Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2015 / Paper 1 / 2 / b

For steady inviscid motion of a barotropic fluid in a conservative gravitational potential $\Phi$, the Euler equations for an inviscid fluid imply the Bernoulli equationalong each streamline. The barotropic enthalpy function $h$ has units of energy per unit mass. For an isentropic ideal gas, $p=K{\rho }^{\gamma }$ gives $h=\gamma p/[(\gamma -1)\rho ]$ up to an additive constant. For constant density, the corresponding term is $p/\rho$. Pressure work converts between kinetic, enthalpy and gravitational energy.

A converging subsonic nozzle can accelerate a fluid at the expense of its enthalpy; in the incompressible horizontal approximation, higher speed accompanies lower static pressure. Together with conservation of mass, this explains a Venturi effect measurement. Compressible nozzle flow additionally needs its equation of state and sonic conditions. In a stellar wind, the potential term is $-GM/r$, so enthalpy or mechanical energy can be spent escaping gravity. A pressure-negligible marginal-escape wind has $v^2/2-GM/r=0$.

The constant need not be the same on different streamlines. Irrotational barotropic flow in a connected region gives a common value; a general rotational flow does not. A pump, viscosity, a radiating shock, or heat addition changes the relevant energy balance. In particular, adiabaticity alone does not allow one to drop the enthalpy term.