Too fast for a profile

Poiseuille’s law is the most-used result in this subject and it has a hypothesis that is almost never stated: the flow has had time to become the profile the pressure gradient asks for.

Make the pressure oscillate and that hypothesis becomes a question about two times — the period of the oscillation, and how long viscosity needs to reach the middle of the tube. Their ratio is one number, and it decides everything.

One number, and what is in it #

Viscosity diffuses momentum, so in a time $t$ it reaches a depth of order $\sqrt{\nu t}$ — the same depth an oscillating wall drags fluid to. Over one cycle of an oscillation at frequency $\omega$ that depth is $\sqrt{\nu/\omega}$, and comparing it with the tube’s radius gives

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

which is the Womersley number. It is a ratio of lengths written as a number, and its square is the more physical statement: $\alpha^2$ is the ratio of the unsteady inertia to the viscous force.

Below one, viscosity crosses the tube many times per cycle, the profile is the parabola the instantaneous gradient calls for, and the flow is in step with the pressure. Above ten it does not get anywhere near the axis, and the core simply obeys Newton’s law with no viscosity in it at all.

The solution, and the function in it #

The velocity is

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

with a Bessel function of a complex argument, which is where the phase comes from and is the one place this calculation could quietly be wrong. The series is checked here against the equation it is supposed to satisfy — $J_0'' + J_0'/z + J_0 = 0$, differenced in the complex plane — rather than against a table, and the whole solution is checked against Poiseuille’s in the limit α → 0, where it must reproduce the parabola at every radius and the flow rate $\pi P R^4/8\mu$ exactly.

That limit is the assertion that matters. A solution with the wrong argument in the Bessel function is smooth, periodic, satisfies no-slip, and looks entirely plausible; it does not reduce to Poiseuille’s law when the frequency goes to zero.

Richardson’s ring #

The off-axis maximum is the strangest thing here and it was measured before it was explained. Richardson and Tyler reported in 1929 that the velocity amplitude in an oscillating pipe peaked near the wall — the annular effect — and it stayed a curiosity until Womersley solved this problem in 1955 for the arterial circulation.

The mechanism is a reversal of the usual intuition about viscosity. In a steady flow the fluid near the wall is the slowest, because viscosity holds it back. In a rapidly oscillating one the fluid near the wall is the quickest to respond, because it is the only fluid connected to anything: the core is being pushed by pressure alone and its response is limited by its own inertia, while the annulus within a diffusion depth of the wall feels the wall’s constraint and reaches its equilibrium within the cycle.

Neither statement is about the average speed, which is why the effect is invisible in any measure that averages over a cycle.

A resistor, then an inductor #

The electrical analogy is exact rather than decorative, and it is the one the cardiovascular literature runs on. A tube’s relation between pressure gradient and flow is a complex impedance: its magnitude falls with frequency and its phase runs from zero to ninety degrees. At low α the impedance is a resistance — energy is dissipated and the flow is in step. At high α it is an inductance — energy is stored and returned, and the flow lags.

The aorta at rest sits at α ≈ 18, so arterial flow is inertia-dominated: the heart is pushing a pressure wave into an inductive load, and what arrives at the far end has almost nothing to do with Poiseuille’s law.

What the wall feels, which is not what the flow does #

That difference of one power is the practically important result in this essay, and it explains why wall shear stress is the quantity measured in arteries rather than flow.

The reason is the diffusion depth again. The shear is set by the velocity divided by the layer thickness $\sqrt{\nu/\omega}$; the flow is set by the velocity times the tube’s area. Raising the frequency thins the layer and lowers the velocity, and the two effects partly cancel in the shear and reinforce in the flow.

So a tube in a pulsatile regime carries little flow and plenty of shear, and the two cannot be estimated from one another with a steady formula. Endothelial cells respond to wall shear stress, atherosclerotic plaques form preferentially where it is low or oscillating, and every one of those measurements lives in exactly this solution.

The same circulation contains both regimes #

The list of cases carries a fact worth stating on its own. The human circulation spans the whole range of this number. The aorta is at 18 and inertia-dominated; a femoral artery at 5.9 and firmly in the transition; an arteriole at 0.044 and quasi-steady, so Poiseuille’s law applies there to within a fraction of a per cent.

