Pulse in an artery

How blood moves across an artery during one heartbeat: the velocity profile, the shear on the wall and the moments it turns back. Womersley's exact solution, for the aorta, a smaller artery or any vessel you enter.

FREE · NO SIGN-UP v1.0.0 · checked against 10 published values

INPUTexact solution

1 · Flow over the beat

2 · Vessel and blood

Vessel diameter [mm] Heart rate [/min]

Peak mean velocity [m/s] Pulse amplitude [m/s]

Blood viscosity [mPa·s]

Defaults: the ascending aorta De Tullio et al. (2009) simulated: 27 mm, 69.3 beats a minute (866 ms), 0.81 m/s at peak. Try a smaller artery or a faster heart and watch the profile change.

3 · Playback

VELOCITY ACROSS THE VESSEL

Blue: the profile now. Dashed: Poiseuille's parabola for the same flow, as if the flow were steady. Grey: the profile at eight moments of the beat.

OVER ONE BEATflow · wall shear stress

WOMERSLEY NUMBER20.0α = R·√(ω/ν)

CENTRE / MEAN AT PEAK1.21 ×Poiseuille: 2

PEAK WALL SHEAR6.40 Pa0.83 Pa if steady

OSCILLATORY SHEAR INDEX0.4370 one way · 0.5 back and forth

What this beat shows

Womersley number 20.0: inertia rules the core. The profile is flat across most of the vessel, and viscosity acts only in a layer about 0.95 mm thick by the wall.
At peak flow the centreline runs at 1.21× the mean velocity; Poiseuille flow would give 2.
Peak wall shear stress 6.40 Pa, against 0.83 Pa if the flow were taken as steady at every instant: a difference of +674 %.
The wall shear stress leads the flow: by 42° at the heart rate's frequency (0° for steady flow, approaching 45° at large Womersley numbers).
Time-averaged wall shear stress 0.15 Pa; oscillatory shear index 0.437 (0 = always one way, 0.5 = as much back as forth).
For 70 % of the beat the blood by the wall flows backwards while the core still flows forward.

CHECKED AGAINST PUBLISHED VALUES · RUN IN OUR TEST SUITE
What	Published	This calculator
Waite & Fine §7.7, harmonic 1: 1 − 2J₁/(ΛJ₀)	0.5783 · 0.3292	0.5783 · 0.3292	agrees
Waite & Fine §7.7, harmonic 2: 1 − 2J₁/(ΛJ₀)	0.7088 · 0.2467	0.7088 · 0.2467	agrees
De Tullio et al. 2009: mean flow of the aortic wave	5 L/min	5.03 L/min	agrees
Waite & Fine §7.7: steady centreline velocity	0.35525 m/s	0.35536 m/s	agrees
Womersley 1955, dog femoral artery: M′10 at α = 4.72	0.7436	0.7437	agrees
Womersley 1955, dog femoral artery: M′10 at α = 5.78	0.7839	0.7839	agrees
Womersley 1955, dog femoral artery: M′10 at α = 6.67	0.8096	0.8096	agrees
Womersley 1955, dog femoral artery: M′10 at α = 7.46	0.8278	0.8278	agrees
Womersley 1955, dog femoral artery: M′10 at α = 8.17	0.8416	0.8414	agrees
Womersley 1955, dog femoral artery: M′10 at α = 3.34	0.6651	0.6554	see note

Womersley's α = 3.34 entry is printed as 0.6651; the same formula, and scipy, give 0.6554, while the other five entries agree to 0.0002. We take it for a misprint.
Small α: the parabola of steady flow, with the wall shear in step with the flow. Large α: a flat core, and the wall shear 45° ahead of the flow. Both limits are in the test suite.
For learning and comparison only. Not for diagnosis or treatment.
Elastic walls, branches and the developing flow behind a valve need a full simulation, the kind we build Tessera for. See Tessera →

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HOW IT'S CALCULATED · WITH YOUR NUMBERS

Step by step, with your numbers

01
Womersley numberα = R·√(ω/ν) = 13.50 mm × √(2π / 0.866 s / 3.302 mm²/s) = 20.01The ratio of the vessel's radius to the depth the wall's drag reaches in one beat.
02
Viscous layer by the wallδ = √(2ν/ω) = 0.954 mmStokes layer of the fundamental harmonic; the higher harmonics have thinner ones.
03
Mean flow (Poiseuille)ū₀ = 0.1450 m/s · Q₀ = 4.981 L/min · u₀(r) = 2ū₀(1 − r²/R²)The steady part of the beat.
04
Each harmonic (Womersley 1955)u_n(r) = Re{ C_n [1 − J₀(Λ_n r/R) / J₀(Λ_n)] e^(inωt) }, Λ_n = α√n · i^(3/2)15 harmonics; for the first, α₁ = 20.01 and F = 1 − 2J₁(Λ)/(ΛJ₀(Λ)) = 0.9318∠4.19°.
05
Wall shear stressτ_w = 4μū₀/R − Re Σ μ C_n (Λ_n/R) J₁(Λ_n)/J₀(Λ_n) e^(inωt)Peak 6.40 Pa; quasi-steady 4μū/R would give 0.83 Pa.
06
Oscillatory shear indexOSI = ½·(1 − |∫τ dt| / ∫|τ| dt) = 0.437Ku et al. (1985): 0 when the wall shear never reverses, 0.5 when it spends as long backwards as forwards.
07
Reynolds numberRe = ρ ū d / μ: 6 526 at peak 1 186 meanThe solution assumes laminar flow; it does not say where a real beat turns turbulent.

References: Womersley, J. Physiol. 127 (1955); Waite & Fine, Applied Biofluid Mechanics (McGraw-Hill 2007), ch. 1 and 7; De Tullio et al., J. Fluid Mech. 622 (2009); Ku et al., Arteriosclerosis 5 (1985).

ASSUMPTIONS & LIMITS

A straight, rigid, circular tube, far from branches and bends, with the flow fully developed: no entrance length, no wave travelling along an elastic wall.
Blood as a Newtonian fluid, 1060 kg/m³; laminar flow.
The flow wave is the ascending aorta's (De Tullio et al. 2009, read off their Fig. 3b) in 15 harmonics, stretched to the heart rate and scaled to the peak velocity. Other arteries have waves of their own: the carotid keeps flowing forward in diastole, the femoral turns back.
Just past the valve the real flow is far from fully developed: the layer by the wall is still growing from the annulus, which this solution does not see. Read the wall shear as the fully-developed limit, for comparison.
In a rigid tube the motion set up in systole outlives the beat (it decays over R²/ν, many seconds in the aorta), so in diastole the core still drifts forward and the blood by the wall back, with no net flow. An elastic, branching aorta behaves differently.
For learning and comparison only. Not for diagnosis or treatment.

DISCLAIMER · For education and preliminary sizing only. Results must be reviewed by a qualified engineer before use in design.

BEYOND A RIGID TUBE

Elastic arteries, branches, stents and the flow behind a valve: we model them with CFD and fluid–structure interaction.

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