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The thread: A smooth picture proves nothing — page 31

Page 31 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 31 of 31.

The air pulls on the crests, and short ripples start to grow. The growth rate of a ripple on an inviscid jet against its wavenumber times the radius, at gas Weber numbers of 0, 0.4, 2, 6 and 13 on the diameter. The air flowing over a rippled jet is faster over the crests and its pressure lower there, which pulls them further out. Without it nothing shorter than the circumference grows; at a gas Weber number of 2 ripples up to ka = 1.45 grow, at 13 up to 6.19, and the fastest moves with them.

The air shortens a jet's fastest ripple, and the drops follow

A jet in a vacuum breaks into drops nearly twice its own width, whatever its speed. A jet in air does not, and the reason is a pressure the air puts on its surface: flowing over a rippled jet it is faster over the crests and its pressure lower there, which pulls them further out. That pull lets ripples shorter than the jet's circumference grow, shortens the fastest one, shrinks the drops, and puts a ceiling on how long a fast jet can be — at the gas Weber number where the measured break-up regimes change.

$A spread of sizes smears the stop band and pushes the fast band up. The phase speed of sound against frequency in water carrying air bubbles at a void fraction of 10⁻³: all of one millimetre radius, and spread from 10 µm to 1 mm as n ∝ R^(−q) with q = 0, 3/2 and 10/3. Every cloud starts at Wood's speed, 312 m/s, because at low frequency every bubble is driven below its resonance. The single size jumps through its stop band near 3 to 13 kHz into a fast band. The distributions climb through a band as wide as their resonances, 3.2 kHz for the largest to 301 kHz for the smallest, and the more small bubbles they carry the later they emerge faster than water.$ Compressible flow

The more small bubbles a cloud holds, the later it turns fast

Bubbles of one size give water three sound speeds: slow below their resonance, nothing in a stop band, faster than water above it. A cloud of many sizes has every resonance at once, so at any frequency its small bubbles soften it and its large ones stiffen it. The stop band becomes a wide band of attenuation, the fast band survives above it, and where the fast band starts is set by how much of the gas is in the smallest bubbles.

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