A crevice keeps the nucleus a free bubble loses

A crown dissolves its nuclei or breaks on them, and fast followed a single free bubble held at the crown of a siphon running with degassed water. A siphon taller than ten metres runs with its crown under tension, below the vapour pressure and below zero, and it breaks when some bubble in the water passes Blake’s threshold and grows without limit. That essay found that the bubble decides fast. Its gas exchanges with the water in milliseconds, and every stable free bubble dissolves unless the water holds more dissolved gas than half the crown’s tension. In degassed water under a crown at tens of kilopascals of tension, a free bubble is gone almost as soon as it arrives.

That left a puzzle the essay stated plainly, and it is the same puzzle that runs under the siphon that does not break and the air that breaks a siphon nothing else can: what, in water with no bubbles left, is there to break. Degassed siphons run for hours and then break, and they break at heights that vary from run to run and hose to hose. Something keeps a nucleus alive at the crown. The essay named the candidate the cavitation literature has relied on since the 1940s: a pocket of gas held in a crevice of the wall, behind a meniscus that the wall’s own contact angle curves, so that the pocket’s gas pressure is not the bubble’s and need not exceed what the water can supply. It asked at what tension such a pocket escapes, how that depends on the crevice’s angle and the wall’s wettability, and whether a crevice nucleus under tension gains air or loses it — which decides whether a siphon’s height limit drifts up or down as it runs.

A pocket behind a meniscus #

The crevice here is a cone in the hose wall: a mouth of radius $a$ on the surface, a half-angle of ten degrees, and an apex inside the wall. Gas and water vapour fill it from the apex out to a meniscus, a spherical cap whose rim touches the cone where its radius is $r$. Laplace’s law, written with the cap’s signed curvature $s$ — positive when it bulges towards the water — gives

$$p_g + p_v - p_l = 2\sigma s,$$

and inside the cone the wall’s contact angle $\theta$, measured through the water, fixes the curvature at $s = \cos(\theta - \beta)/r$. That one expression carries the whole of the crevice’s virtue. On a wall the water wets, $\theta < 90° + \beta$, the cap bulges into the water and the pocket’s pressure exceeds the water’s, as a free bubble’s does. On a wall it does not wet, the cap bulges into the pocket, and the pocket’s pressure is below the water’s — the arrangement in which a free bubble cannot exist at all, because a free bubble’s surface tension always squeezes its gas above the liquid’s pressure. At $\theta = 90° + \beta$ the meniscus is flat and the pocket carries the water’s pressure exactly.

The gas in the pocket is isothermal: its pressure times its volume is fixed by how much gas the pocket holds. At rest, in the hose at one atmosphere, the pocket is placed with its rim half-way out of the cone, which fixes its gas.

The path the pocket follows #

As the siphon starts and the crown’s pressure falls, the pocket’s gas expands, and it follows a definite path of equilibria. First the rim slides out of the cone, at the receding contact angle, the meniscus keeping its curvature in proportion to the cone’s widening radius. Then the rim reaches the mouth and stops, pinned at the sharp edge where the crevice meets the hose wall — an edge, as an edge holds any angle it is given showed, can hold a meniscus at any angle between its two faces’ — and the cap bulges out into the water, its curvature rising until it is a hemisphere on the mouth. Past the hemisphere it grows and its curvature falls again.

Each state needs a particular liquid pressure, and along the path that pressure first falls and then rises. Its minimum is the escape threshold. A crown held above it has a state for the pocket to sit in; a crown below it has none, and the pocket grows into a cavity that fills the hose. For a one-micrometre mouth the threshold is −106 kilopascals on a wall wetted at 40° and −138 on a wall at 110°. The free bubble of the earlier essay, with no wall, had no such path to follow at all once its gas had gone.

The mouth decides the threshold #

The threshold is set almost entirely at the mouth. A crevice with no gas in it, on a wall that does not let it fill, escapes when the cap on its mouth is a hemisphere: $p_l = p_v - 2\sigma/a$, −142.7 kilopascals for a one-micrometre mouth. With its gas at rest the threshold is a few per cent less negative, −138 kilopascals, because the gas helps the cap out. Turned into the height of the tallest siphon whose crown stays above it — with the earlier essays’ hose, ten millimetres bore and its outlet two metres below the source — that is about 23 metres for a micrometre mouth, 58 for a third of a micrometre and over 150 for a tenth.

That makes a crevice behave, at the moment it escapes, like a free bubble the size of its mouth — which is why a degassed siphon is as tall as its largest nucleus gave sensible heights when it assumed a fixed nucleus. The difference is what happens before that moment. A free bubble the size of a micrometre mouth would have dissolved long before the crown reached −140 kilopascals. A crevice does not dissolve, because its gas is not squeezed above the water’s pressure, and the mouth’s size stays the nucleus’s size for as long as the crack is there.

The wall’s wettability, at rest #

At rest, the wall’s wettability matters modestly. A wall the water wets curves the meniscus so the pocket’s gas is at a higher pressure than the water’s, which helps it out, and the threshold is −99 kilopascals at a receding angle of 20°. A wall the water does not wet holds the pocket in, and at the largest angles the threshold is the empty hemisphere’s, −142.7. Between them there is a factor of under one and a half. The crevice’s mouth, which moves the threshold in inverse proportion, is the far larger lever, and a siphon builder who wants height should ask first how smooth the hose’s bore is.

The crevice’s own shape matters less still. From a needle-sharp cone of three degrees to a broad one of forty-five, the threshold of a wetting crevice moves by one kilopascal and a non-wetting one’s by ten. The half-angle changes how much gas the cone holds and tilts the meniscus by β, but at the moment of escape the cap is on the mouth, and the mouth is the same.

Gaining gas, or losing it #

The threshold at rest is not the end, because the pocket is not sealed. Its gas exchanges with the gas dissolved in the water, and the direction is set by the comparison the earlier essay made for the free bubble: gas leaves the pocket if its pressure is above the dissolved gas’s partial pressure, and enters if below. The figure draws the pocket’s gas pressure in the state it holds at each crown pressure, beside the partial pressure of water a third saturated with air — the degassed water of the siphon essays — and of saturated water.

On a wall the water wets, at 20° to 60°, the pocket’s gas pressure is above a third of an atmosphere at every crown pressure, from 56 kilopascals to 240. Gas leaves. As it leaves, the meniscus retreats into the cone, its curvature grows, the pressure rises further, and the loss accelerates: the pocket empties and the crevice fills with water. A wetting crevice in degassed water ends with no nucleus at all. On a wall at 80° the pocket’s gas pressure crosses the water’s under tension, and the outcome depends on the crown. On a wall at 110° the pocket’s gas pressure is below a third of an atmosphere everywhere — 9 to 24 kilopascals — and gas enters.

That is the answer to the earlier essay’s question, and it divides by the wall. A siphon whose hose is wetted by its water loses its crevice nuclei one by one as they fill, and its height limit drifts upwards as it runs, towards whatever nucleus is left. A siphon whose hose is not wetted — a waxed, greasy or fluoropolymer bore — feeds its crevice nuclei from the water, and its height limit drifts downwards.

Where the feeding stops #

A non-wetting crevice does not feed for ever. As gas enters, the pocket grows towards the mouth and its gas pressure rises, until it equals the water’s partial pressure and the exchange stops. The threshold it then has depends on how much gas the water holds. Under a crown held at −50 kilopascals, in water a twentieth saturated, the settled threshold is −139 kilopascals, a margin of 89 beyond the crown. In water a third saturated it settles at −119; at nine-tenths saturated, at −62, a margin of twelve. And in saturated water there is no settled state short of escape: the pocket grows until it escapes at whatever tension the crown holds, which is the reason a siphon of ordinary tap water cannot sustain tension at all.

So the dissolved gas, which the free bubble needed to exceed half the crown’s tension to survive, matters to a crevice in a different way. It does not decide whether the nucleus survives — on a non-wetting wall it always does — but how close to breaking the nucleus settles. Degassing buys margin in proportion.

Why a free bubble cannot do this #

The contrast with the free bubble is the essay’s core, and it is worth stating once without geometry. A free bubble’s surface is curved one way only, convex towards the water, and its surface tension therefore always adds to its gas pressure. Whatever the water does, a small free bubble’s gas is at a higher pressure than the water around it, and in water that is not supersaturated its gas leaves. That is where a liquid does pull turned inside out: the tension that would grow the bubble cannot act until the bubble has survived long enough to meet it.

A crevice lets the surface curve the other way. The wall’s contact angle sets the meniscus’s tilt, and on a non-wetting wall in a narrow cone the curvature is reversed, the surface tension subtracts from the gas pressure, and the pocket can hold its gas below the water’s pressure indefinitely. It is the same geometry that keeps a drop on a waxed leaf rather than spreading, used here to keep gas in the water rather than out of it.

A worked hose #

Take an ordinary flexible hose of a fluoropolymer the water does not wet, its bore scored by the extrusion die with cracks whose mouths are, say, a micrometre across, and fill it with water degassed to a third of saturation. At rest, its crevices hold their gas with thresholds near −138 kilopascals, enough for a siphon about 23 metres tall. Run the siphon at fifteen metres, with its crown near −55 kilopascals. The crevices at the crown draw gas from the water and settle with thresholds near −119: still a margin of over sixty kilopascals, and the siphon runs. Run it at twenty metres instead, crown near −104 kilopascals, and the margin at rest is under forty kilopascals; the settling eats much of it, and the next crevice to arrive at the crown with a slightly larger mouth breaks the column.

The same hose in a material the water wets behaves the other way. Its crevices empty and fill within the first minutes of the siphon’s running, and the column is then held by whatever nucleus is left — a larger defect, a particle, the hose’s seal — at a height the crevices no longer decide. This is one reading of why careful degassed-siphon experiments condition their tubing by running water through it under pressure before a run: a breaking strength that is the size of a flaw applies to the hose as much as to the water.

Checks on the crevice #

Three limits are checked. A non-wetting crevice with no gas must escape at the hemisphere on its mouth, $p_v - 2\sigma/a$, and it does, to rounding. A wetting crevice with no gas must have no pocket at all — its meniscus at rest runs to the apex — and the calculation reports it filled. The pocket placed at rest must need exactly one atmosphere of water around it, and does. And at $\theta = 90° + \beta$ the meniscus must be flat, with no curvature and no pressure difference. The tests also refuse a mouth of zero radius, a cone with no angle and a contact angle of 180°.

What the cone leaves out #

Diffusion’s pace. The direction of gas exchange is computed; its rate is not, and the same crevices seed the cavities a body tears in the water and the bubbles that hammer, where the time a nucleus spends in low pressure is far shorter. A crevice pocket exchanges gas through a meniscus a micrometre across, and its approach to the settled state takes seconds to minutes rather than the free bubble’s milliseconds. A crevice that arrives at the crown with the flow may pass through before it has settled.

Contact-angle hysteresis. The rim slides at a single receding angle. Real walls pin and release, and a rim can stick inside the cone at any angle between the advancing and receding ones.

One crevice. A real bore has many, with a distribution of mouths, and the siphon breaks on the largest that reaches the crown. The calculation gives each one’s threshold; it does not give the distribution.

A rigid crack. The hose’s wall flexes under the crown’s tension, and a soft wall’s crevice opens as the water pulls on it.

The convention: tension as negative pressure #

Pressures are absolute: one atmosphere is 101.3 kilopascals, zero is no pressure, and a crown “under tension” has a negative liquid pressure. The contact angle is measured through the water, so a wetting wall has a small angle. The dissolved gas is given as a fraction of saturation with air at one atmosphere, and its partial pressure is that fraction of an atmosphere. Heights are computed with the earlier essays’ siphon: a ten-millimetre hose, its outlet two metres below its source.

Harvey’s crevice #

Harvey and his colleagues proposed the crevice nucleus in 1944 to explain why water in clean containers cavitates at tensions far below the liquid’s own, and why pressurising water before a test raises the tension it will stand — the pressure pushes the meniscus into the crevice and dissolves its gas. Apfel made the model quantitative in 1970, and Atchley and Prosperetti in 1989 gave the full path of a crevice’s equilibria with contact-angle hysteresis, the form followed here. That a wetting crevice in degassed water fills and a non-wetting one feeds, and what that does to a running siphon, is what this calculation adds to the siphon’s story.

Still open: the crevice that arrives #

The calculation holds the crevice at the crown. In a running siphon the crown’s water is replaced every fraction of a second, but the crevices are in the wall and stay put — except that the crown is a place, and the wall there is the same wall for the whole run. The next calculation follows the crown’s own crevices through a run: their gas exchange at its actual rate, the meniscus moving as gas enters, and the threshold drifting, with a distribution of mouth sizes in the wall. It asks how long a siphon of a given height runs before its crown’s largest non-wetting crevice settles within reach of the crown’s tension — a time the earlier essays’ free bubbles could not have, and the one experiments with degassed siphons actually report.