HPC Tier Module
Womersley Pulsatile Arterial Flow
Evaluate pulsatile arterial flow from the exact Womersley solution, anchored to the Poiseuille limit.
See it run - a worked example, 100% in this browser tab
The problem
Hemodynamics teaching needs an exact Womersley evaluation whose velocity profile, phase lag, and flow rate can be checked against the steady Poiseuille limit, not a numerical approximation of pulsatile flow with unstated error.
The local-first solution
This plugin evaluates the cited closed-form Womersley solution via a complex-Bessel series in exact f64, reports the velocity profile, centerline phase lag, and flow rate, and benchmarks them against the alpha-to-0 Poiseuille limit - in your browser with no upload.
What it does
Closed-form Womersley velocity profile from the complex J0 Bessel series
Womersley number alpha = R*sqrt(omega/nu) as the single governing dimensionless group
Recovers the parabolic Poiseuille profile and u_max in the alpha-to-0 limit
Centerline phase lag growing from 0 toward pi/2 with rising alpha
Plug-like high-alpha core with a thin near-wall Stokes layer
Cross-sectional flow-rate integral with a measured benchmark residual
Honest scope
Exact: the Womersley closed-form solution, the complex J0 series, the alpha-to-0 Poiseuille limit, the centerline phase lag, and the flow-rate integral, evaluated in f64 with a bounded, reported series-truncation error. It assumes a rigid straight circular tube, an incompressible Newtonian fluid, fully-developed axisymmetric laminar flow, and a single pressure-gradient harmonic; it does not model wall elasticity and wave reflections, tapering or bifurcations, entrance-length development, non-Newtonian blood rheology, turbulence, or the full multi-harmonic waveform. Research and education only - not for clinical, diagnostic, or treatment decisions.
Authorities cited
- Womersley, J. R. (1955). Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. Journal of Physiology 127(3), 553-563. DOI 10.1113/jphysiol.1955.sp005276. - the original closed-form solution and the Womersley number alpha.
- Womersley, J. R. (1957). An Elastic Tube Theory of Pulse Transmission and Oscillatory Flow in Mammalian Arteries. WADC Technical Report TR 56-614. - extension to elastic walls (the wall elasticity NOT modeled here).
- Zamir, M. (2000). The Physics of Pulsatile Flow. Springer / AIP / Biological Physics Series. Ch. 4 (oscillatory flow in a rigid tube), eqns 4.5.x. ISBN 978-0-387-98925-3. - derivation, the alpha -> 0 Poiseuille limit, and the high-alpha annular effect.
- Nichols, W. W., O'Rourke, M. F., & Vlachopoulos, C. (2011). McDonald's Blood Flow in Arteries, 6th ed., Ch. 7 (Wave reflection) and the Womersley analysis. CRC Press. ISBN 978-0-340-98501-4.
- Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions. NBS Applied Mathematics Series 55, eqn 9.1.10 - the entire power series of the Bessel function J0(z).
- Loudon, C., & Tordesillas, A. (1998). The use of the dimensionless Womersley number to characterize the unsteady nature of internal flow. Journal of Theoretical Biology 191(1), 63-78. DOI 10.1006/jtbi.1997.0564. - physiological alpha ranges across the arterial tree.
Evaluate the flow
Compute the velocity profile, phase lag, and flow rate in your browser, with nothing uploaded to anyone's cloud. Save the run to Sandbox, attach it to a Worklog case, or share parameters through a Gate portal.