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

Swimming — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

    A swimmer that cannot go backwards

    Taylor's waving sheet is the simplest self-propelled object in a viscous fluid, and its arithmetic contains a result that reads like a mistake — the work it does to travel a metre does not depend on how big its waves are. Doubling the amplitude quadruples both the speed and the power, and changes the bill for the journey by nothing at all.

    part 1 · viscous
  2. Three strokes, and only the flat one is a theorem. Three cycles drawn in the swimmer's shape space: a square, a circle of the same width, and an out-and-back along the diagonal. The first two enclose area and carry the swimmer forward; the third encloses none and carries it exactly nowhere, which is the scallop theorem with no symmetry argument in it. What the third lacks is not a broken symmetry but an interior.

    A stroke is worth the area it encloses

    The usual account of swimming without inertia is a symmetry argument about reciprocal strokes, which says what cannot work and nothing about what does. Draw the stroke in the space of the swimmer's own shapes and the displacement is a line integral — so it is an area, it does not depend on how fast the stroke is played, and the scallop theorem is Stokes' theorem.

    part 2 · viscous
  3. The only term that turns spin into thrust, and what it is made of. The coupling term of the propulsion matrix against the drag anisotropy, with the geometry held fixed. It is exactly proportional to the difference of the two drag coefficients, so it is zero when they are equal — not small, zero — and no helix of any pitch turned at any rate would move. A real filament sits at 1.66, which is a third of the way from useless to the unreachable limit.

    Two drags, or nothing swims

    A bacterium turns a corkscrew and goes forward, and the reason is not the corkscrew. It is that a thin filament dragged broadside resists more than the same filament dragged end-on. Make the two resistances equal and the thrust is not small but exactly zero, for every pitch and every rate.

    part 3 · viscous

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