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

Viscosity lets a jet break, later and into bigger drops put the jet’s own viscosity into Rayleigh’s instability and found that it slows every ripple, shifts the fastest to longer wavelengths and makes the drops bigger — but cannot stop the jet breaking, and leaves its break-up length growing in proportion to its speed. Where a jet stops being a jet had already found, from measurement, that this is true only at low speed: past a gas Weber number of about 0.4 a jet leaves the Rayleigh regime, its drops shrink towards its own size, and at higher speeds far below it. The measured map did not say why. This essay puts the air into the same relation and asks whether the air’s pressure on the surface is the reason.

What the air does to a ripple #

A jet moving through still air at speed $U$ is, in its own frame, a stationary column with air streaming past it. Put a ripple on the column and the air’s path is narrowed over each crest and widened over each trough, so it speeds up over the crests and slows over the troughs. By Bernoulli the pressure over a crest falls, which pulls the crest further out, and over a trough rises, which pushes it further in. That is the Kelvin–Helmholtz mechanism, acting on a cylinder rather than a sheet, and unlike surface tension it does not care whether the ripple is longer or shorter than the circumference.

Solving for the air as potential flow outside the rippled cylinder gives the pressure perturbation over a ripple of height $\eta$ as $-\rho_g U^2 k\,\eta\,K_0(ka)/K_1(ka)$, with $K$ the modified Bessel functions. Added to the relation of the previous essay, in the same capillary units,

$$\omega^2 + 3\,\mathrm{Oh}\,\xi^2\omega = \xi\,\frac{I_1}{I_0}\,(1 - \xi^2) + \mathrm{We}\,\xi^2\,\frac{I_1 K_0}{I_0 K_1},$$

with the gas Weber number $\mathrm{We} = \rho_g U^2 a/\sigma$ on the radius. The figures quote it on the diameter, twice as large, as the measured regime map does. The new term is positive for every ripple, so it can only make the jet less stable.

Short ripples start to grow #

Without the air, nothing shorter than the circumference grows: the drive changes sign at $ka = 1$, Plateau’s limit, because a ripple shorter than the circumference increases the surface rather than reducing it. With the air the curve rises everywhere and stretches past $ka = 1$. At a gas Weber number of 2, ripples up to $ka = 1.45$ grow; at 13, up to 6.2. The air has made the jet unstable to short ripples that surface tension alone would flatten, and since a short ripple, if unstable, grows faster than a long one, the fastest moves with them.

On a short ripple the jet is a flat sheet #

The relation has a limit that shows what the air’s term is. For a ripple much shorter than the radius, $ka \gg 1$, both ratios of Bessel functions tend to one and the surface’s curvature round the jet stops mattering: the relation becomes $\omega^2 = \mathrm{We}\,\xi^2 - \xi^3$, which in ordinary units is $\rho_g U^2 k^2/\rho - \sigma k^3/\rho$. That is the Kelvin–Helmholtz relation for a flat interface between a dense fluid at rest and a light one streaming over it, with surface tension holding back the short waves — the same instability that makes a vortex sheet unable to stay a sheet, with the tension supplying the cut-off the bare sheet lacks. Its cut-off is at $k = \rho_g U^2/\sigma$, which is $ka$ equal to the radius-based gas Weber number, and the figure’s cut-off at a diameter-based 13, $ka = 6.2$, is within a few per cent of it.

So the air’s term is the bridge between two instabilities usually treated separately. At long wavelength the jet is Rayleigh’s column, broken by its own surface tension; at short wavelength it is a sheet in a wind, broken by the wind; and the gas Weber number says which end of the relation the fastest ripple sits at.

The fastest ripple, and the measured regimes #

In jet diameters, the fastest ripple shortens steadily as the gas Weber number rises. The measured boundaries on the figure come from experiment and from the first essay, not from this calculation, and they line up with what it computes. At 0.4, the measured edge of the Rayleigh regime, the fastest ripple is 4.17 diameters, 7.4 per cent shorter than in a vacuum — the edge of the regime is where the air has begun to matter at the level of a few per cent. At 13, the measured edge of the first wind-induced regime, it is 0.72 diameters, shorter than the jet’s own circumference: the ripples that win are no longer varicose necks the jet’s width apart but short waves on its surface, which is what the second wind-induced regime is described as. At 40, the measured start of atomisation, the fastest ripple is a quarter of a diameter and the linear varicose picture has run out of meaning.

So the measured regime map is, to a first approximation, a map of where the air’s term passes the surface tension’s. The boundaries are not sharp in the calculation — the fastest wavelength changes smoothly — and the measured ones are not sharp either; what the calculation supplies is the reason the regimes are ordered by the gas Weber number and not by the jet’s own.