That is a consequence of the definition and of nothing else: α goes as the radius, and the radius falls by three orders of magnitude between the aorta and an arteriole. The pulsatility of the flow is therefore progressively erased on the way down the tree, which is what the compliant large vessels are for, and by the capillaries there is a steady flow with a steady profile — which is the condition Taylor dispersion needs and one reason exchange happens there rather than higher up.

Reading the number backwards: what it says about a design #

The Womersley number is one of the few dimensionless groups in this subject that a designer can move in four different ways, and each of them has a different cost. It is worth laying them out, because the group’s structure decides which lever is worth pulling.

Radius. α goes as R, so halving a tube’s radius halves it — and quarters the steady flow it can carry, by Poiseuille’s fourth power. This is the expensive lever and it is the one the circulation uses, because it has no choice.

Frequency. α goes as √ω, so a fourfold change in frequency is only a doubling. A hydraulic line at 50 Hz sits at α = 13 and the same line at 200 Hz at 26; the regime does not change, and the flow amplitude falls by four.

Viscosity. α goes as 1/√ν, so a fluid ten times thicker lowers it by a factor of three. This is why a pulsating flow of oil in the same pipe as water is much closer to quasi-steady, and why hydraulic systems designed on steady formulae work better than they should.

Nothing else. There is no density in α and no pressure amplitude: the shape of the flow is independent of how hard it is driven, which is a consequence of the problem’s linearity and is checkable — the profiles above are drawn at unit pressure amplitude and scaling it changes every velocity by the same factor and nothing else.

Where the energy goes when nothing is transported #

There is a last consequence worth drawing out, because it is the reason an oscillating flow is expensive.

At large α the flow rate is small and the shear is not, so the ratio of dissipation to delivered volume is high: the fluid is being sheared vigorously in the wall layer while the net transport per cycle is nearly nothing. A tube driven at high α is mostly a heater.

The arithmetic is the two slopes already drawn. Dissipation scales with the shear times the velocity, so it falls as 1/α³ against a flow falling as 1/α² — meaning the energy cost per unit of fluid delivered rises as α. Every pulsatile pumping system pays that, and it is why pulsation is used where the pressure wave itself is the point (the arterial pulse, a hydraulic accumulator) rather than where fluid needs moving.

The same length, in four places on this site #

The quantity doing all the work here is $\delta = \sqrt{\nu/\omega}$, and it is worth collecting where else it has appeared, because the collection is an argument that it is the right length rather than a convenient one.

It is the depth to which an oscillating wall drags fluid, where the amplitude falls by $e$ every δ and the phase lags by one radian every δ. It is the thickness of the acoustic boundary layer that damps a sound wave at a solid surface — 0.07 mm in air at 440 Hz, which is why an organ pipe’s losses are a surface effect. It is the layer in which a vortex spreads by diffusion, with the clock in place of the frequency. And it is the layer here, into which all of a pulsatile tube’s shear is squeezed.

In every case the same statement holds: viscosity is a diffusion, and a diffusion given a time gives a length. What differs between the four is only what supplies the time — a wall’s period, a sound wave’s, a heart’s, or the age of the flow.

What a measurement in a pulsatile flow actually gets #

There is a practical consequence worth stating for anybody measuring a flow that pulses, and it follows from the phase rather than from the amplitude.

A flow meter that averages over many cycles reports the mean flow correctly whatever the Womersley number is, because the oscillating part integrates to zero. A meter that samples — a Doppler probe taking an instantaneous velocity at one radius, say — does not, and the error it makes depends on where in the section it looked and when in the cycle.

At small α that error is manageable, because the profile is parabolic and the centreline speed is twice the mean at every instant. At α = 13 there is no such relation: the profile is a plug whose edge structure changes through the cycle, the centreline speed is a poor proxy for the mean at some phases and a good one at others, and the ratio between them is a function of the phase rather than a constant.

So the quantity a clinical measurement can trust is the cycle-averaged flow, and the quantity it wants is often the instantaneous wall shear — which are at opposite ends of exactly the difficulty this essay describes.

What the model does not contain #

Rigid walls. The tube here does not move. Real arteries are compliant, the pressure travels along them as a wave at a few metres a second, and Womersley’s own papers treat the elastic case — which adds a wave speed and a reflection coefficient and changes the impedance completely. Nothing here has either.

