Where the parabola goes

Steady flow in a round pipe has a parabolic velocity profile, a flow rate proportional to the fourth power of the radius, and no phase to speak of. Make the driving pressure oscillate and the profile changes: above a Womersley number of about ten the core moves as a plug, a quarter of a cycle behind the pressure, with the fastest fluid in a ring near the wall rather than on the axis.

The number that decides is

$$\alpha = R\sqrt{\frac{\omega}{\nu}},$$

which compares the pipe’s radius with the depth viscous information reaches in a cycle, or equivalently the unsteady term in the momentum equation with the viscous one. It is one when they are equal, and the textbook sentence is that below one the flow is quasi-steady.

Womersley’s solution is exact, so that sentence can be checked rather than believed.

What the solution says #

For a rigid tube driven by a harmonic pressure gradient, the velocity amplitude at radius $r$ is

$$A(r) = \frac{P}{i\omega\rho}\left[1 - \frac{J_0(kr)}{J_0(kR)}\right], \qquad k = \sqrt{\frac{\omega}{\nu}}\,e^{-i\pi/4}.$$

Everything follows from that: the profile, the flow rate, the phase, the wall shear. The Bessel function has a complex argument, and it is computed here by its own series and checked against the differential equation it satisfies rather than against a table — a solution with the wrong argument is smooth, periodic, satisfies no-slip and is entirely plausible, which is why the check has to be an identity.

Two limits must hold. As $\alpha \to 0$ the profile must become the parabola the instantaneous pressure gradient would produce, with no lag, and the flow amplitude must approach $\pi P R^4/8\rho\nu$. As $\alpha \to \infty$ the lag must approach a quarter cycle and the amplitude must fall as $1/\alpha^2$.

The two thresholds, and why the phase goes first #

Expanding the solution for small $\alpha$ gives the shape of both errors:

$$\frac{|Q|}{Q_{\text{Poiseuille}}} = 1 - \mathcal{O}(\alpha^4), \qquad \text{lag} = \mathcal{O}(\alpha^2).$$

The lag is second order in $\alpha$ and the amplitude deficit is fourth. So the phase error appears first, by a wide margin, and any threshold quoted on the amplitude alone is systematically too generous.

Numerically:

Two things fall out of that column pair, and they pull in opposite directions.

The amplitude threshold is very nearly one. A per cent of flow rate is lost at α = 0.911, which makes the Womersley number the best-behaved group in this collection: the value it is named for and the value at which its most obvious observable departs are within ten per cent of each other. That is the folklore working, and it deserves saying.

The phase threshold is not. One degree of lag arrives at α = 0.324 and one per cent of a right angle at 0.307 — a factor of three below. At the amplitude threshold the flow is already eight degrees late, and at α = 1 it is nine and a half.

The pattern is the general one for any linear system driven harmonically, and it is the same as a wing’s unsteady lift: the transfer function’s argument moves before its magnitude does, so a system that looks quasi-steady in amplitude is already substantially out of phase.

Why the phase matters more than the amplitude #

Because the flow rate and the pressure gradient are not in phase, the work done on the fluid over a cycle is not the product of their amplitudes. It is that product times the cosine of the lag, and the part that is out of phase is stored and returned rather than dissipated.

In a rigid tube that stored part is the fluid’s own inertia. In an elastic one it is the vessel wall, and the phase relationship between pressure and flow becomes the impedance — the quantity that decides how much of a pulse is transmitted and how much reflected, and therefore what the pressure waveform looks like anywhere downstream.

So a calculation that gets the amplitude right and the phase wrong gets the flow rate right and the energetics wrong, which is the opposite of the usual expectation about which half of a complex number matters.

Where real vessels sit #

The Womersley number was invented for blood flow, and the arterial tree spans the whole interesting range at a single heart rate.

The aorta is firmly in the plug-flow regime, with the profile flat in the core and all the shear at the wall. The small arteries are near one, which is where both errors are appreciable and neither limit applies. And the arterioles and capillaries are genuinely quasi-steady — below the 0.22 threshold, so the profile is a parabola and the lag is under a degree.

The middle row is the awkward one and it is also where most of the resistance of the circulation sits. A calculation of shear stress at the wall of a small artery — which is the quantity that decides where atherosclerotic plaque forms — is being done at a Womersley number where the profile is neither a parabola nor a plug.

