Fig. 5

Hydraulic resistance of concentric elliptical annuli modeling PASs surrounding pial arteries. a Hydraulic resistance \(\mathcal {R}\) as a function of \((\alpha - \beta )/K\) for various fixed values of the area ratio K ranging in steps of 0.2. The black circles indicate the analytic value for the circular annulus, provided by Eq. (11). Red dots indicate optimal shapes, which have minimum \(\mathcal {R}\) for each fixed value of K. b Plots of the hydraulic resistance (red dots) for the optimal concentric elliptical annulus as a function of the area ratio K. Also plotted, for comparison, is the hydraulic resistance of the concentric circular annulus for each value of K. The shaded region indicates the range of K observed in vivo for PASs. The two curves in the shaded region are well represented by the power laws shown. For larger values of K (larger than actual PASs) the influence of the inner boundary becomes less significant and the curves converge to a single power law. c–e Velocity profiles for the optimal shapes resulting in the lowest hydraulic resistance, with fixed \(K=0.4\), 1.4, and 2.4, respectively. The optimal shapes look very similar to the PASs surrounding pial arteries (Fig. 1b–d)