We consider a steady, isentropic, laminar fluid flow through a pipe whose cross sectional area changes continuously with time. We assume the flow is always along the axis of the pipe, and disregard motion perpendicular to the axis. In the next sections we derive relations between the area and the hydrodynamic variables for an ideal gas.
In steady state, the conservation of momentum reduces to
where is the velocity, is the density, is the pressure and is the position along the pipe. We recall that the specific enthalpy is given by
Hence, for an isentropic flow
So the total enthalpy is conserved along the pipe. For an ideal gas the enthalpy is given by
where is the adiabatic index. Hence the following quantity is conserved along the pipe
The mass current is also constant along the pipe
Plugging into the conservation of total momentum yields a relation between the pressure, velocity and cross sectional area
If the flow is adiabatic, then , when is a constant the represents the entropy. Under this assumption the equation reduces to
The particle current is constant throughout the pipe
where is the number density in the rest frame, is the dimensionless velocity and is the Lorentz factor. Another quantity which is conserved is the energy current
where is the relativistic enthalpy density (in the rest frame), is the pressure in the rest frame and is the energy density in the rest frame. By dividing the two conserved quantities we get a third conserved quantity
In case of an ideal gas with an adiabatic index
Eliminating , we obtain
It can be shown that this expression reduces to the Newtonian result in the limit .