Womersley Number (pulsatile flow)

John Womersley was a mathematician who spent the early 1950s at Bristol working on arterial haemodynamics with Donald McDonald, and in 1955 he published in the Journal of Physiology the solution for oscillatory flow of a viscous fluid in a rigid tube. The solution is written in Bessel functions and is not something to carry around; the parameter it depends on is, and it is now universally called $\alpha$.

The question it answers is whether the flow has time to sort itself out between beats. In steady flow in a tube, viscosity has all the time in the world to diffuse momentum inward from the wall, and the answer is Poiseuille's parabola. In pulsatile flow it may not. Viscous information diffuses inward a distance of roughly $\sqrt{\nu/\omega}$ in one radian of the cycle — the Stokes layer thickness — and $\alpha$ is just the vessel radius divided by that:

\[\alpha = R\sqrt{\frac{\omega\rho}{\mu}} = \frac{R}{\sqrt{\nu/\omega}}\]

with $\omega = 2\pi f$ and $f$ the heart rate. That is the whole of it. Everything else is consequence.

• $\alpha \ll 1$: quasi-steady. The Stokes layer is far thicker than the vessel, viscosity reaches the centreline in a fraction of a beat, and at every instant the profile is Poiseuille's parabola scaled up and down. Flow and pressure gradient stay in phase, and steady-flow resistance is a fair model. Arterioles are near $\alpha = 0.1$, capillaries near 0.005.
• $\alpha \gg 1$: inertia dominated. The Stokes layer is a thin annulus at the wall, the core moves as a plug driven by its own inertia, and the profile is flat with steep gradients only at the edges. Flow lags the pressure gradient by close to 90°, because the fluid is behaving like a mass being accelerated rather than a resistance being driven. The human aorta sits at $\alpha \approx 15\text{–}20$ at rest.
• $\alpha \sim 1\text{–}10$: the transition, which is where most named arteries live — femoral, carotid, coronary.

The consequence that matters clinically is wall shear stress. In a plug-like profile at high $\alpha$, the shear is concentrated in the thin wall layer, so peak wall shear is higher than a Poiseuille calculation at the same mean flow would suggest, and it oscillates hard through the cycle. Endothelial cells read wall shear stress and respond to it, and regions of low or oscillating shear — the outer wall of a bifurcation, the inner curve of an arch — are exactly the regions where atherosclerotic plaque preferentially forms. That connection is the reason the number is taught in medicine rather than only in fluid mechanics.

Notice how weakly $\alpha$ depends on heart rate: it goes as $\sqrt{f}$, so doubling the rate raises it by only 41 %. Exercise does not move a vessel out of its regime. What changes $\alpha$ is the radius, which enters linearly, and that is why the interesting variation is along the arterial tree rather than across the day — the same blood and the same beat give a fifty-fold range of $\alpha$ from aorta to arteriole simply because the vessels get smaller.

Three honesty points sit under this number. Womersley's solution assumes a rigid, straight tube. Real arteries are elastic and tapered, and the elasticity is not a small correction — it is what allows a pressure pulse to propagate at all, and pulse-wave velocity, wave reflection and the augmentation of central pressure are entirely outside this model. It assumes a Newtonian fluid, and blood is not: below shear rates of about 100 s⁻¹ red cells aggregate and the apparent viscosity climbs steeply, which is good enough in the aorta and poor in venules and in low-flow states. And the radius, which enters linearly and therefore sets the answer, is a choice: an artery is neither circular nor of constant calibre, and a radius measured in diastole differs from one measured in systole by ten percent or more.

Where the number earns its keep beyond diagnosis is in scaling. Any bench model of an artery — a flow phantom for validating an ultrasound method, a mock circulation for testing a valve — has to match $\alpha$, not just the geometry, or the velocity profile in the rig is not the profile in the patient. That usually means choosing a glycerol-water mixture for its viscosity and a pump rate to suit, and the algebra on this page is how those two are picked.