Ripples shorter than the jet #

The cut-off makes the change of regime precise. In a vacuum it is $ka = 1$ exactly. With air it rises as the gas Weber number grows, roughly in proportion to it at large values, because the air’s pull on a short ripple grows as $k^2$ while surface tension’s restoring push grows as $k^3$, and the two balance at a wavenumber proportional to the gas Weber number. The fastest ripple follows the cut-off at a fixed fraction of it, and crosses $ka = 1$ at a gas Weber number of about 1.87. Past that point the ripple that grows fastest is shorter than the circumference, and a drop made from one wavelength is smaller than the jet.

That crossing is the cleanest computed marker of the end of the Rayleigh regime the relation offers, and it sits inside the measured band — above the 0.4 at which the measured regime ends and well below 13. Where exactly an experimenter places the boundary depends on what is measured, and the next figure is the measurement most often used.

The drops shrink, and viscosity holds them back #

One drop per fastest wavelength makes drops of $(12\pi/ka)^{1/3}/2$ jet diameters. For water from a millimetre nozzle, at an Ohnesorge number of 0.004, the drops fall from Rayleigh’s 1.89 jet diameters in a vacuum to 1.04 at a gas Weber number of 13. For a liquid at an Ohnesorge number of 0.3 they start larger, 2.11, and fall more slowly, to 1.22. The viscosity damps the short ripples the air would favour, holding the fastest one long, so a viscous jet stays in the Rayleigh-like regime to higher speeds and makes larger drops at a given gas Weber number. That is why the measured boundary of the first wind-induced regime moves up with the Ohnesorge number, as the regime map’s correlations say it does, and why viscous liquids are harder to atomise finely.

One number for every liquid in every gas #

The relation contains the gas only through the gas Weber number, and the liquid’s density only through the capillary time that sets the units. So a mercury jet and a water jet at the same gas Weber number and the same Ohnesorge number have the same fastest ripple in diameters and make the same drops in diameters; only the time it takes, and so the break-up length, differs. A jet into pressurised gas — fuel into a combustion chamber at twenty atmospheres — reaches a given gas Weber number at a far lower speed than the same jet in air, because the gas density is twenty times greater, and it does so with the liquid’s own behaviour unchanged. That is the content of one number deciding here: the map the first essay drew with real nozzles is a map in two numbers because the relation has only two.

The ceiling on a jet’s length #

Without the air, the previous essay found the break-up length growing in proportion to the speed. With it, the length grows more slowly, reaches a maximum and falls. For a millimetre water jet the maximum is 314 diameters, about thirty centimetres, at 10.6 metres a second; at twenty metres a second the jet breaks after 173 diameters against 923 in a vacuum. The faster the jet, the harder the air pulls, the faster the fastest ripple grows — fast enough, past the peak, to outrun the jet’s own speed.

That curve, rising, peaking and falling, is the classic measured break-up curve of a liquid jet, and its peak is the “critical point” experimenters use to mark the end of the Rayleigh regime. In this calculation the peak falls at a gas Weber number of 1.87 — the same value at which the fastest ripple becomes shorter than the circumference. The two markers coincide because both are the point at which the air’s term starts to dominate the drive: below it the growth rate barely depends on speed and the length grows with it; above it the growth rate grows faster than the speed and the length falls.

Three jets from a garden #

The numbers place familiar jets. A kitchen tap’s six-millimetre stream at a metre a second has a gas Weber number of 0.1: the air does nothing, the fastest ripple is 4.44 diameters, and the drops at the bottom of a long fall are about 11 millimetres across, Rayleigh’s. A fountain’s five-millimetre jet at three metres a second is at 0.75, just past the measured edge of the Rayleigh regime; its fastest ripple is 3.9 diameters, twelve per cent shorter, and its drops nine millimetres instead of nine and a half. A garden hose’s three-millimetre jet at ten metres a second is at 5, in the first wind-induced regime, with a fastest ripple of 1.8 diameters and drops of about four millimetres — and its break-up length is near the top of its curve, which is why a hose jet holds together for a few metres and then breaks abruptly rather than gradually. A diesel injector, at a gas Weber number of about twenty thousand in compressed air, is so far past every boundary on the map that none of this applies: its jet is atomised at the nozzle.

