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

Suspension — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where Einstein's line stops being the measurement. The viscosity of a suspension of rigid spheres relative to the liquid's, against the volume fraction. Einstein's 1 + 5φ/2 is exact for one sphere and holds while the spheres cannot feel one another, which is up to about five per cent by volume. Batchelor and Green's two-sphere term takes it a little further; beyond about a fifth nothing derived works and the curve drawn is a fit, which diverges at a maximum packing that is itself a measurement.

    A viscosity made of particles

    Stir rigid spheres into a liquid and the mixture is thicker, by five halves of the volume fraction. Einstein's coefficient is not an empirical constant — it is a dissipation calculation on one sphere — and doing it as an energy rather than as a stress shows that four-fifths of it comes from somewhere nobody mentions.

    part 1 · viscous
  2. The orbits a rod's axis can be on, seen along the vorticity. The tip of the unit vector along a rod of aspect ratio five, tumbling in a simple shear, seen from along the vorticity axis, for five values of the orbit constant. Every orbit is closed and every one takes the same time. A rod near the centre is spinning about the vorticity axis; a rod on the outer circle tumbles end over end in the plane of shear. The flow never moves a rod from one orbit to another.

    A viscosity the flow cannot decide

    Spheres stirred into a liquid thicken it by a definite amount. Rods do not. A rod in a shear flow tumbles round a closed orbit, the flow never moves it to another, and the extra viscosity depends on which orbit it is on. The equations of slow flow permit a whole range of values and choose none of them. The smallest amount of noise chooses one, and it does not matter how small.

    part 2 · viscous

All series