One frequency. The solution is linear, so a real waveform is handled by summing harmonics, and the arterial pulse needs about six of them. Each harmonic has its own α — the fourth harmonic of the aorta is at α = 36 — so the profile is a superposition of regimes rather than in one.

Laminar throughout. The peak Reynolds number in the aorta is several thousand, which would be turbulent in a steady pipe; what saves it is that the flow accelerates and decelerates too quickly for transition to complete, so the flow is conditionally laminar and this solution is a better approximation than the Reynolds number alone suggests. Nothing here computes that, and it is the weakest assumption in the essay.

No entrance. The profile is fully developed, and a real vessel is a few diameters from a branch — where the entrance length for a pulsatile flow is itself a function of α.

No secondary flow, no curvature, no taper, all of which arteries have.

Which Reynolds number an oscillating flow is actually judged by #

The list above names the laminar assumption as the weakest thing in the essay, and it is worth resolving rather than leaving as a worry, because the resolution is the same argument the whole page is built on.

A peak Reynolds number of several thousand formed on the aorta’s diameter would be firmly turbulent in a steady pipe. But a Reynolds number is a statement about a disturbance growing in a shear layer, and the shear layer here is not the tube — it is the thin annulus of thickness $\delta = \sqrt{\nu/\omega}$ that the whole essay is about. The core is a plug with no shear in it, and a plug has nothing for a disturbance to feed on.

So the number that governs stability is formed on the layer:

$$\mathrm{Re}_\delta = \frac{U\delta}{\nu},$$

and measurements of oscillating boundary layers put transition somewhere near 500. For the aorta, with a peak velocity around a metre a second and a Stokes layer of about two-thirds of a millimetre, $\mathrm{Re}_\delta$ is close to 200 — below the threshold, on the correct length, while the diameter-based figure was above it by a factor of ten on the wrong one.

The length in a Reynolds number has to be the length the disturbance lives in, which is this collection’s standing complaint about the group and is here worth an order of magnitude in the answer.

There is a second stabilising mechanism and it is specific to a flow that oscillates. Transition takes time — a disturbance has to grow through several e-foldings before it is turbulence — and a pulsatile flow does not hold still long enough. Worse for the disturbance, the two halves of the cycle are not equivalent. During acceleration the profile is being filled from the wall outwards and is becoming fuller, which is a favourable pressure gradient and is strongly stabilising, for the same reason an accelerating boundary layer has no inflection point to work with. During deceleration the gradient reverses, the profile develops an inflection, and disturbances grow rapidly.

What is observed in a large artery matches that exactly: the flow is quiet through systolic acceleration, becomes disturbed in late systole as the flow decelerates, and relaminarises during diastole because the disturbances have nothing to sustain them. It is neither laminar nor turbulent in the steady sense; it is conditionally laminar, spending part of each cycle in a state that would not survive if it lasted.

Which is why the solution in this essay works better than any steady criterion would predict, and why it is quoted with a caveat rather than a correction. A linear solution is a good description of a flow that never has time to become nonlinear, and how much time it has is set by the same frequency that sets $\alpha$ — so the number governing the profile and the number governing whether the profile survives are built from the same two quantities, arranged differently.

Who found it, and when #

John Womersley was a mathematician who spent 1954 and 1955 at Bristol working with the physiologist Donald McDonald on the arterial pulse, and the solution above is from that collaboration. What makes it a landmark is not the mathematics — the equation is a Bessel equation and had been solved before — but the recognition that the profile in an artery is not Poiseuille’s, and that the number deciding it is a property of the vessel and the heart rate together.

Richardson and Tyler’s measurement of the annular effect came twenty-six years earlier and was the kind of result that sits unexplained in the literature until someone solves the right problem. Sexl had in fact published the same solution in 1930, in a different context and in German, and it is a fair example of how a result can exist and not be available.

Where the ladder goes next #

Three essays in this field have each been about a number deciding a regime — a length for surface tension, a Péclet number for dispersion, a Womersley number for pulsation. The next question is what happens when the flow that a number describes is turbulent and the turbulence itself is confined to a plane, where the cascade that carries energy to small scales runs backwards.