## FANDOM

157 Pages

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 Edit

### General Formulation Edit

Conservation of mass

$\frac{\partial \rho}{\partial t} + \nabla\left({\rho \mathbf{v}}\right) = 0$

conservation of momentum

$\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} + \frac{\nabla p}{\rho} = 0$

conservation of energy

$\frac{\partial}{\partial t}\left(e+\frac{1}{2}\rho v^2 \right) + \nabla \left[ \mathbf{v} \left(e+p+\frac{1}{2}v^2 \right) \right] = 0$

## Special Relativistic Edit

### Planar, One Dimensional Edit

Conservation of particle number

$\frac{\partial n}{\partial t} + \frac{\partial}{\partial x} \left( n \beta \gamma \right)= 0$

Conservation of momentum

$\frac{\partial}{\partial t}\left[\gamma^2 \beta \left(e+p\right)\right] + \frac{\partial}{\partial x} \left[ \gamma^2 \beta^2 \left(e+p\right) \right] + \frac{\partial p}{\partial x} = 0$

conservation of energy

$\frac{\partial}{\partial t}\left[\gamma^2 \left( e + \beta^2 p \right)\right] + \frac{\partial}{\partial x}\left[ \gamma^2 \beta \left( e+p\right)\right] = 0$

### Spherical, One Dimensional Edit

Conservation of particle number

$\frac{\partial n}{\partial t}+\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 n \beta \gamma \right ) = 0$

Conservation of momentum

$\frac{\partial}{\partial t}\left[\gamma^2 \beta \left(e+p\right)\right] + \frac{1}{r^2} \frac{\partial}{\partial r} \left[ r^2 \gamma^2 \beta^2 \left(e+p\right) \right] + \frac{\partial p}{\partial r} = 0$

conservation of energy

$\frac{\partial}{\partial t}\left[\gamma^2 \left( e + \beta^2 p \right)\right] + \frac{1}{r^2} \frac{\partial}{\partial r}\left[ r^2 \gamma^2 \beta \left( e+p\right)\right] = 0$

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