What the profile is doing while the flow rate is fine #

It is worth looking at the profile itself over the range where the flow rate is still within a per cent, because “the profile is a parabola” and “the flow rate is the Poiseuille one” turn out to be different claims with different thresholds — and the second is much more forgiving than the first.

The flow rate is an integral over the section, and an integral is insensitive to the shape of what it integrates. A profile can be visibly flattened in the core and visibly steepened near the wall and still deliver very nearly the same flux, because the two changes are in opposite directions and the weighting $2\pi r\,dr$ is unkind to the core.

So at $\alpha = 0.9$, where the flow rate is one per cent short, the peak velocity has already fallen by rather more than that and the position of the maximum has begun to move off the axis. The integral hides the shape, which is why a threshold set on flow rate is generous and a threshold set on wall shear or on peak velocity is not.

The annular effect, which arrives long before the plug #

Richardson’s annular effect is the signature of the pulsatile regime: at high Womersley number the fastest fluid is in a ring near the wall rather than on the axis, because the thin layer there responds to the pressure gradient promptly while the core is still catching up.

It is quoted as a high-$\alpha$ phenomenon, and the peak’s departure from the axis is continuous. It begins as soon as there is any lag at all, which is to say from $\alpha \approx 0.3$, and by $\alpha = 3$ the peak sits at about three quarters of the radius.

That is one more observable with its own threshold, and it is the one an experimentalist sees first — a velocity profile measured by ultrasound or by particle tracking shows the off-axis peak long before the flow rate departs measurably from the quasi-steady value.

The wall stress, which is the quantity anybody wants #

Wall shear stress in steady Poiseuille flow is $4\mu Q/\pi R^3$ — proportional to the flow rate, in phase with it, and easy. In pulsatile flow it is neither proportional nor in phase.

At small $\alpha$ the stress leads the flow rate slightly and tracks its amplitude. At large $\alpha$ the stress is set by the thin Stokes layer at the wall rather than by the core, so it scales as $\alpha$ times the quasi-steady value and leads the flow by up to 45 degrees.

That is the same layer an oscillating wall produces, and its thickness $\sqrt{2\nu/\omega}$ is the length hiding inside the Womersley number: $\alpha$ is the pipe radius divided by that depth, times $\sqrt2$. Reading the number that way makes the whole regime picture obvious — small $\alpha$ means the layer fills the pipe, large $\alpha$ means it is a skin.

The threshold is not one, and the reason is ordinary #

Everything above fits the general pattern this collection keeps finding. The Womersley number is a term ratio: it compares $\rho\,\partial u/\partial t$ with $\mu\nabla^2 u$, and it is one where they are equal. The observable is smooth in it, so the threshold is the tolerance divided by a slope, and the slope is whatever the solution supplies.

What is unusual is how close the two numbers are for the amplitude — a factor of 1.1, where the Knudsen number’s is 594. That is entirely because the amplitude deficit is quartic in $\alpha$: a fourth power climbs so fast that a hundredfold change in tolerance moves the threshold by only a factor of three, and the threshold therefore sits wherever the coefficient puts it rather than wherever the tolerance does.

A steeply-varying observable makes a well-behaved threshold. That is the general rule this case supplies to the rest of the collection, and it explains the pattern: the groups whose folklore thresholds are worst — Knudsen, reduced frequency, Bond — all have observables linear in the group, where the tolerance is the whole story. The ones whose folklore is nearly right have steep ones.

And the phase spoils it, for the same reason in reverse. The lag is quadratic rather than quartic, so its threshold moves further per decade of tolerance and lands three times below. One group, one solution, two observables of different order, and two thresholds that cannot both be one.

A heartbeat is not a sine wave, and each harmonic has its own regime #

Everything above is one frequency, and no artery is driven at one frequency. A pressure pulse has a sharp systolic upstroke and a dicrotic notch in it, which is to say it has substantial content at several multiples of the heart rate — and the Womersley number of the $n$-th harmonic is not the fundamental’s.

$$\alpha_n = R\sqrt{\frac{n\omega}{\nu}} = \alpha_1\sqrt{n}.$$

The square root is the whole of the consequence. The problem is linear — a straight rigid tube has no convective term, so harmonics do not interact and the solution is the sum of Womersley’s solution evaluated at each one — but the regime is different for each term in the sum. In the aorta, where the fundamental sits at $\alpha_1 = 13$, the second harmonic is at 18, the fifth at 29 and the tenth at 41. Every one of them is deep in the plug-flow regime and each is deeper than the last.

