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 τ=(A∕P)²(1∕ν), 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⪡τ we find a universal linear dependence, Q(t)=Q_∞(α∕C)(t∕τ), where Q_∞ is the asymptotic steady-state flow rate, α is the geometrical correction factor, and C=P²∕A is the compactness parameter. For the long-time dynamics $Q(t)$ approaches Q_∞ exponentially on the time scale τ, but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation.