A crown dissolves its nuclei or breaks on them, and fast

A degassed siphon is as tall as its largest nucleus allows found that a siphon of degassed water is not limited by the atmosphere at all. Its crown can sit below the vapour pressure and below zero, held by the water’s cohesion, and it runs until its crown’s tension reaches Blake’s threshold for the largest speck of gas the water carries: 14.4 metres for a largest nucleus of a micron, 27 for three-tenths of a micron. The water’s own strength, hundreds of atmospheres, never enters.

That calculation, like the siphon that does not need the air before it, took the largest nucleus as given and fixed. A nucleus lodged at the crown — in a crevice of the hose wall, or clinging to a mote caught there — is not fixed. It sits in water that still holds some dissolved air, and air crosses its surface. The essay’s closing question was how long a siphon running at a given height has before its largest nucleus gathers enough gas to reach the threshold. The answer is that the question has the wrong shape: the nucleus does not wait, and it does not necessarily grow.

A nucleus’s gas, and the water’s #

The nucleus is Blake’s: a small sphere of gas and vapour, its gas isothermal so that its content is the product of its gas pressure and its radius cubed, in mechanical balance with the water around it. At the crown’s pressure p the balance is

$$p = p_v + \frac{K}{R^3} - \frac{2\sigma}{R},$$

with K the gas content, $p_v$ the vapour pressure and σ the surface tension. For a given K there are two radii that balance; the smaller is stable, and as K grows the two approach and meet at a fold. Past the fold there is no balance, the nucleus grows without limit, and the water breaks. Blake’s threshold is that fold, reached by lowering the pressure at fixed K; the question here is the same fold, reached by raising K at fixed pressure.

Air enters or leaves the nucleus by diffusion through the water around it, and the direction is set by a single comparison. Dissolved air has a partial pressure — the pressure of air it would be in equilibrium with — and if the water holds a fraction f of what it would hold saturated at one atmosphere, that partial pressure is f times an atmosphere. Air flows into the nucleus if its own gas pressure, $K/R^3$, is lower than the water’s, and out of it if higher. With steady diffusion to a sphere,

$$\frac{dK}{dt} = 3LDR\left(f p_0 - \frac{K}{R^3}\right),$$

where D is air’s diffusivity in water and L its Ostwald coefficient, about 0.02: the ratio of air’s concentration dissolved to its concentration as a gas.

Half the tension #

The figure draws the gas pressure of every stable nucleus at three crown pressures. Along each curve the smallest nuclei have the highest gas pressure — surface tension squeezes them — and the gas pressure falls as the nucleus grows, ending at the fold. At the fold it has a value that can be written down exactly: differentiating the balance and setting it to zero gives

$$\frac{K}{R_c^3} = \frac{p_v - p}{2},$$

half the crown’s tension, plus half the vapour pressure. At −10 kPa that is 6.2 kPa, at −30 kPa 16.2, at −60 kPa 31.2 — the dots at the curves’ ends.

The consequence follows at once. A nucleus can gain air only where the water’s dissolved-air pressure is above its gas pressure, and every stable nucleus has a gas pressure at least as high as the fold’s. So if the water holds less air than half the crown’s tension, no nucleus on its stable branch can gain air at all; every one of them loses it and shrinks. The dashed lines are water 10, 30 and 60 per cent saturated. Against a crown at −60 kPa, water 10 or 30 per cent saturated lies wholly below every curve: every nucleus dissolves. Only in water 60 per cent saturated can the largest nuclei, near the fold, gain air.

That is the essay’s first answer, and it runs against the intuition the question carried. A crown under tension is not a place where gas gathers; in degassed water it is a place where small nuclei lose gas, because their surface tension holds their gas at a pressure the water cannot match.

A verdict in milliseconds #

The second answer is about time. The figure lodges four nuclei at a 14-metre crown, where the pressure is −36.7 kPa, in water half saturated with air, and follows their radius. The two smaller, 0.7 and 0.9 microns at the reservoir, have gas pressures at the crown above the water’s and shrink away, in 11 and 35 milliseconds. The two larger, 0.95 and 1.03 microns, have gas pressures below it; they gain air, their gas pressure falls further as they grow, and they run away to the fold and break the water, in 57 and 6.7 milliseconds.

Milliseconds, because diffusion across a micron is fast: the time for a molecule to diffuse a distance R is about $R^2/D$, half a millisecond for a micron. A nucleus lodged at a crown reaches its verdict — dissolve, or break the siphon — within a tenth of a second. There is no slow gathering and no borrowed time. The siphon’s life at a given height is either for ever, if no nucleus that can grow ever reaches the crown, or as long as it takes one to arrive. Which is a question about the water’s nucleus population, not about diffusion.

The slowest verdicts are the marginal ones. The nucleus of 0.95 microns takes 57 milliseconds because its gas pressure at the crown is only just below the water’s, and it starts gaining air very slowly — the same slowing that every threshold has near its edge.

Where growth is possible at all #

The half-tension rule turns into a line against the siphon’s height. Taller crowns have more tension, so the water may hold more air before any nucleus there can use it: at a 12-metre crown, 9.6 per cent of saturation; at 14 metres, 19 per cent; at 20 metres, 48 per cent. Below the line, only a nucleus that arrives already past its fold — Blake’s static threshold, the earlier essay’s limit — can break the water, and diffusion plays no part.

The line is a necessary condition, not a sufficient one. Above it, only nuclei near enough to their fold can grow, and the small ones still dissolve, as the 0.7- and 0.9-micron nuclei did. What decides a given nucleus is its own gas pressure at the crown, set by its size, against the water’s.

The height dissolved air takes #

Putting both routes to breaking together gives the siphon’s limit with diffusion included: the tallest crown at which the largest nucleus neither breaks the water at once nor has a gas pressure below the water’s. The figure draws it for five gas contents. With the water fully degassed it is the earlier essay’s static curve. For a one-micron largest nucleus, water up to about a fifth saturated changes nothing — 14.41 metres either way — because at that crown’s tension the nucleus’s gas pressure is above what such water can supply. At 30 per cent the limit is 14.31 metres, at half saturation 13.61, and in water saturated with air 10.71.

Larger nuclei are more sensitive, because their surface tension holds their gas less tightly. A two-micron nucleus loses height from about a tenth of saturation and falls from 11.87 to 5.88 metres in saturated water. A five-micron nucleus in saturated water breaks a siphon at 2.4 metres, and that number is not new: it is the air that breaks a siphon nothing else can, the ordinary siphon of ordinary water whose crown releases its dissolved air, arrived at here from the other end. Small nuclei are the opposite: a three-tenths-of-a-micron nucleus loses only 10 centimetres of its 27 metres even in saturated water.

Degassing has a knee #

Plotted against gas content, each curve is flat and then falls. The knee is where the water’s dissolved-air pressure reaches the gas pressure of the largest nucleus at its static limit — near a fifth of saturation for a one-micron nucleus, near a tenth for two microns. Degassing below the knee cannot raise the crown, because diffusion has already stopped mattering there; only removing the largest nuclei can. Degassing above the knee is worth exactly the height the figure shows.

That is a practical rule for anyone trying to run a tall siphon, or a water tunnel, or a high-tension experiment on water. Boiling and cooling water under a vacuum takes it to a few per cent of saturation, well below either knee; beyond that, effort spent on degassing buys nothing, and effort spent on filtering and on clean surfaces buys everything. It is the same conclusion a breaking strength that is the size of a flaw reached about measurements of water’s tensile strength, where every result was really a statement about the largest flaw in the sample.

Three clocks at a crown #

A running siphon has three times in it, and the millisecond verdict is the shortest by far. The slowest is the siphon’s own: water at a few centimetres a second, the speed at which the tallest siphons run, takes tens of seconds to climb the rising leg, and a nucleus carried in the flow spends of order a second in the stretch of hose near the crown where the pressure is lowest. Next is the diffusive verdict, tens of milliseconds. Fastest is a nucleus’s own inertia: once past its fold, a micron bubble grows to break the column in about a fifth of a microsecond, the time a threshold that is also a duration found governs short pulses of tension.

The ordering matters twice. Because diffusion is fast compared with the transit, even a nucleus carried through the crown rather than lodged at it has time to reach its verdict: in the second it spends near the crown, a nucleus that can gain air will grow to its fold, and one that cannot will shrink. So the half-tension rule applies to a flowing siphon’s passing nuclei as well as to its lodged ones. And because inertia is fast compared with diffusion, the break itself is abrupt: a nucleus creeps towards its fold over milliseconds and then leaves it in a fraction of a microsecond. The siphon runs until it suddenly does not, which is what siphons of degassed water are reported to do.

A siphon’s hydraulics move the verdict too. The margin friction lends a siphon found that friction in the hose always raises the crown’s pressure above what a frictionless hose would have there, by an amount decided by where along the hose the crown sits. Every pascal of that margin is a pascal of tension a nucleus at the crown does not have to hold, and it shifts every verdict towards dissolving; a siphon whose crown sits where friction lends it most can carry slightly more dissolved air at the same height before its largest nucleus turns.

