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 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 $ν$ 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_oo(alpha/C)(t/tau), where Q_oo is the asymptotic steady-state flow rate, alpha is the geometrical correction factor, and C=P^2/A is the compactness parameter. For the long-time dynamics Q(t) approaches Q_oo exponentially on the timescale $τ$, but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation.