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

Knudsen — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. How wrong the ordinary answer already is. The error in a no-slip continuum calculation of the flow through a channel, against the Knudsen number, both logarithmic. The rule at Kn = 1 is where a molecule crosses the whole channel between collisions — the value the number is named for. The error is one per cent at Kn = 1/594, five per cent at 1/114 and ten at 1/54, so the continuum regime of the usual classification, which runs to Kn = 0.01, is a region in which the continuum answer is already six per cent out at its far end.

    Where a fluid stops being one

    The Knudsen number is the mean free path over the size of the thing, and at one a molecule crosses the whole channel between collisions. The continuum equations with a no-slip wall are already one per cent wrong at one part in five hundred and ninety-four, which is a factor nothing about the definition would suggest.

    part 1 · regimes
  2. The gas that does not stop at the wall. Channel flow profiles with and without slip, at a Knudsen number of a twentieth. The slipping profile does not reach zero at the wall: the gas there is moving, by an amount proportional to the mean free path times the velocity gradient.

    Slip is a memory of one mean free path

    A molecule arriving at a wall last collided about a mean free path away and carries the velocity from there. Averaged over arrivals and departures, that leaves the gas at the wall moving — by two per cent of the centreline speed at a Knudsen number of a hundredth, and sixty per cent more flow through a microchannel at a tenth.

    part 2 · regimes
  3. Between plates the flow rate falls with rarefaction to a minimum near one free path across, then climbs without limit. The reduced flow rate G of a gas driven by a pressure gradient between two plates, against the rarefaction parameter δ = √π/(2Kn), from the linearised BGK equation; beside it Navier–Stokes with no slip (δ/6), with Maxwell's slip (δ/6 + 1.016) and the uncollided flux of molecules straight from a wall. The least flow rate is 1.538 at δ = 1.07, Kn = 0.827; at δ = 0.01 the solve gives 3.049, at δ = 0.001 4.449 and at δ = 10 2.769.

    A thinning gas flows least at one free path across, and then more without end

    Pump the gas out of a narrow channel and, for the same pressure gradient, less of it flows — until the channel is about one mean free path high. Thinner than that, more flows, and between infinite plates the flow rate climbs without limit, because the molecules carrying it are the ones skimming along the walls, whose flights only collisions with other molecules can end. Knudsen measured the minimum in 1909. The linearised kinetic equation puts it at a reduced flow rate of 1.538, at a Knudsen number of 0.83, and shows it is not a dip below the two closed forms but a place far above both.

    part 3 · regimes

All series