The layer every airway restarts bends Murray's cube towards the turbulent pipe's

The cheapest fork loses only its sideways speed priced the split in a turbulent network as Borda would have — an abrupt change of velocity, undone by mixing, at a cost of the kinetic energy of the difference — and found that at the cheapest fork the loss is only the sideways part of the parent’s motion. That essay ended on the case its pricing did not cover. In a laminar network at Reynolds numbers of a few hundred, a small artery or an airway past the first few generations, the flow does not mix at the split. It turns smoothly into each daughter, and the energy it loses there is lost by viscosity, in a different place and at a different rate.

The place is the daughter itself. A daughter airway begins at the flow divider, the sharp ridge where the parent’s flow is cut in two, and the fluid arriving on its inner wall has just been sheared against that ridge. The profile has to be rebuilt from there, with a fresh boundary layer growing from the divider like the layer on a flat plate in a stream. An airway is only about three diameters long; the layer never reaches the axis before the next divider cuts it again. So the daughter carries a developing flow all its length, with a thinner layer and a steeper wall gradient than Poiseuille’s parabola, and it dissipates more.

Pedley’s factor, and the radius that cancels from it #

Pedley, Schroter and Sudlow measured the dissipation in model bifurcations in 1970, with the profiles inside them, and wrote the result as a multiple of Poiseuille’s,

$$Z = \frac{C}{4\sqrt 2}\left(\mathrm{Re}\,\frac{d}{L}\right)^{1/2}, \qquad C = 1.85,$$

whenever that exceeds one, with $d$ and $L$ the daughter’s diameter and length. The form comes from boundary-layer growth: the layer’s thickness after a distance $L$ goes as $\sqrt{\nu L/U}$, the wall gradient as $U$ over that, and the excess dissipation over the developed flow’s as the ratio of the airway’s radius to the layer’s thickness, which is $\sqrt{\mathrm{Re}\,d/L}$ up to a constant. The constant $C$ is measured. The factor is therefore a model with one fitted number in it, and it describes inspiration through a tree of reasonably smooth Y-shaped junctions; expiration, where two streams merge, has its own coefficient.

Write the Reynolds number in terms of the flow, $\mathrm{Re} = 4\rho Q/(\pi\mu d)$. Then $\mathrm{Re}\,d = 4\rho Q/(\pi\mu)$ and the diameter cancels:

$$Z = \frac{C}{4\sqrt 2}\left(\frac{4\rho Q}{\pi\mu L}\right)^{1/2}.$$

The excess dissipation of a restarted layer depends on how much flows and how far, and not on the width of the pipe it flows in. A wider airway carrying the same flow has a smaller Reynolds number, but its diameter is larger in the same proportion, and the ratio of its radius to the layer’s thickness at the far end comes out unchanged. That cancellation is the whole of what follows.

The restart, seen from inside a daughter #

The profile a daughter receives is not a flat one with a layer to grow on it. The parent’s fastest fluid is on its axis, and the divider cuts the axis; so each daughter receives its fastest fluid against its inner wall, the one beside the divider, and its slowest against the outer wall. The measured profiles in model bifurcations show exactly that: a velocity peak skewed hard towards the divider wall at the entry, a thin layer under it, and a pair of counter-rotating secondary vortices carried round the curve of the daughter as in a bent pipe. The inner wall’s shear just past the divider is several times Poiseuille’s, and the fourth power of the diameter behind which Poiseuille’s law hides everything has nothing to do with it.

The angle a junction chooses fixed the geometry of a cheapest fork from a balance of tensions; it assumed, as Murray did, that each segment carries developed flow from end to end. Pedley’s factor is the price of that assumption failing. It is not a junction loss in Borda’s sense — nothing is destroyed in an instant of mixing at the split — but a loss distributed along the whole daughter, at a rate that grows with the flow faster than Poiseuille’s and slower than Borda’s. The power a daughter dissipates goes as $Q^2$ by Poiseuille, as $Q^3$ by Borda’s mixing, and as $Q^{5/2}$ by Pedley: the laminar restart sits halfway between the developed laminar pipe and the turbulent junction, and that is where the exponent of the cheapest tree ends up too.

Where the layer finally fits #

The factor falls to one when $\mathrm{Re}\,d/L = (4\sqrt2/C)^2 = 9.35$. The classical entrance length of a round pipe, the distance a laminar profile needs to come within a per cent of Poiseuille’s, is $0.0575\,\mathrm{Re}\,d$ — the length a duct needs to forget what was fed into it — which is $\mathrm{Re}\,d/L = 17.4$ for a pipe exactly one entrance length long. So Pedley’s excess vanishes when an airway is a little under twice its own entrance length, which is the consistency a reader should want: the dissipation’s excess is concentrated near the start, where the layer is thinnest, and is paid off well before the profile is fully developed by the strict one-per-cent standard.

