Universal dynamics in the onset of a Hagen-Poiseuille flow

The dynamics in the onset of a Hagen-Poiseuille flow of an incompressible liquid in a channel of circular cross section is well-studied theoretically. We use an eigenfunction expansion in a Hilbert space formalism to generalize the results to channels of an arbitrary cross section. We find that the steady state is reached after a characteristic time scale tau=(A/P)2(1/nu), where A and P are the cross-sectional area and perimeter, respectively, and nu is the kinematic viscosity of the liquid. For the initial dynamics of the flow rate Q for t<