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

Rate of strain — where it appears

The symmetric half of the velocity gradient, describing how fast a parcel of fluid is changing shape rather than how fast it is moving or rotating. It is what a viscous stress responds to, and it is what the dissipation function is built from.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted.

The price of a gradient

Viscosity does not charge for motion. It charges for the rate at which a parcel is being deformed, and a fluid in solid-body rotation at any speed whatever destroys nothing at all. What is charged for is a sum of squares, which is why the bill can be computed twice.

viscous · Dissipation Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

kinematics · Material lines 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 Drop deformation in simple shear, against the capillary number. The shape model's steady deformation for six viscosity ratios. Every curve is linear in the capillary number at small Ca — which is Taylor's result — and every one of them saturates, at 5/2(2λ+3), because the shear's own rotation turns the drop out of the stretching direction. A drop in simple shear cannot be deformed beyond that however hard it is sheared.

The number that cannot break a drop

The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.

regimes · Capillary 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 figure on this site; 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.

viscous · Bulk viscosity

Named alongside it

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

DissipationVorticityCreeping flowDissipation functionModel limitStokes flowViscosityAveragingBulk viscosityCapillary numberCompressibilityConservation

All concepts