Measured from the other end, the deepest generation whose layer does not fit moves with the flow. At a quarter of a litre a second, quiet breathing, the factor is above one down to generation 8; at one litre a second, to generation 11; at five, the flow of hard exercise, to generation 14, nearly the end of the conducting tree. The developing-flow region is not a fixed part of the lung but a front that moves down the tree as the breathing deepens, and the airways it covers are the ones that set the cost.

What it does to a lung #

Weibel’s model A, from 1963, is the symmetric tree the respiratory literature still uses for arithmetic: each generation doubles the number of airways, and the diameter and length of each generation are measured averages from casts. At a total flow of a litre a second — a brisk walk — generation 1 carries half a litre a second in each main bronchus at a Reynolds number of 3500, generation 5 a thirty-second of it at 760, and generation 10 at 64. The restarted layer is thinnest where the flow per airway is largest, so the factor is largest at the top: 9.8, 5.2 and 1.4 in those three. It falls below one, where the layer finally has room to develop, at the twelfth generation.

By Poiseuille, the pressure the conducting airways cost is spread over the middle generations, where the airways are narrow but still few, and peaks near the sixth. Priced, it moves upward, to the generations with the most flow per airway: generations 1 to 5 carry two-thirds of it at every flow from a quarter of a litre to five. The narrow airways deep in the lung, which the fourth power of the diameter in Poiseuille’s law makes look expensive, are cheap because there are so many of them in parallel and their layers do develop; the expensive ones are the few large ones whose layers do not.

The resistance this gives is not a number the airways have. Poiseuille’s resistance of generations 1 to 16 is 0.159 cmH₂O per litre a second at any flow; priced, it is 0.348 at 0.25 L/s, 0.657 at 1 and 1.45 at 5, rising as the square root of the flow because the factor does.

That is the shape physiologists measure. The pressure needed to drive air through the lungs grows faster than the flow — Rohrer wrote it in 1915 as a linear term plus a quadratic one — and the classical reading attributes the excess to turbulence in the large airways. The trachea and the main bronchi at a few litres a second are turbulent, and Pedley’s factor is not meant for them; generation 1 at 5 L/s has a Reynolds number of 17,000. But most of the excess sits in generations 2 to 8, at Reynolds numbers of a few hundred to a few thousand, where it is laminar and comes from the restarted layers. A flow-dependent resistance does not need turbulence. It needs only a tube too short for its layer.

The quiet zone, explained by the loud one #

Clinicians have a name for the consequence. Airways narrower than about two millimetres — generation 8 and below in Weibel’s tree — are called the lung’s quiet zone, after Mead’s observation in 1970 that disease can narrow them substantially before the measured airway resistance notices. By Poiseuille that is puzzling: generations 8 to 16 carry 45 per cent of the tree’s Poiseuille resistance at any flow, and narrowing them all by a fifth in diameter multiplies their share by 2.44 and the total by about five-thirds. With the restarted layers priced, generations 8 to 16 carry 18 per cent of the resistance at half a litre a second and 17 per cent at one, and the same narrowing raises the total by about a quarter. The deep airways are not quiet because they are cheap; they are quiet because the shallow ones are loud, and the shallow ones are loud because their layers never fit. A measurement of total resistance is a measurement of the first few generations, and a disease of the small airways has to be looked for with a test that isolates them — the uneven emptying of a single-breath washout, for instance — rather than with a pressure and a flow at the mouth.

The cheapest airway, priced for its layer #

The argument this essay extends began with the radius that costs least: a vessel’s pumping power, $8\mu L Q^2/\pi r^4$ by Poiseuille, plus an upkeep proportional to its volume, $b\pi r^2 L$. The sum is least where $r^6 \propto Q^2$, which is Murray’s cube law, $Q \propto r^3$, and with it the rule that the cubes of the radii at a junction add. The turbulent pipe replaced Poiseuille’s dissipation with Blasius’s friction and found $Q \propto r^{27/11}$, an exponent of 2.45.

Multiply the pumping term by $Z$. Since $Z$ does not depend on $r$, the minimisation over $r$ is the same as Murray’s with a larger coefficient, $r^6 \propto Z\,Q^2$. But $Z$ grows with the flow, as $Q^{1/2}$ at a fixed length, so

$$r^6 \propto Q^{5/2} \quad\Longrightarrow\quad Q \propto r^{12/5}.$$

If the segment’s length is not fixed but proportional to its diameter — as in a self-similar tree, and Weibel’s airways are about three diameters long throughout — then $Z \propto (Q/r)^{1/2}$, the minimisation picks up the extra half-power of $r$, and $Q \propto r^{13/5}$. The two cases bracket the exponent between 2.4 and 2.6, and the turbulent pipe’s 2.45 is inside that band.

That is the surprising part. A turbulent pipe and a laminar pipe too short for its layer are different flows with different mechanisms — eddies carrying momentum to the wall in one, a thin laminar layer in the other — and both concentrate the velocity gradient in a thin region at the wall whose thickness falls as the flow rises. The cheapest radius only sees how the dissipation scales with the flow, and both give a dissipation rising faster than Poiseuille’s linear law. The exponent of an optimal network reports how its walls’ layers scale, not whether the flow is turbulent.

