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

Dispersion — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

    Two slow things make a fast one

    Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.

    part 1 · regimes
  2. The spreading per metre has a best speed; the spreading per second has none. The plate height — how much variance a slug picks up per unit length travelled — against the velocity, for four retention factors. Each curve falls as one over the velocity, because molecular diffusion along the tube matters when the flow is slow, and rises with it, because that is Taylor's dispersion. The minimum is what a chromatograph is run at, and Taylor's effective diffusivity has no such point anywhere.

    The best a tube can do is its own radius

    Taylor's dispersion coefficient rises without limit with velocity, so there is no best speed to run a tube at. Ask instead how much a slug spreads per metre travelled and an optimum appears at once — and the spreading there is the tube's radius over the square root of three, with no diffusivity and no velocity left in it.

    part 2 · regimes
  3. What an oscillation carries, against how fast it is asked. The extra axial diffusivity of a zero-mean oscillation, at fixed velocity amplitude, against the Womersley number. At low frequency it is exactly Taylor's dispersion evaluated at the mean square velocity — the profile has time to be Poiseuille's at every instant. At high frequency the shear is confined to a Stokes layer, the tracer in the core is never sheared, and the transport falls as the cube of the Womersley number.

    Transport with nothing transported

    Taylor's mechanism needs shear and diffusion and nothing else — and in particular it does not need a mean flow. Oscillate a tube about a fixed position and the tracer still spreads along it, by hundreds of times the molecular rate, which is how a patient is ventilated with a tidal volume smaller than the airway it goes down.

    part 3 · regimes

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