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

Suspension — where it appears

Solid particles carried in a liquid, dense enough that they affect its flow. Its settling rate, its effective viscosity and the concentration at which it stops flowing are all functions of the volume fraction, and none of them is the single-particle answer.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

viscous · Suspension The line that decides how much a thickener can pass. The batch settling flux against concentration, the same curve the closed column was solved on, with the operating line of a continuous thickener drawn across it. The line runs from the underflow concentration on the axis to the flux the plant can take, and it is placed by one condition: it must touch the flux curve rather than cut it. Where it touches is the concentration at which the thickener is working hardest.

The tangent that sizes a plant

A closed cylinder of settling suspension is a kinematic wave problem with a flux curve. Open the cylinder — feed it at the top and draw thickened slurry from the floor — and the capacity of the whole plant is a straight line laid against that same curve, touching it rather than cutting it.

kinematics · Kinematic waves 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.

viscous · Suspension

Named alongside it

The objects these essays reach for when they reach for this one.

Constitutive lawAveragingCreeping flowModel limitRate of strainStokes flowStressletConservationDesignDiffusionDiscontinuityDissipation

All concepts