Two things follow, and both are visible in any recording of arterial pressure and flow.

The flow waveform is smoother than the pressure waveform. At large $\alpha$ the admittance falls as $1/\alpha^2$, which is $1/n$ for a fixed vessel, so the tube is a first-order low-pass filter on the pressure it is given. The sharp features of a pressure pulse — the upstroke, the notch — are carried by the high harmonics and are attenuated in proportion to their harmonic number. A flow trace therefore looks like a smoothed version of the pressure trace, and it looks that way for a reason that has nothing to do with the measurement.

And the lag is nearly the same for all of them. Every harmonic above $\alpha \approx 5$ is approaching the quarter-cycle limit, so each is delayed by a quarter of its own period — which is $T/4n$ in absolute time. The harmonics do not all shift by the same interval, so the waveform does not merely arrive late; it changes shape, with the higher components moving forward relative to the fundamental. A pulse in a large artery is filtered and dispersed at once, and both effects come out of one square root.

The mean flow is the exception and it is exact. The zero-frequency component has $\alpha = 0$ by construction, so it obeys Poiseuille’s law with no deficit and no lag at all, however violent the pulsation on top of it. In a rigid tube the steady and oscillatory parts of the flow simply add, and the cardiac output is what the mean pressure gradient says it is — a statement that survives only because the equation is linear, and one that fails as soon as the vessel is elastic or the flow detaches at a branch.

That is also why Womersley’s papers are tables rather than a formula. What a physiologist needed was not the solution at one $\alpha$ but the ability to take a measured pressure waveform, decompose it, apply the right complex factor to each harmonic, and sum — which is a transfer function used as an instrument. The number that carries his name is the argument of that function, and the reason it is quoted with a threshold at one is that somebody summarised a filter by its corner.

The harmonic decomposition also explains a measurement artefact worth knowing about. A probe that resolves the flow waveform poorly loses the high harmonics first, which is exactly the content the tube has already attenuated — so a crude measurement of arterial flow looks more like the theory than a good one does, and agreement with Womersley’s prediction is not by itself evidence that the prediction was tested.

What the picture cannot show #

The tube is rigid. Real arteries are elastic, so a pressure pulse propagates as a wave rather than acting instantaneously along the length, and the local pressure gradient is not what a rigid-tube calculation assumes. The wave speed brings in a second dimensionless group entirely.

The flow is fully developed. The solution assumes the profile depends only on radius, which requires the tube to be long compared with an entrance length. In the arterial tree the branches come faster than that, so much of the circulation is permanently developing.

It is laminar and linear. Peak Reynolds numbers in the aorta exceed 4,000 during systole, and the flow there is transitional; nothing in a linear solution has any of that.

And blood is not a Newtonian fluid. Its viscosity depends on shear rate and on vessel diameter, falling in small vessels because red cells migrate to the axis. Every threshold here is computed with a constant viscosity, and the small-vessel rows of the table above are the ones where that assumption is worst.

Who found it, and when #

Womersley published the solution in 1955, working with the physiologist Donald McDonald on arterial pressure and flow measurements that had just become possible. The mathematics was not new — the same solution appears in Sexl’s work of 1930 and in Lambossy’s of 1952 — but Womersley’s papers connected it to measurement, and the number carries his name because his tables were the ones people used.

The threshold “quasi-steady below one” is nowhere in those papers. It appears later, as a summary, and it is a summary of the regime rather than of the accuracy — which is the same slippage that happens to every group in this collection: a statement about which terms matter is transmitted as a statement about which answers are right.

The surprising connection is with a wing. Womersley’s function and Theodorsen’s function are both transfer functions of a linear system driven harmonically, both are built from Bessel functions, both have a magnitude that falls and a phase that leads or lags, and both are quoted with a threshold at order one that belongs to their term balance rather than to their accuracy. The two problems share no physics at all — one is viscous and internal, the other inviscid and external — and they share their entire structure.

Where the ladder goes next #

Above this rung is the elastic tube, where the pressure pulse becomes a wave and the impedance becomes the object of study. That needs a wall model and a wave speed, and it is where the arterial tree stops being a set of pipes and becomes a transmission line.

Beside it sits the wing with the same structure, and the oscillating wall that supplies the length inside the number. Below it is the plug-flow regime, which is where the number was invented to be large, and which this essay approaches from the other end.