A water tunnel’s nuclei #

The same verdict runs in every water tunnel. Whether a propeller or a hydrofoil tears the water in a test depends on the nuclei the tunnel’s water carries, and tunnels are run with their water deliberately degassed or deliberately seeded to control them. The rule here says what degassing can and cannot do to that population. In the low-pressure region on a blade, as at a siphon’s crown, a nucleus gains air only if the water’s dissolved air is at a higher pressure than the nucleus’s gas; and the time a nucleus spends in that region — a millisecond on a fast blade — is short enough, for nuclei of a few microns, that diffusion only begins to act. So in a tunnel it is the nuclei’s size distribution, set by filtering and by the tunnel’s resorber, that decides inception, and the dissolved-gas content that decides how fast the cavities that do form fill with gas and how violently the bubble that hammers collapses. The two knobs a tunnel has correspond to the two curves in the last figure: one moves the knee, the other moves along the flat part.

Why small free nuclei cannot be the ones #

The model has an uncomfortable consequence that is worth stating rather than hiding. A free spherical nucleus of a micron, sitting in water saturated with air at one atmosphere — the reservoir itself — dissolves: its surface tension holds its gas at 2.4 atmospheres, far above what the water can match, and the ledger’s third check finds it gone in 14 milliseconds. In degassed water it goes faster. Free nuclei of this size cannot persist in any water long enough to be carried to a crown.

This is an old puzzle, and its answer is that real nuclei are not free spheres. Harvey argued in 1944 that the nuclei that break water are pockets of gas in hydrophobic crevices, where the meniscus curves the other way and the gas can sit at a pressure below the water’s, stable indefinitely; later models add skins of surface-active material that stop a bubble dissolving. A crevice nucleus’s threshold, its gas pressure and its response to tension all differ from a free sphere’s. The free-sphere calculation is exact for what it describes; what it describes is the simplest nucleus, and the crevice is the one a real siphon meets. The half-tension rule does carry over in spirit: whatever holds a nucleus’s gas, it can gain air only when its gas pressure is below the water’s, and near its threshold that gas pressure is set by the tension.

What was checked #

Three checks. At three crown pressures the fold’s gas pressure equals half the tension plus half the vapour pressure to rounding error, and the stable branch ends exactly where the gas content reaches the fold’s. The growth time, marched in the gas content with the radius found at every step, was computed a second way: the same time written as an integral over the bubble’s radius along its stable branch, where the gas content is known in closed form, and done by Simpson’s rule. For a one-micron nucleus at −35 kPa in water 60 per cent saturated the two give 13.72 milliseconds to three parts in ten thousand. And a free one-micron nucleus at one atmosphere in saturated water dissolves, as the Laplace pressure says it must. The tests refuse a negative gas content, a fold asked for above the vapour pressure, and a negative saturation.

What the picture cannot show #

Convection. The diffusion is to a sphere in still water. A nucleus lodged at a crown sits in flowing water, which renews the dissolved air at its surface and speeds the exchange. That shortens every time drawn and changes no verdict, since the direction of the exchange is set by the two pressures alone.

Crevices, as above: the nucleus is a free sphere.

One gas. Air is two gases, oxygen and nitrogen, with different solubilities and diffusivities; treating it as one gas with averaged properties changes the times by tens of per cent and the thresholds not at all.

The convention: saturation at one atmosphere #

A gas content is quoted as a fraction of what the water would hold saturated with air at one atmosphere and 20 °C, so that f = 1 is tap water left to stand and f = 0.05 is water well degassed. The dissolved air’s partial pressure is then f atmospheres, and every comparison in the essay is between that pressure and a nucleus’s gas pressure.

Who worked it out #

The critical radius at which a gas nucleus in a liquid under tension becomes unstable is Blake’s, from 1949, following Harvey and colleagues’ crevice model of 1944. Epstein and Plesset gave the diffusive growth and dissolution of a gas bubble in 1950, including the finding that a free bubble in saturated liquid always dissolves because of its surface tension. Tall siphons of degassed water were demonstrated in the nineteenth century and have been run to heights well above the barometric one since; where a liquid does pull is the essay on what water’s cohesion allows before any nucleus is involved.

Still open: a crevice at the crown #

The free sphere cannot be the nucleus that breaks a real siphon, and the next calculation should be the one that can. A conical crevice in the hose wall holds a pocket of gas behind a meniscus whose contact angle is set by the wall; the pocket’s gas pressure depends on where the meniscus sits in the cone and on its curvature, which changes sign as the meniscus advances. Under the crown’s tension the meniscus retreats towards the crevice’s mouth. The calculation asks at what tension the meniscus reaches the mouth and the pocket escapes, how that threshold depends on the cone’s angle and the wall’s contact angle, and whether a crevice nucleus in degassed water under tension gains air or loses it — which decides whether a siphon’s height limit drifts upwards or downwards as it runs.