Why the gas’s Weber number, not the jet’s #

The first essay remarked that the regimes are ordered by the gas Weber number and that the jet’s own Weber number, $\rho U^2 d/\sigma$, a thousand times larger for water in air, decides nothing. The relation says why. The liquid’s inertia appears only as the $\omega^2$ term, the resistance of the liquid to being moved; it does not drive anything. What drives the ripple is surface tension at long wavelengths and the air’s pressure at short ones, and the air’s pressure is $\rho_g U^2$, so the comparison that decides which drive wins is $\rho_gU^2$ against $\sigma/a$: the gas Weber number. A jet in a vacuum chamber, the first essay’s refutation, has a gas Weber number of zero whatever its speed, and the relation gives it Rayleigh’s drops at every speed — which is what the chamber showed. The same comparison decides when a drop falling through air is torn apart by the air rather than held round by its tension.

How it was checked #

The relation is the previous essay’s with one term added, so its checks are the same ones plus one for the new term. The Bessel functions agree with tables to $2\cdot10^{-15}$; with the gas Weber number set to zero the fastest ripple is Rayleigh’s 0.69702. The gas term itself is checked against its long-wave limit: at $ka = 0.05$ the ratio $K_0/K_1$ is close to $ka\,(\ln(2/ka) - \gamma)$ and $I_1/I_0$ to $ka/2$, and the full term is 1.005 times the limit’s product, approaching one as the ripple lengthens.

What the measurement would look like #

The peak of the break-up curve is the easiest thing here to test, because it needs only a ruler and a pressure gauge. A millimetre nozzle fed from a regulated supply, the jet photographed against a scale at a sequence of speeds, should show the break-up length rising in proportion to the speed to about ten metres a second and falling beyond it, with the drops, photographed at the same time, shrinking from about two millimetres towards one across the peak. The same nozzle in a chamber of helium at atmospheric pressure, whose density is a seventh of air’s, should move the peak to about 2.7 times the speed and lengthen the jet there accordingly — the gas Weber number, not the speed, being what the peak is attached to. That second measurement separates the air’s effect from everything else that changes with speed, the nozzle’s turbulence included, which is the separation the vacuum chamber made by removing the gas altogether.

The convention: the gas Weber number on the diameter #

Figures quote the gas Weber number $\rho_g U^2 d/\sigma$ on the jet’s diameter, as the regime map does, and the relation uses it on the radius, half as large. The air is inviscid and at rest far from the jet; the jet moves at $U$ through it with a flat velocity profile; break-up is the fastest ripple grown from $e^{-12}$ of the radius to the radius. Drops are one fastest wavelength of jet each.

What the picture cannot show #

The air here is inviscid potential flow, and real air over a jet has a boundary layer that thickens with distance from the nozzle; the pressure it puts on a ripple depends on the ripple’s length against that layer’s thickness, and for short ripples on a thick layer it is weaker than the potential-flow value. That is the standard reason the measured boundaries sit at somewhat higher gas Weber numbers than inviscid theory’s. The jet is also given a flat velocity profile: a jet leaving a long nozzle carries the nozzle’s profile, which relaxes downstream and drives instabilities of its own — a jet’s own shear, which this relation does not contain. A jet leaving a short nozzle at high Reynolds number is also often turbulent inside, and its own velocity fluctuations disturb the surface far more strongly than any seed this calculation imagines — which shortens the break-up length without changing which ripple is fastest, since the relation sets the selection and the seed only the starting amplitude. And the linear relation says nothing about atomisation proper, the regime past a gas Weber number of about forty, where the surface is stripped into ligaments and drops directly at the nozzle and the varicose picture has no meaning; that regime needs a nonlinear calculation of the stripping itself.

Who found it, and when #

Weber’s 1931 paper included the air’s effect with the viscous one, in a long-wave form; the inviscid relation with the full Bessel functions for both the liquid and the gas is in Lin’s treatment of the subject. Sterling and Sleicher in 1975 corrected Weber’s aerodynamic term for the gas boundary layer, which brought the computed break-up lengths into line with measurement at moderate speeds. Reitz’s 1978 classification of break-up regimes by the gas Weber number, with the boundaries quoted here, is the measured map this relation explains, and Lin and Reitz’s review of 1998 sets the two side by side.

Still open: the boundary layer in the air #

The weakest link here is the air’s pressure on the surface, computed as if the air slipped past the jet without friction. The next calculation gives the air a boundary layer growing from the nozzle, computes the pressure perturbation over a ripple of each wavelength through a sheared air flow — weaker for ripples short against the layer — and asks how far that moves the fastest ripple’s crossing of the circumference and the break-up curve’s peak towards the measured values, and whether a jet issued into a co-flowing stream of air, which thins the layer, breaks closer to the inviscid prediction, as Sterling and Sleicher’s correction implies it should.