Hydrodynamics
Mass
Total mass in volume:
$m = \int_\Omega \rho dV$ = mass, in $[kg]$
$\rho$ = density, in $[\frac{kg}{m^3}]$
$\Omega$ = spatial domain, within $\textbf{R}^3$
Rate-of-change of mass:
$\partial_t m = \int_\Omega \partial_t \rho dV$, in $[\frac{kg}{s}]$
Rate-of-change only considering boundary:
$\partial_t m = \int_{\partial\Omega} \rho v^j dS_j$, in $[\frac{kg}{s}]$
$v^i$ = velocity, in $[\frac{m}{s}]$
Apply Green's theorem to rate-of-change of boundary:
$\partial_t m = -\int_\Omega \nabla_j (\rho v^j) dV$ in $[\frac{kg}{s}]$
Consider rate-of-change at any point:
Conservation of mass:
$\partial_t \rho + \nabla_j (\rho v^j) = 0$ in $[\frac{kg}{m^3 \cdot s}]$
Momentum
Conservation of momentum:
$\int_\Omega \Pi_C^i dV = \int_\Omega \rho v^i dV$ in $[\frac{kg \cdot m}{s}]$
$\partial_t \int_\Omega \Pi_C^i dV$ = change in momentum due to conservation, in $[\frac{kg \cdot m}{s^2}]$
Rate-of-change of momentum
$\partial_t \int_\Omega \Pi_C^i dV = \int_\Omega \partial_t (\rho v^i) dV$, in $[\frac{kg \cdot m}{s^2}]$
Rate-of-change of only considering boundary (neglecting forces like gravity, pressure, viscosity, etc)
$\partial_t \int_\Omega \Pi_C^i dV = \int_{\partial\Omega} \rho v^i v^j dS_j$, in $[\frac{kg \cdot m}{s^2}]$
Apply Green's theorem:
$\partial_t \int_\Omega \Pi_C^i dV = -\int_\Omega \nabla_j (\rho v^i v^j) dV$, in $[\frac{kg \cdot m}{s^2}]$
Consider at any point:
(TODO this should either be a different symbol, (as $\rho$ relates to $m$), or annotation of like $\Pi$ vs $\tilde\Pi$).
$\partial_t \Pi_C^i = -\nabla_j (\rho v^i v^j)$, in $[\frac{kg}{m^2 \cdot s^2}]$
Equation of state: ideal gas law.
Hydro pressure from internal energy:
$P = (\gamma - 1) \rho e_{int} = (\gamma - 1) E_{int}$, in $[\frac{kg}{m \cdot s^2}]$
$\gamma = $ heat capacity ratio, in $[1]$
$P$ = hydro pressure, in $[\frac{kg}{m \cdot s^2}]$
$e_{int}$ = specific internal energy per density, per volume, in $[\frac{m^2}{s^2}]$
$E_{int}$ = internal energy per volume, in $[\frac{kg}{m \cdot s^2}]$
internal energy, per volume:
$E_{int} = \rho e_{int} = \frac{P}{\gamma - 1}$, in $[\frac{kg}{m \cdot s^2}]$
Force due to pressure:
$\partial_t \int_\Omega \Pi_P^i dV = \int_{\partial\Omega} P dS_i$, in $[\frac{kg \cdot m}{s^2}]$
$\partial_t \int_\Omega \Pi_P^i dV = -\int_\Omega \nabla_i P dV$ via Green's theorem, in $[\frac{kg \cdot m}{s^2}]$
$\partial_t \Pi_P^i = -\nabla_i P$ at any point, in $[\frac{kg}{m^2 \cdot s^2}]$
Total change in momemtum:
$\partial_t \Pi^i = \partial_t \Pi_C^i + \partial_t \Pi_P^i$, in $[\frac{kg}{m^2 \cdot s^2}]$
$\partial_t \Pi^i = -\nabla_j (\rho v^i v^j) - \nabla_i P$, in $[\frac{kg}{m^2 \cdot s^2}]$
$\partial_t \Pi^i + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
... where $g^{ij}$ is the inverse metric tensor associated with our manifold tangent-space basis.