What Weibel’s tree actually does #

For a symmetric fork the exponent gives the ratio of a daughter’s diameter to its parent’s, $2^{-1/n}$: 0.794 for Murray’s cube, 0.766 for the restarted layer at fixed shape, 0.749 at fixed length. Weibel’s measured diameters narrow at an average of 0.707 a generation from the trachea to generation 4, and at 0.845 from generation 4 to 16.

So the pricing moves the prediction the right way at the top of the tree and not far enough. The upper airways narrow faster than Murray’s cube, as a network priced for its restarted layers should, and further still. Something else is being bought there: the trachea is wide for reasons that have nothing to do with steady flow — coughing, the swallowing apparatus, the larynx above it — and the first branchings are asymmetric in a real lung, which Weibel’s symmetric averaging hides. Below generation 4 the measured airways are wider than any cost rule asks, and the usual explanation is the opposite of an economy: wider small airways mean a lung whose resistance survives narrowing by disease or constriction, a safety margin in diameter rather than an optimum. The same deep generations are where gas moves by oscillation and diffusion as much as by bulk flow, and a cost made of pumping power alone does not describe them. Arteries, priced with a pulse rather than a steady flow, depart from the cube law for a different reason again: Murray’s law is not the rule for a pulse, because a branching tube that must pass a wave without reflecting it has its own preferred ratio, and it is not the cheapest one.

The convention behind the numbers #

Every quantity here is for inspiration through Weibel’s symmetric model A at steady flow, with Pedley’s coefficient $C = 1.85$ for inspiratory Y-junctions, generations 1 to 16 — the trachea and the respiratory zone beyond generation 16 are both excluded, the first because it is turbulent and its loss is not Pedley’s, the second because its walls carry alveoli and its flow is not a tube flow. Resistances are in cmH₂O per litre a second, the respiratory unit; pressures in pascals. The upkeep $b$ in the cheapest-segment calculation sets the absolute radius and not the exponent, which is all that is compared with the tree.

What the calculation does not price #

Breathing is not steady: at fifteen breaths a minute the Womersley number of the trachea is about three, and the flow reverses twice a cycle, so the layers restart in time as well as space; Pedley’s factor is a steady measurement applied to a quasi-steady flow. The factor’s coefficient was measured in rigid glass models with smooth dividers, and real carinae are rounded, real airways are compliant and lined with mucus. Expiration merges two streams into one and has its own coefficient, which makes the lung’s resistance different in and out. The junction’s own turning loss — the secondary flow in the curved daughter, Dean’s vortices — is inside Pedley’s measured $C$ and not separated from the layer’s development. And Weibel’s tree is symmetric; a real lung’s branching is asymmetric from the first fork, which Horsfield’s models describe, and an asymmetric tree’s generations are not a single number each.

The optimisation is an argument about exponents, and it needs the dissipation’s scaling only. It does not claim that a lung is built to minimise anything; it says what a lung built that way would look like, and the comparison with the casts says where it is not built that way.

Who measured the restart #

Pedley, Schroter and Sudlow published the dissipation measurements and the factor in 1970, from steady flow through rigid models of the bronchial tree, and gave the boundary-layer argument that fixes its form. Weibel’s model came from his 1963 Morphometry of the Human Lung, built from measurements on resin casts; Rohrer’s two-term law for airway resistance is from 1915. Murray derived his law in 1926 for blood vessels, and its application to the airways — and the observation that the upper ones do not obey it — has been argued over ever since, with the size of the trachea, the asymmetry of the branching and the diffusion zone each offered as the reason.

How it was checked #

The factor computed at one flow and length for diameters of 2 and 20 millimetres agrees to rounding, which is the cancellation done numerically rather than by algebra. Each generation’s Poiseuille drop matches $128\mu L Q/\pi d^4$ for its share of the flow to rounding. The cheapest segment, found by golden-section search on the full bill, gives local exponents 3.000000, 2.400000 and 2.600000 — Murray’s, twelve-fifths and thirteen-fifths, each to seven figures — and returns to the cube at flows small enough for the layer to develop. The tree’s resistance at 0.25, 1 and 5 L/s is 2.20, 4.14 and 9.16 times Poiseuille’s, increasing with the flow, and the model refuses a tree with no flow, an airway of negative diameter and a segment with no upkeep.

Still open: the layer that restarts in time as well #

A breath is not steady, and the restart this essay priced in space has a counterpart in time: every reversal of the flow restarts the layer too. The next calculation would solve the developing laminar layer in a daughter airway under an oscillating flow at the Womersley numbers of the upper airways — about three in the trachea and a half in the fifth generation — and ask whether the dissipation over a cycle is the steady factor averaged over the breath, or more, because a layer that has to restart at every reversal is thinner for longer, and so whether the exponent of a tree priced for breathing rather than for steady flow sits lower still in the band between 2.4 and 2.6.