Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Branching — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two bills, and the radius that settles them. The cost of a vessel against its radius: the pumping power, which falls as the inverse fourth power, and the price of owning the fluid and the wall, which rises as the square. Their sum has a minimum, found here by golden-section search and agreeing with the closed form to eight figures. At that radius the pumping bill is exactly a third of the total — for every set of constants, because it follows from the two exponents alone.

    The radius that costs least

    A vessel that carries a flow costs two things to own — the power to push fluid along it and the price of the tissue itself. Minimising the sum gives a best radius, the best radius makes flow proportional to radius cubed, and the rule that follows is a statement about a photograph that came out of a cost function.

    part 1 · applied
  2. The junction the minimisation is over. A parent vessel entering from the left and two daughters leaving to fixed points. The radii are settled by Murray's law; what is left free is where the branch point sits, and the cost of the junction depends on it. The point drawn is the one the minimisation finds.

    The angle a junction chooses

    Murray's law fixes the radii at a branching vessel and is where every account of it stops. The same minimisation fixes the angles completely — 74.93 degrees for a symmetric bifurcation, a right angle for a vanishing side branch — and it does so as a triangle of forces, with tensions proportional to the squares of the radii.

    part 2 · applied
  3. Past transition the cube law gives way to two and a half. The exponent k in Q ∝ r^k obeyed by the cheapest vessel, against that vessel's Reynolds number, for a smooth wall and for a wall with a millimetre of roughness. While the flow is laminar k is 3, Murray's law. Past transition it drops to 2.45 at Re = 2·10⁴ and 2.4 at 1.01·10⁷, near Blasius's 27/11 = 2.455. A fixed absolute roughness barely moves it, because the cheapest pipes for large flows are large and a millimetre is a small fraction of them.

    The cheapest turbulent pipe follows two and a half, not three

    Murray's cube law comes from Poiseuille's resistance, and it fails the moment the flow in a vessel turns turbulent. The same cost minimised with turbulent friction gives a flow that grows as the radius to the power 27/11 in a smooth pipe and 7/3 in a rough one. Murray's invariant, a wall shear the same in every vessel, goes with it, and so does the local rule that made the law credible as biology: a vessel that holds its shear at a set point is twice too wide.

    part 3 · applied
  4. At the cheapest fork each daughter runs at the parent's velocity along it, so the split loses only the sideways part. The power lost at a symmetric split, as a multiple of the flow times the parent's dynamic pressure, against the daughters' radius as a fraction of the parent's, at each of three cheapest angles. For laminar, k = 3 the angle is 37.5°, the curve's minimum is at a ratio of 0.7937 — the cheapest ratio — and its value is sin²θ = 0.37; for smooth turbulent, 27/11 the angle is 28.4°, the curve's minimum is at a ratio of 0.754 — the cheapest ratio — and its value is sin²θ = 0.226; for rough, 7/3 the angle is 25.1°, the curve's minimum is at a ratio of 0.743 — the cheapest ratio — and its value is sin²θ = 0.18.

    The cheapest fork loses only its sideways speed, and pricing it closes the fork

    A turbulent network built at least cost forks at 28.4°, with daughters three quarters of the parent's radius, and those numbers priced the pipes and never the split. Price the split as Borda priced a sudden widening and something exact appears: at the cheapest fork each daughter already runs at the parent's velocity component along it, so the junction loses nothing but the sideways part of the parent's speed, and its loss cannot be reduced by resizing the daughters. It can be reduced by narrowing the fork. A split costing under two per cent of the bill moves the cheapest angle by seven degrees, because the pipes' bill is flat at its best angle and the split's loss is not.

    part 4 · applied
  5. Every airway down to the eleventh generation dissipates several times Poiseuille's loss, because its layer restarts at the fork. Pedley's factor, the developing layer's dissipation over the Poiseuille dissipation of the same airway, generation by generation along Weibel's symmetric tree, at total flows of 0.25, 1 and 5 litres a second. 0.25 L/s: 4.88 in generation 1, 2.57 in generation 5, above one down to generation 8; 1 L/s: 9.77 in generation 1, 5.15 in generation 5, above one down to generation 11; 5 L/s: 21.8 in generation 1, 11.5 in generation 5, above one down to generation 14.

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

    A laminar fork does not lose energy the way a turbulent one does, in a burst of mixing at the split. It loses it all the way down each daughter, because the boundary layer starts again at every flow divider and an airway a few diameters long ends before the layer has grown across it. Pedley measured that extra dissipation in 1970, and in the form he wrote it the radius cancels: it depends on the flow and the length alone. Price it into the cheapest tree and Murray's cube law gives way to a flow growing as the radius to the power twelve-fifths or thirteen-fifths, the band the turbulent pipe's exponent sits in, across the first eleven generations of a resting lung.

    part 5 · applied

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