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

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

Occasionally barotropic relation substitutes the energy equation (2.3).
Especially polytropic relation
is often used on behalf of the energy equation.
In the case that the gas is isothermal or isentropic,
the polytropic relations of

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