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 $\Omega$ (time-periodic Poiseuille flow). In this talk 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 $\Omega$ 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.