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

Dilatation — where it appears

The rate at which a material element's volume grows, per unit volume, which is the divergence of the velocity. Continuity makes it minus the material rate of change of density, so an incompressible flow is one in which each parcel keeps its own density rather than one whose density is uniform.

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

A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

kinematics · Transport theorem A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Incompressible is not a property of the fluid

A flow whose density varies three to one across it, with a divergence of 10⁻¹⁰ everywhere. And a flow of air at Mach 0.1, whose divergence is three per cent of U/a and which every textbook calls incompressible. The word is about what the flow does to a parcel's volume, and about nothing else.

kinematics · Dilatation

Named alongside it

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

Mass conservationMaterial derivativeModel limitCompressibilityConservationControl volumeDeformation gradientDensityDivergenceEnergy equationEulerian and LagrangianIncompressible

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