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<<tau we find a universal linear dependence, Q(t)=Q(infinity) (alpha/C)(t/tau), where Q(infinity) is the asymptotic steady-state flow rate, alpha is the geometrical correction factor, and C=P2/A is the compactness parameter. For the long-time dynamics Q(t) approaches Q(infinity) exponentially on the time scale , but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation.