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

Continuity — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A streamtube narrows and the flow speeds up. Two neighbouring streamlines bound a tube that no fluid crosses. Where the tube pinches, the same mass has to pass through a smaller gap every second, so it must move faster — which is mass conservation with no equations in sight.

    Mass has nowhere to go

    Squeeze a stream of fluid and it speeds up, not because anything pushes it but because the same amount has to get through a smaller gap every second. Almost every result in the subject is that observation with more machinery attached.

    part 1 · kinematics
  2. The number on a streamline is a flow rate. Two streamlines and a crooked line drawn between them. The volume of fluid crossing that line every second, integrated from the velocity field, is the difference between the two streamfunction values at its ends — which is what makes a streamline a label rather than merely a curve.

    The number on a streamline is a flow rate

    Streamlines get drawn as decoration — curves the flow follows, spaced however the plotting looked best. Each one carries a number, the difference between two of those numbers is the volume of fluid passing between them every second, and it does not matter what route the measurement takes.

    part 2 · kinematics
  3. In the frame of the wave the walls stand still, and a bolus rides between them. Streamlines of a peristaltic channel of amplitude ratio 0.7 over two wavelengths, drawn in the frame moving with the wave, where the flow is steady and the walls are themselves streamlines. The time-mean flow is Θ = 0.5904 of the wave speed times the mean half-width, so the flow rate between centreline and wall in this frame is q = −0.4096 and the pressure rise per wavelength is 0.000 in units of μcλ/a². The centreline velocity changes sign at 0.106π and 0.894π, and the streamline through those points closes round a bolus holding 30.5 per cent of the fluid in each wavelength, which travels with the wave.

    A wave on the wall is a pump

    A channel whose wall only moves in and out, in a wave travelling along it, delivers a steady net flow with no part of the wall moving along the channel. In the frame of the wave the walls stand still and are streamlines, so continuity alone fixes how the laboratory flow rate follows the wall shape — and the momentum equation is needed only for one number, which also decides whether fluid rides along with the wave or leaks back against it.

    part 3 · kinematics
  4. The best efficiency runs from nine-eighths of φ² to one. The best efficiency a peristaltic pump can reach, against the fraction of the channel its wave closes, with its two limits. For a shallow wave it is 9φ²/8, which is small — a wave closing a fifth of the channel is at best 4.5 per cent efficient. As the wave closes the channel the best efficiency tends to one, and its shortfall shrinks in proportion to the remaining gap: about 1.9(1 − φ). Nothing in between is independent of the amplitude.

    The pump that is better the more it squeezes

    A waving sheet swims at a cost per metre with no amplitude in it. A waving wall pumping fluid is the same mechanism turned round, and its efficiency is nothing like that: it starts at nine-eighths of the amplitude ratio squared, is exactly 2 − √3 at half closure, and rises towards one as the wave closes the tube — where the pump stops being a wave and becomes a piston.

    part 4 · kinematics

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