Concept

Growth rate — where it appears

The rate at which a small disturbance grows, the real part of the exponent in an e^(st) solution of the linearised equations. A positive growth rate for any wave makes the flow unstable; comparing rates across wavelengths says which pattern appears first.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not.

Past three, an ellipse is a shear layer

Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.

inviscid · Vortex patch

A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

kinematics · Dilatation

Viscosity slows every ripple, and the short ones most. The growth rate of a varicose ripple on a liquid jet against its wavenumber times the jet's radius, at Ohnesorge numbers of 0, 0.1, 1 and 10, in units of the capillary time. Every ripple longer than the circumference grows. Viscosity damps each one in proportion to the square of its wavenumber, so the short ones lose most: the fastest moves from ka = 0.697 inviscid to 0.344 at Oh = 1 and 0.123 at Oh = 10, and its rate falls from 0.343 to 0.0114.

Viscosity lets a jet break, later and into bigger drops

A thread of honey falls for metres before it breaks, and a thread of water for centimetres, which suggests that viscosity holds a jet together. It does not: it cannot stop any ripple longer than the jet's circumference from growing. It slows them, the short ones most, so the ripple that wins is longer, it takes the viscous time rather than the capillary one to win, and each drop it makes is bigger. The wavelength grows as the square root of the Ohnesorge number and the drop as its sixth root.

regimes · Atomisation

Long waves grow, short ones are held flat, and one in between grows fastest. The growth rate of a ripple on the interface where air displaces oil in a Hele-Shaw cell, against its wavenumber, at displacement speeds of 0.5, 1 and 2 mm/s. The viscosity contrast drives every wavelength at a rate proportional to its wavenumber; surface tension holds back short ones as the cube. At 1 mm/s the cut-off is 0.245 per mm and the fastest wavenumber 0.141 per mm, a wavelength of 44.4 mm, growing at 0.0942 per second.

The fastest finger is set by the gap and one number

Push air into oil between two glass plates and the interface breaks into fingers. The averaged law of the cell makes the instability exact in its linear stage: every wavelength is driven in proportion to its wavenumber, surface tension holds back the short ones as the cube, and the fastest finger is π gap widths divided by the square root of a capillary number. A narrow channel stays flat; a wide one chooses how many fingers to make; and a growing bubble of air makes more of them the larger it gets.

inviscid · Hele-shaw

Named alongside it

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

Model limit Instability Dispersion relation Surface tension Wavenumber Atomisation Capillary number Combustion Contour dynamics Darcy's law Drop Equilibrium

All concepts