Basic Equations of Hydrodynamics

The basic equation of hydrodynamics are (1) the continuity equation of the density [equation (A.11)],

$$\frac{\partial \rho}{\partial t}+{\rm div} (\rho{\bf v})=0,$$ (2.1)

(2) the equation of motion [equation (A.7)]

$$\rho \left[ \frac{\partial {\bf v}}{\partial t}+\left({\bf v}\cdot \nabla\right) {\bf v} \right]=-\nabla p +\rho {\bf g},$$ (2.2)

and (3) the equation of energy [equation (A.23)]

$$\frac{\partial \epsilon}{\partial t}+{\rm div}(\epsilon+p){\bf v}=\rho {\bf v}\cdot {\bf g}.$$ (2.3)

Occasionally barotropic relation $p=P(\rho)$ substitutes the energy equation (2.3). Especially polytropic relation $p=K\rho^\Gamma$ is often used on behalf of the energy equation. In the case that the gas is isothermal or isentropic, the polytropic relations of

$$p=c_{is}^2\rho \rm (isothermal)$$ (2.4)

and

$$p=c_s^2 \rho^\gamma \rm (isentropic)$$ (2.5)

are substitution to equation (2.3). [We can replace equation (2.3) with equations (2.4) and (2.5).]