Where a jet stops being a jet

A kitchen tap turned down gently produces a stream that becomes a chain of drops a few centimetres below the spout. A garden hose produces one that stays a stream for a metre or more. An agricultural sprayer produces a mist at the nozzle, and a diesel injector produces something closer to a fog.

All four are the same liquid leaving a round hole, and all four break up by the same mechanism. What separates them is one dimensionless number, and it is not the one most people would guess.

What this essay does not do #

The mechanism by which a liquid cylinder breaks — that every disturbance longer than the circumference grows, that the fastest-growing one is at 4.51 diameters, and that this fixes the drop size — is not derived here. It is illustrated-physics.com’s argument, which owns surface tension as a property of a liquid: the skin, the pressure inside a drop, how far water climbs a tube, and the selection of the wavelength.

What this site owns, and what this essay is, is the flow: what a nozzle and a stream of moving liquid do with that mechanism. How far the jet gets before it breaks, which of four regimes it is running in, and which number decides.

Saying that plainly is worth a paragraph because the two questions look like one question and are answered by different quantities. The mechanism contains no velocity at all; everything below is velocities.

Four numbers, three of them independent #

A jet issuing into a gas has one thing holding it together and two pulling it apart:

$$\mathrm{We}_l = \rho_l U^2 d/\sigma, \qquad \mathrm{We}_g = \rho_g U^2 d/\sigma, \qquad \mathrm{Re} = \rho_l U d/\mu, \qquad \mathrm{Oh} = \mu/\sqrt{\rho_l \sigma d}.$$

Only three are independent, and the relation between them is exact:

$$\mathrm{Oh} = \frac{\sqrt{\mathrm{We}_l}}{\mathrm{Re}}.$$

That identity is asserted rather than stated, and it is the cheapest possible check that the four numbers were built from the same quantities — because the standing hazard in this subject is putting the gas’s density where the liquid’s belongs, and an Ohnesorge number formed that way passes every other test.

It also explains why the classical diagram can have Reynolds and Ohnesorge on its axes and Weber-number boundaries drawn on it. A fixed Weber number means a fixed product Oh·Re, which in logarithmic coordinates is a straight line of slope exactly −1, and the assertion checks that a decade in Reynolds number moves each boundary by exactly one decade.

The number that decides is the gas’s #

The four regimes and their borrowed boundaries, in the gas Weber number:

• Rayleigh break-up, below 0.4. The jet breaks a long way downstream into drops larger than itself — 1.89 diameters, from the mechanism linked above — and the air plays no part.
• First wind-induced, from 0.4 to 13. The gas’s pressure fluctuations amplify the same disturbance, the break comes sooner, and the drops are about the size of the jet.
• Second wind-induced, from 13 to 40.3. Short-wavelength disturbances grow on the surface and the drops are much smaller than the jet.
• Atomisation, above 40.3. The jet breaks at the nozzle exit and there is no coherent length at all.

The density in that number is the gas’s, and the consequence is a clean prediction that a vacuum chamber can check. A 1 mm water jet at 40 m/s in air has a gas Weber number of 26 and is in the second wind-induced regime; the same jet at the same speed in air at a thousandth of an atmosphere has a gas Weber number of 0.03 and is a Rayleigh jet, breaking far downstream into drops twice its own diameter. The jet’s own Reynolds number and Weber number are unchanged. What was removed was the only thing that differed.

That is why “fast enough to atomise” is a statement about the surroundings as much as about the jet, and why altitude matters to a fuel injector.

How far the jet gets #

On the Rayleigh branch the break-up length is the one quantity in this subject that is a pure flow result:

$$\frac{L}{d} = \frac{U}{\omega_{\max} d}\ln\frac{a}{\varepsilon_0},$$

a growth time multiplied by a speed, multiplied by the number of e-foldings the initial disturbance needs to reach the jet’s own radius. The first factor comes from the dispersion relation, the second is the flow, and the third is the only unmeasurable in the whole calculation.

And it enters as a logarithm, which is what makes the prediction usable. A disturbance a hundred times larger at the nozzle removes ln(100) = 4.6 e-foldings out of about eleven, shortening the jet by forty per cent rather than by a factor of a hundred. So the break-up length is a robust measurement that depends almost entirely on things that can be known.

Five nozzles #

The span of that axis is the argument. The same three boundaries separate a dripping tap from a fuel injection at three hundred metres a second, and between the two extremes nothing about the mechanism changes: one dimensionless number moves by seven orders of magnitude.

Two of the rows are worth a sentence each.

