Hydrodynamics describe the flow of a fluid. It is governed by three to five (depending on the dimension) conservation law for mass momentum and energy. The main difficulty in this area is that the equations are nonlinear, so it is difficult to obtain analytic results.
In the following section we will write several useful explicit forms of the hydrodynamic equations.
Newtonian [ ] General Formulation [ ] Conservation of mass
∂
ρ
∂
t
+
∇
(
ρ
v
)
=
0
{\displaystyle \frac{\partial \rho}{\partial t} + \nabla\left({\rho \mathbf{v}}\right) = 0 }
conservation of momentum
∂
v
∂
t
+
v
⋅
∇
v
+
∇
p
ρ
=
0
{\displaystyle \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} + \frac{\nabla p}{\rho} = 0 }
conservation of energy
∂
∂
t
(
e
+
1
2
ρ
v
2
)
+
∇
[
v
(
e
+
p
+
1
2
v
2
)
]
=
0
{\displaystyle \frac{\partial}{\partial t}\left(e+\frac{1}{2}\rho v^2 \right) + \nabla \left[ \mathbf{v} \left(e+p+\frac{1}{2}v^2 \right) \right] = 0 }
Special Relativistic [ ] Planar, One Dimensional [ ] Conservation of particle number
∂
n
∂
t
+
∂
∂
x
(
n
β
γ
)
=
0
{\displaystyle \frac{\partial n}{\partial t} + \frac{\partial}{\partial x} \left( n \beta \gamma \right)= 0 }
Conservation of momentum
∂
∂
t
[
γ
2
β
(
e
+
p
)
]
+
∂
∂
x
[
γ
2
β
2
(
e
+
p
)
]
+
∂
p
∂
x
=
0
{\displaystyle \frac{\partial}{\partial t}\left[\gamma^2 \beta \left(e+p\right)\right] + \frac{\partial}{\partial x} \left[ \gamma^2 \beta^2 \left(e+p\right) \right] + \frac{\partial p}{\partial x} = 0 }
conservation of energy
∂
∂
t
[
γ
2
(
e
+
β
2
p
)
]
+
∂
∂
x
[
γ
2
β
(
e
+
p
)
]
=
0
{\displaystyle \frac{\partial}{\partial t}\left[\gamma^2 \left( e + \beta^2 p \right)\right] + \frac{\partial}{\partial x}\left[ \gamma^2 \beta \left( e+p\right)\right] = 0 }
Spherical, One Dimensional [ ] Conservation of particle number
∂
n
∂
t
+
1
r
2
∂
∂
r
(
r
2
n
β
γ
)
=
0
{\displaystyle \frac{\partial n}{\partial t}+\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 n \beta \gamma \right ) = 0 }
Conservation of momentum
∂
∂
t
[
γ
2
β
(
e
+
p
)
]
+
1
r
2
∂
∂
r
[
r
2
γ
2
β
2
(
e
+
p
)
]
+
∂
p
∂
r
=
0
{\displaystyle \frac{\partial}{\partial t}\left[\gamma^2 \beta \left(e+p\right)\right] + \frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \gamma^2 \beta^2 \left(e+p\right) \right] + \frac{\partial p}{\partial r} = 0 }
conservation of energy
∂
∂
t
[
γ
2
(
e
+
β
2
p
)
]
+
1
r
2
∂
∂
r
[
r
2
γ
2
β
(
e
+
p
)
]
=
0
{\displaystyle \frac{\partial}{\partial t}\left[\gamma^2 \left( e + \beta^2 p \right)\right] + \frac{1}{r^2} \frac{\partial}{\partial r}\left[ r^2 \gamma^2 \beta \left( e+p\right)\right] = 0 }
All items (13)
![{\displaystyle {\frac {\partial }{\partial t}}\left(e+{\frac {1}{2}}\rho v^{2}\right)+\nabla \left[\mathbf {v} \left(e+p+{\frac {1}{2}}v^{2}\right)\right]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/9f72d41abb81f2dd465cb483cab91624fcf40ee1)
![{\displaystyle {\frac {\partial }{\partial t}}\left[\gamma ^{2}\beta \left(e+p\right)\right]+{\frac {\partial }{\partial x}}\left[\gamma ^{2}\beta ^{2}\left(e+p\right)\right]+{\frac {\partial p}{\partial x}}=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/c22f5e01910f064050080f2815f16b646d7364bf)
![{\displaystyle {\frac {\partial }{\partial t}}\left[\gamma ^{2}\left(e+\beta ^{2}p\right)\right]+{\frac {\partial }{\partial x}}\left[\gamma ^{2}\beta \left(e+p\right)\right]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/b92fcef1a3ffd95c45c62a97f99df7d3eb0b1b5c)
![{\displaystyle {\frac {\partial }{\partial t}}\left[\gamma ^{2}\beta \left(e+p\right)\right]+{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left[r^{2}\gamma ^{2}\beta ^{2}\left(e+p\right)\right]+{\frac {\partial p}{\partial r}}=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/708715c9926578f67266f236c2418fecdf781aee)
![{\displaystyle {\frac {\partial }{\partial t}}\left[\gamma ^{2}\left(e+\beta ^{2}p\right)\right]+{\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left[r^{2}\gamma ^{2}\beta \left(e+p\right)\right]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/6ef2578cf4c8a779b245dd85e796a695c0ba1080)