$\partial_t \Pi^i + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
For $\Pi^i = $ hydro momentum $ = \rho v^i$ we find
$\partial_t (\rho v^i) + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
Energy
specific kinetic energy, per density, per volume:
$e_{kin} = \frac{1}{2} v^2$, in $[\frac{m^2}{s^2}]$
... where $v^2 = v^i v^j g_{ij}$.
kinetic energy, per volume:
$E_{kin} = \rho e_{kin} = \frac{1}{2} \rho v^2$, in $[\frac{kg}{m \cdot s^2}]$
Total hydro energy:
$\int_\Omega E_{hydro} dV = \int_\Omega (E_{int} + E_{kin}) dV$, in $[\frac{kg \cdot m^2}{s^2}]$
$\int_\Omega E_{hydro} dV = \int_\Omega (\frac{1}{2}\rho v^2 + \frac{P}{\gamma - 1}) dV$, in $[\frac{kg \cdot m^2}{s^2}]$
change in total energy due to conservation of hydro energy:
$\partial_t \int_\Omega E^C_{total} dV = \int_\Omega \partial_t E_{hydro} dV$, in $[\frac{kg \cdot m^2}{s^3}]$
consider velocity moving through boundaries:
$\partial_t \int_\Omega E^C_{total} dV = \int_{\partial\Omega} E_{hydro} v^i dS_i$, in $[\frac{kg \cdot m^2}{s^3}]$
apply Green's theorem:
$\partial_t \int_\Omega E^C_{total} dV = -\int_\Omega \nabla_j (E_{hydro} v^j) dV$, in $[\frac{kg \cdot m^2}{s^3}]$
for any point:
$\partial_t E^C_{total} = -\nabla_j (E_{hydro} v^j)$, in $[\frac{kg}{m \cdot s^3}]$
Change in energy due to hydro pressure force:
$\partial_t \int_\Omega E^P_{total} dV = \int_{\partial\Omega} P v^j dS_j$, in $[\frac{kg \cdot m^2}{s^3}]$
$\partial_t \int_\Omega E^P_{total} dV = -\int_\Omega \nabla_j (P v^j) dV$ via Green's theorem, in $[\frac{kg \cdot m^2}{s^3}]$
$\partial_t E^P_{total} = -\nabla_j (P v^j)$ at any point, in $[\frac{kg}{m \cdot s^3}]$
Total change in total energy:
$\partial_t E_{total} = \partial_t E^C_{total} + \partial_t E^P_{total}$, in $[\frac{kg}{m \cdot s^3}]$
$\partial_t E_{total} = -\nabla_j (E_{hydro} v^j) - \nabla_j (P v^j)$, in $[\frac{kg}{m \cdot s^3}]$
$\partial_t E_{total} + \nabla_j ((E_{hydro} + P) v^j) = 0$, in $[\frac{kg}{m \cdot s^3}]$
$\partial_t E_{total} + \nabla_j (H_{hydro} v^j) = 0$, in $[\frac{kg}{m \cdot s^3}]$
for $H_{hydro} = E_{hydro} + P$, in $[\frac{kg}{m \cdot s^2}]$
Conservative form of Euler:
$\partial_t \rho + \nabla_j (\rho v^j) = 0$, in $[\frac{kg}{m^3 \cdot s}]$
$\partial_t (\rho v^i) + \nabla_j (\rho v^i v^j + g^{ij} P) = 0$, in $[\frac{kg}{m^2 \cdot s^2}]$
$\partial_t E_{hydro} + \nabla_j (H_{hydro} v^j) = 0$, in $[\frac{kg}{m \cdot s^3}]$
the hyperbolic conservation law equation:
$\partial_t U^I + \nabla_j F^{Ij} = 0$
Euler fluid PDE conserved variables:
$U^I = \downarrow I \left[\begin{matrix}
\rho \\ \rho v^i \\ E_{hydro}
\end{matrix}\right]$
Euler fluid PDE flux vector:
