Spectral spacetime-geometry of Womersley flow

TL;DR

This work formulates a rigorous bridge between frequency-domain impedance analysis and time-domain loop geometry for pulsatile Womersley flow by deriving a spectral spacetime representation of the pressure gradient and mean velocity. By decomposing the driving pressure into harmonics and applying the exact Womersley solution, the authors reconstruct the instantaneous $(G^*,U^*)$ trajectory, showing that the loop area equals the mean hydraulic power per cycle while the loop shape evolves with the Womersley number and harmonic content. The key contributions include a complete analytic framework, a dimensionless atlas of loop energetics, and a topology map that predicts when simple loops become complex via cusps and self-intersections, all tied to physical energy exchange and dissipation. This framework enables principled interpretation of arterial pressure–flow loops, provides a baseline for connecting impedance to time-domain geometry, and offers a reproducible, open-source toolkit for exploring pulsatile hemodynamics in both research and clinical contexts.

Abstract

We revisit the classical Womersley solution for pulsatile viscous flow in a circular tube and reconstruct its full time-domain geometry from first principles. By combining harmonic decomposition with exact Bessel solutions, we derive a unified spectral spacetime analytical solution in which the instantaneous relationship between pressure gradient and velocity can be visualized as a loop in phase space. The enclosed loop area is shown to equal the mean hydraulic power per cycle, establishing an exact geometric-energetic identity that holds for arbitrary Womersley number and harmonic composition. We further show that higher harmonic content and increasing Womersley number induce topological transitions in these loops, producing cusps, self-intersections, and curvature hot spots that correspond to inertial-viscous phase dispersion. This framework provides a rigorous baseline for interpreting arterial pressure-flow loops, connecting frequency-domain impedance analysis to measurable time-domain geometry.

Figures (6)

• Figure 1: Framework validation for a representative case ($\alpha_1=8.0$). (a) Magnitude and phase of the transfer function $\hat{H}^*(n)$. (b) Bar chart confirming that the two methods for calculating mean power agree, and demonstrating that the hysteretic work (loop area) is a distinct physical quantity.
• Figure 2: Evolution of loop morphology with increasing Womersley number for a single-harmonic input. The transition from a narrow ellipse (viscous-dominated) to a wide, open loop (inertia-dominated) is evident.
• Figure 3: Effect of harmonic content on loop topology at a fixed $\alpha_1=8.0$. (a) A moderate case with simple skew. (b) A transitional case showing the onset of cusps. (c) A pathological case with fully developed cusps and a self-intersection.
• Figure 4: Curvature analysis of the "Pathological" loop ($\alpha_1=8.0$). The trajectory is colored by local curvature, highlighting "hot spots" of rapid deceleration. The marked cusps represent points of instantaneous flow stagnation.
• Figure 5: Dimensionless atlas showing the universal scaling of (a) hysteretic work per cycle and (b) fundamental phase lag as a function of the Womersley number, $\alpha_1$.