The Relation Between Flow Rate and Axial Pressure Gradient for Time-Periodic Poiseuille Flow in a Pipe

Consider a fully developed, time-periodic motion of a Navier–Stokes fluid in an infinite straight pipe of constant cross section Ω (time-periodic Poiseuille flow). In this note we show that the axial pressure gradient and the flow rate associated to this motion are uniquely connected through a very simple relation involving parameters depending only on Ω and, therefore, independent of the particular velocity field. One immediate and important consequence of this property is that it allows for a very elementary proof of existence of time-periodic Poiseuille flow under a given flow rate.