The inkjet nozzle is the odd one out, and its Ohnesorge number is why. At 25 µm and 8 m/s its gas Weber number is small — it is a Rayleigh jet — and its Ohnesorge number is a hundred times a tap’s, because Oh goes as one over the square root of the diameter. Viscosity is what stops the drop from having satellites, which is why inkjet formulation is a viscosity problem rather than a speed problem.

The diesel injector is not really described by any of this, and the essay says so below. At a gas Weber number in the hundreds of thousands the jet has no coherent length, the break-up is at the nozzle, and what happens next — droplet collision, secondary break-up, evaporation — needs a great deal more than a regime name.

A group with no velocity in it #

The Ohnesorge number is the odd one of the four, and what makes it odd is worth a section: it contains no velocity. It is built from the liquid’s viscosity, density and surface tension and from the nozzle’s diameter, and from nothing about how hard the pump is pushing.

So it labels the nozzle rather than the operating point. Turning the pressure up moves a jet horizontally across the Ohnesorge diagram and never vertically, which is exactly why those are the diagram’s coordinates: every nozzle is a horizontal line, and asking which regime it is in becomes asking how far along its own line it has been pushed.

That is also why the diagram survived. A map whose axes both contained the velocity would put a single nozzle at a single point and tell an engineer nothing about what changing the pressure would do; this one puts a nozzle on a track.

Why small nozzles are viscous #

The Ohnesorge number goes as the inverse square root of the diameter, which has a consequence for everything small.

For water, a 1 mm nozzle has Oh = 0.0037 and a 25 µm nozzle has Oh = 0.0235 — six times larger, for the same liquid. Shrink to a 1 µm orifice and it is 0.117. So viscosity matters more the smaller the nozzle, at the same liquid and the same speed, and a microjet is a viscous object in a way that a hose is not.

This is the same scaling argument that runs through the rest of this field, and it has the same shape as the capillary comparison: a force that scales with a length competes with forces that scale with volumes, so the balance is decided by size rather than by judgement. Here the small-scale winner is viscosity, and the practical consequences are specific.

An inkjet drop must detach cleanly with no satellites, and satellites are suppressed by viscosity — which is why inkjet inks are formulated to an Ohnesorge number rather than to a viscosity, and why the printable window is usually quoted as roughly 0.1 < Oh < 1 in the inverse convention the industry uses. A fuel injector’s orifice is 150 µm and its Ohnesorge number is 0.0096: still small, because the speed does not enter and the fuel is thin.

The refutation, in a vacuum chamber #

The claim this essay refutes is that a jet atomises when it is fast enough, with the jet’s own Reynolds number deciding. The cleanest test is to remove the air and change nothing else.

Take the 1 mm water jet at 40 m/s. In air it has a gas Weber number of 26.4, which is second wind-induced: the surface develops short-wavelength disturbances and the drops are much smaller than the jet. Pump the chamber down to a thousandth of an atmosphere and the gas Weber number becomes 0.026, below Reitz’s first boundary — the jet is a Rayleigh jet, breaking far downstream into drops nearly twice its own diameter.

The jet’s Reynolds number is 40,000 in both cases. Its own Weber number is 21,900 in both cases. Its Ohnesorge number, containing no velocity and no gas, is 0.0037 in both cases. Three of the four numbers cannot tell the two experiments apart, and the fourth changes by a factor of a thousand.

That is what it means for a regime to be set by one number, and it is why this site insists on naming which. A figure quoting a Reynolds number beside a picture of a spray is quoting the wrong hypothesis in a way no reader could detect — the number is correct, the picture is correct, and the hypothesis attached to them is about a competition that is not happening. The same experiment run the other way is just as instructive: raising the chamber pressure at constant jet speed atomises a jet that was making drops, which is why a diesel engine’s spray depends on when in the compression stroke it is injected.

What the model does not contain #

The boundaries are measurements. Reitz’s 0.4, 13 and 40.3 are transitions read off experiments, they are quoted with a spread in the literature, and they are drawn everywhere in these figures in the colour this site keeps for a borrowed claim. What is computed is where a nozzle sits relative to them, and the geometry of the boundaries in the Ohnesorge plane. Nothing here derives a transition.

No drop-size distribution. A spray is characterised by a distribution — a Sauter mean diameter, a spread — and none of that follows from a regime name. The correlations that supply it are fits, and this site does not carry them.

No gas boundary layer and no gas turbulence. The gas Weber number treats the surrounding air as a density and nothing else. The actual mechanism of wind-induced break-up is a pressure fluctuation on a wavy surface, which is an aerodynamic stability problem with a shear layer in it.

