Fig. 3

Hydraulic resistance and velocity profiles in eccentric circular annuli modeling PASs surrounding penetrating arteries. a Plots of hydraulic resistance \(\mathcal {R}\) for an eccentric circular annulus, as a function of the relative eccentricity \(\epsilon /(\alpha - 1)\), for various fixed values of the area ratio \(K= \alpha ^2 - 1\) ranging in steps of 0.2, computed using Eq. (12). b Plots of the hydraulic resistance (red dots) for the tangent eccentric circular annulus (defined as \(\epsilon /(\alpha -1)=1\)) 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. Power laws are indicated that fit the points well through most of the shaded region. c–e Velocity profiles for three different eccentric circular annuli with increasing eccentricity (with \(K=1.4\) held constant): (c) \(\epsilon =0\) (concentric circular annulus), (d) \(\epsilon =0.27\) (eccentric circular annulus), and (e) \(\epsilon =0.55\) (tangent eccentric circular annulus). The black circle, purple asterisk, and red dot in a indicate the hydraulic resistance of the shapes shown in c–e, respectively. The volume flow rates for the numerically calculated profiles shown in c–e agree with the analytical values to within 0.3%. As eccentricity increases hydraulic resistance decreases and volume flow rate increases