$F^{Ij} = \downarrow I \left[\begin{matrix}
\rho v^j \\
\rho v^i v^j + g^{ij} P \\
H_{hydro} v^j
\end{matrix}\right]$
Euler fluid PDE primitive variables:
$W^K = \downarrow K \left[\begin{matrix}
\rho \\ v^k \\ P
\end{matrix}\right]$
$\frac{\partial E_{hydro}}{\partial \rho}$
$= \frac{\partial}{\partial \rho} ( \frac{1}{2} \rho v^2 + \frac{P}{\gamma-1})$
$= \frac{1}{2} v^2$, in $[\frac{m^2}{s^2}]$
$\frac{\partial E_{hydro}}{\partial v^k}$
$= \frac{\partial}{\partial v^k} ( \frac{1}{2} \rho v^2 + \frac{P}{\gamma-1})$
$= \rho v_k$, in $[\frac{kg}{m^2 \cdot s}]$
$\frac{\partial E_{hydro}}{\partial P}$
$= \frac{\partial}{\partial P} ( \frac{1}{2} \rho v^2 + \frac{P}{\gamma-1})$
$= \frac{1}{\gamma-1}$, in $[1]$
Derivative of conservatives wrt primitives:
$\frac{\partial U^I}{\partial W^J} = \downarrow I \overset{\rightarrow J}{ \left[\begin{matrix}
1 & 0 & 0 \\
v^i & \rho \delta^i_j & 0 \\
\frac{1}{2} v^2 & \rho v_j & \frac{1}{\gamma-1}
\end{matrix}\right] }$
Derivative of primitives wrt conservatives:
$\frac{\partial W^J}{\partial U^I} = \downarrow J \overset{\rightarrow I}{ \left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^j & \frac{1}{\rho} \delta_i^j & 0 \\
\frac{1}{2} (\gamma-1) v^2 & -(\gamma-1) v_i & \gamma-1
\end{matrix}\right] }$
Derivative of flux with respect to primitive variables:
$\frac{\partial F^{Ij}}{\partial W^K} = \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
v^j & \rho \delta^j_k & 0 \\
v^i v^j & \rho (\delta^i_k v^j + v^i \delta^j_k) & g^{ij} \\
\frac{1}{2} v^2 v^j & \rho v^j v_k + H_{hydro} \delta^j_k & \frac{\gamma}{\gamma-1} v^j
\end{matrix}\right] }$
Derivative of flux with respect to conservative variables:
$\frac{\partial F^{Ij}}{\partial U^K} = \frac{\partial F^{Ij}}{\partial W^L} \frac{\partial W^L}{\partial U^K}
= \downarrow I \overset{\rightarrow L}{ \left[\begin{matrix}
v^j & \rho \delta^j_l & 0 \\
v^i v^j & \rho (\delta^i_l v^j + v^i \delta^j_l) & g^{ij} \\
\frac{1}{2} v^2 v^j & \rho v^j v_l + (E_{hydro} + P) \delta^j_l & \frac{\gamma}{\gamma-1} v^j
\end{matrix}\right] } \cdot \downarrow L \overset{\rightarrow K}{ \left[\begin{matrix}
1 & 0 & 0 \\
-\frac{1}{\rho} v^l & \frac{1}{\rho} \delta^l_k & 0 \\
\frac{1}{2} (\gamma-1) v^2 & -(\gamma-1) v_k & \gamma-1
\end{matrix}\right] }$
$= \downarrow I \overset{\rightarrow K}{ \left[\begin{matrix}
0 & \delta^j_k & 0 \\
-v^i v^j + \frac{1}{2} g^{ij} (\gamma-1) v^2 &
\delta^i_k v^j + \delta^j_k v^i - g^{ij} (\gamma-1) v_k &
g^{ij} (\gamma - 1) \\
\frac{1}{2} v^j v^2 (\gamma - 1) - v^j h_{hydro} &
-(\gamma-1) v^j v_k + \delta^j_k h_{hydro} &
\gamma v^j
\end{matrix}\right] }$
for $h_{hydro} = \frac{1}{\rho} H_{hydro}$, in $[\frac{m^2}{s^2}]$