Nothing about the nozzle’s interior. A real injector’s flow separates at the inlet, may cavitate, and leaves with a velocity profile and a turbulence level that strongly affect the break-up. The number used here contains only the mean speed, and the discharge coefficient of a sharp-edged hole is a separate essay on this site precisely because that interior matters.

Linear theory upstream of everything. The break-up length rests on a growth rate from linear theory, and the pinch-off itself is violently nonlinear — a finite-time singularity, and one of the few real ones in fluid mechanics.

No gravity. A falling jet thins as it accelerates, so its diameter, and therefore every number here, changes along it. For a tap that is a substantial effect within the break-up length.

What happens to the drops afterwards, and the size they stop at #

The list above notes that no drop-size distribution follows from a regime name, which is true and leaves out the one thing about drop size that does follow from the same number. A drop, once made, is a body in a stream, and if the stream is fast enough relative to it the drop breaks again.

The criterion is the Weber number formed on the drop:

$$\mathrm{We}_d = \frac{\rho_g u_{\text{rel}}^2 d}{\sigma},$$

with the gas’s density again, the relative velocity between drop and gas, and the drop’s own diameter. Below a critical value the drop merely oscillates and recovers; above it, it cannot hold together. The critical value is between about six and twelve depending on how it is defined and how viscous the liquid is, and the modes above it are a well-documented sequence — a vibrational deformation, then a bag blown downstream from a stabilising rim, then multiple bags, then a sheet stripped from the equator, and finally a catastrophic disintegration.

Rearranged, that criterion is a maximum stable drop size:

$$d_{\max} \approx \frac{12\,\sigma}{\rho_g\,u_{\text{rel}}^2},$$

and it is the closest thing to a drop-size prediction that the gas Weber number can supply without a correlation. Anything larger breaks; anything smaller survives; so a spray’s coarse end is set by the local relative velocity, and it falls as the square of it.

Two examples make the range plain. For diesel in compressed air, with a surface tension near 0.025 N/m and a gas density of twenty kilograms per cubic metre, a drop moving at thirty metres a second relative to the charge cannot exceed about seventeen microns — which is the right order for a diesel spray’s coarse fraction. And for water falling through ordinary air at nine metres a second, the same expression gives a few millimetres, which is why raindrops have a largest size of five or six millimetres and why anything bigger breaks on the way down rather than arriving.

There is an important subtlety in evaluating it, and it is the reason a spray’s size distribution is not simply this formula. The relative velocity is not the injection velocity: a drop decelerates sharply once it leaves the coherent jet, so $u_{\text{rel}}$ collapses within a few millimetres and the bound relaxes as it does. Break-up therefore stops rather than continuing indefinitely, and where it stops is a competition between the drop’s aerodynamic deceleration and the time each break-up mode takes — which is a calculation about trajectories rather than about regimes, and is what a spray model actually spends its effort on.

But the direction of the whole argument is the essay’s own. The final size of a droplet is set by the density of the gas around it, through the same competition between an aerodynamic pressure and a surface tension that decided the regime the jet was in. A liquid injected into a vacuum makes large drops and keeps them; the same liquid injected into a compressed charge is broken twice — once at the nozzle and again in flight — and both breakings are the surrounding gas doing the work.

Who found it, and when #

Rayleigh gave the mechanism in 1878 and Plateau the geometry before him; both belong to the essay linked at the top. The regime structure came later and from engineering.

Haenlein published photographs of jets in 1931 that show the four behaviours plainly and named the transitions. Wolfgang von Ohnesorge’s 1936 paper is the one the diagram is named after, and its contribution is exactly the coordinates: by plotting the Reynolds number against a combination containing no velocity, he separated what the liquid brings to the problem from what the flow does. Rolf Reitz’s 1978 thesis supplied the gas-Weber-number boundaries that are drawn here, in the context of diesel injection, where the question is which of the four regimes an engine’s spray is in.

The order is worth noticing. The mechanism was understood sixty years before the regimes were mapped, because the mechanism is a stability calculation and the regimes are a question about a machine — and nobody needed the map until fuel had to be atomised reliably.

Where the ladder goes next #

This essay’s numbers are all about a jet’s outside: the air it moves through, the surface it has, the speed it carries. The next question in this field is about a flow’s inside — what happens to a slug of dye carried along a pipe by a profile that moves its middle faster than its edge, and why the answer involves a molecular diffusivity that makes the spreading slower as it grows.