Flow velocity out of a pipe For a compressible homoentropic gas is the flow velocity out of the pipe the same as the velocity into the pipe?
 A: No.
If the fluid is compressible and the flow is not stationary we can choose initial conditions with an inhomogeneous density. Then the flow will equilibrate and the density will become become stationary but during this time anything can happen. One can build a counter example simply by choosing appropriate initial conditions.
If your are asking about a stationary state, I can not prove anything but I can argue that the answer is no as well: A stationary flow along the tube, must be driven by a pressure gradient. The definition of a homoentropic fluid is that, up to an additive constant, the pressure and the density are proportional to each other (and the proportionality constant is positive). Then there is as well a gradient of density along the tube. The density is greater upstream and lower downstream. The mass flux is the product of the density and velocity and must be the same at both ends of the tube. If not the system is gaining or loosing particles and is not stationary. Then if there is a high density at the inlet and a low density at the outlet the velocity must compensate in order for the mass fluxes to be the same. We get a small velocity upstream and a large velocity downstream.
A: I suggest that it would oscillate. Friction against the walls of the pipe would cause the average velocity profile across the pipe to decrease which would increase pressure which would then increase velocity.
