k,ikj}
- \frac{2}{3} \delta_{ik} v_{l,lkj}
)
= g_{i,j}$