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 $\tau ={(\mathcal{A}∕\mathcal{P})}^2(1∕\nu )$, where $\mathcal{A}$ and $\mathcal{P}$ 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, $Q\left(t\right)=Q_{\infty }(\alpha ∕\mathcal{C})(t∕\tau )$, where $Q_{\infty }$ is the asymptotic steady-state flow rate, $\alpha$ is the geometrical correction factor, and $\mathcal{C}={\mathcal{P}}^2∕\mathcal{A}$ is the compactness parameter. For the long-time dynamics $Q\left(t\right)$ approaches $Q_{\infty }$ exponentially on the time scale $\tau$, but with a weakly geometry-dependent prefactor of order unity, determined by the lowest eigenvalue of the Helmholz equation.

Figure 1

A log-log plot of the flow rate $Q\left(t\right)∕Q_{\infty }$ as a function of time $t∕\tau$. The dashed line is the short-time approximation Eq. (27), while the dashed-dotted line is the long-time approximation Eq. (28), both for the case of a circle, i.e., using $\mathcal{C}∕\alpha =2$ and $a_1=1.45$ as listed in Table . The data points are the results of time-dependent finite-element simulations for the cases of the cross section being a circle (white circles), a square (gray squares), and an equilateral triangle (black triangles).