Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Bulk viscosity — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The coefficient Stokes threw away is not a correction. What absorbs sound in dry air, split into the three transport effects that add to make the diffusivity of sound. Shear viscosity is the largest single contribution and it is not most of it; the bulk viscosity — which Stokes set to zero in 1845, saying in the same paper that he had no argument for doing so — is a quarter, and thermal conduction is a fifth. Setting ζ to zero does not make a small error in the absorption; it makes a 24 per cent one.

    The viscosity nobody uses

    A fluid has two viscosities. One resists a change of shape and appears in every viscous calculation; the other resists a change of volume, was set to zero by Stokes in 1845 in a paper that says he had no argument for doing so, and for carbon dioxide is fifteen hundred times larger than the one everybody quotes.

    part 1 · viscous
  2. On Mars the bulk viscosity is two speeds of sound. The speed of sound in carbon dioxide at 610 pascals and 240 kelvin against frequency, with the bulk viscosity scaled from its value at one atmosphere by pressure alone. Low notes travel at 245 m/s, with the bending vibration keeping up; high ones at 252 m/s, with it frozen. The step is centred near 342 Hz. The Perseverance rover's microphones reported the same thing in 2022: two speeds of sound, near 240 and 250 metres a second, either side of a few hundred hertz.

    The largest bulk viscosity is the first to expire

    A bulk viscosity is not a separate property of a gas. It is the time a molecule's internal motion takes to catch up with a compression, multiplied by the pressure and by how much heat capacity lags, and read from below that time's frequency. So the coefficient that is largest is the one that stops being a coefficient soonest: carbon dioxide's fifteen-hundred-fold value is ten per cent wrong at 24 kHz, and on Mars it is two speeds of sound in the audible band.

    part 2 · viscous
  3. Smooth below one Mach number, a jump and a tail above it. The velocity change through steady shocks in carbon dioxide, as a fraction of the whole change, against distance in relaxation lengths — the equilibrium sound speed times the relaxation time, 0.737 mm at one atmosphere. Shocks slower than the frozen sound speed, M below 1.041 referred to the equilibrium speed, are smooth throughout: the whole rise is the vibration catching up. Faster ones jump first, at the frozen speed's own shock, and relax afterwards: at M = 1.3 the jump takes 83.9 per cent of the change at once.

    A bulk viscosity holds a shock together until it splits

    Carbon dioxide's bulk viscosity, fifteen hundred times its shear viscosity, is a vibrational relaxation seen from below its frequency, and a shock is where that description is tested hardest. Carried through a steady shock, the relaxation reproduces the coefficient exactly for the weakest shocks. At a pressure rise of nine and a half per cent the shock outruns the frozen sound speed and splits into a jump and a tail, and the coefficient then draws a shock that does not exist.

    part 3 · viscous

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