The bottle that pours at one rate however full it is

The pressure that only knows depth set out the rule that every essay on still water since has used: the pressure at a point is the pressure at the surface plus the weight, per unit area, of the column of water above the point. The shape of the vessel does not enter, nor the amount of water, only the depth. The rule has two terms, and nearly every application treats the first as fixed — the surface is open to the air, and the air presses on it at one atmosphere. The floodgate that opens at a chosen depth used the second term to make a gate swing open at a level of its choosing, and the gate’s whole design was a statement about where the weight of the water acts.

The first term can be put to work too. Close the top of a vessel and the pressure on the water’s surface is no longer the atmosphere’s; it is whatever the air trapped above the water has, and that can change as the water moves. Edme Mariotte described a bottle in which this is arranged deliberately, in his treatise on the motion of water published in 1686, two years after his death. It is a bottle with a hole near its foot, sealed at the top with a stopper, and a straight tube pushed through the stopper and down into the water, open at both ends. Water runs out of the hole. Air comes in, bubble by bubble, through the bottom of the tube. And the flow from the hole does not slow down as the bottle empties.

The one point that is pinned #

The rule of depth says how pressure changes from one height to another in still water. It does not say what the pressure is anywhere until one point is fixed, and in an open vessel the fixed point is the surface. The trick of Mariotte’s bottle is that the fixed point is somewhere else.

While air is bubbling in through the tube, the bottom of the tube is a place where the water touches the outside air. The air inside the tube is at atmospheric pressure — it is connected straight through the open top — and the water at the tube’s mouth is in contact with that air, separated only by the curved surface of the bubble that is forming there. So the pressure in the water at the height of the tube’s tip is atmospheric, less a small correction for the bubble’s curvature that will matter later. From that one pinned point the rule of depth fixes every other pressure in the bottle. Below the tip the pressure rises by $\rho g$ per unit of depth, as in any still water, and the outlet, at a depth $h$ below the tip, is at $\rho g h$ above atmospheric. Above the tip the pressure falls by the same rate going upward, and the water’s surface, a height $L - z_{\text{tip}}$ above the tip, is below atmospheric by $\rho g (L - z_{\text{tip}})$.

That is all there is to it, and the first figure shows it at three levels. Each line is the same straight hydrostatic profile, shifted so that it passes through atmospheric pressure at the tube’s tip. As the bottle empties, the line simply gets shorter at the top; the bottom end, at the outlet, does not move. An open bottle’s lines are pinned at the surface instead, and their bottom ends move left as the level falls — 2,937 pascals at the outlet when the bottle is full, 1,273 when it has fallen to 13 cm.

The water that hangs from a partial vacuum #

The air pocket at the top of the sealed bottle is what makes the arrangement possible, and it is doing something worth stating plainly: it is holding the water above the tip up.

A column of water standing above a point where the pressure is atmospheric, with less than atmospheric pressure on top of it, is a column hanging from a partial vacuum. It is the same arrangement as the water in a drinking straw closed with a finger, or the mercury in Torricelli’s barometer, and the height a siphon cannot pass found its limit: the atmosphere can hold up about ten metres of water and no more. A Mariotte bottle uses a tiny fraction of that. A 30 cm bottle with its tip 10 cm from the bottom needs its pocket to be 20 cm of water below atmospheric, two per cent of an atmosphere, and the pocket gets there by expanding by two per cent of its own volume. In the calculation behind the figures, with a 40 cm³ pocket, that takes a tenth of a second of outflow, after which the suction has caught up with the water above the tip and the first bubble breaks away from the tube.

From then on the suction is not set by the pocket at all. It is set by the water. Every time the outflow lowers the level a little, the pocket expands a little and its pressure falls a little further below the hydrostatic requirement; the water at the tube’s mouth is then below atmospheric, the air in the tube pushes in, a bubble rises through the water into the pocket, and the pocket’s pressure is restored. The tube is a valve that opens whenever the pocket is too far below the pressure the water’s weight demands, and it admits exactly the volume of air that replaces the water that has left. The pocket’s suction falls in a straight line as the level falls, because the water above the tip gets shorter. When the level reaches the tip there is no water above it left to hold up, the tube’s mouth is in the air, and the bottle is simply an open vessel.

Hanging is not pulling #

The phrase “hangs from a partial vacuum” invites a confusion that is worth heading off, because there is a case where water really is pulled. The column that is pulled, not pushed followed water up a tree, where the absolute pressure in the sap falls below zero — the water is under tension, held together by the cohesion of its molecules, and stretched like a rope. Nothing of that kind happens in the bottle. The pocket is below atmospheric pressure but far above zero: at its most extreme, in the full bottle, it is at about 99.3 kilopascals against the atmosphere’s 101.3. Every point in the water is still being pushed, from every side, by a pressure that the push that has no direction showed has no preferred direction; the water at the top is simply being pushed less hard from above than the water at the tube’s mouth is from below, and the difference is the weight of the water between them.

The tube shows this directly, because it is a manometer. While the bottle bubbles, the water inside the tube stands no higher than the tip — the column of air above it is at atmospheric pressure, and the water at the tip is too. Before the first bubble, while the suction is still building, the water in the tube stands somewhere between the tip and the bottle’s level, below the bottle’s surface by exactly the pocket’s suction expressed in centimetres of water. Watching the tube’s water fall to the tip and the first bubble appear is watching the pocket’s pressure reach its working value.

In a real bottle the tube starts full of water to the level of the water outside it, and the start-up is slower than the figure’s: the outflow must first push the water in the tube down to the tip — the tube’s level falling below the bottle’s by exactly the suction that has been built — before air can reach the bottom. That is a matter of seconds for a narrow tube and changes nothing afterwards.

A steady pour from an emptying bottle #

The outlet does not know any of this. It knows only the pressure just inside it, and that pressure is pinned at $\rho g h$ above atmospheric, with $h$ the height of the tube’s tip above the outlet. Water leaving through a small hole under a pressure difference $\Delta p$ does so at a speed close to $\sqrt{2\Delta p/\rho}$, the rule Torricelli found for an open tank, reduced by a coefficient of about 0.61 for the way a jet narrows after leaving a sharp-edged hole. Here the rule serves only as the meter that reads the head. In an open bottle the head is the level and falls as the bottle empties. In the sealed bottle it is the tip’s height and does not.

The open bottle starts at 10.46 millilitres a second and slows steadily, its flow falling in a straight line with time, which is what a flow proportional to the square root of a level that falls in proportion to the flow produces. The sealed bottle delivers 5.81 millilitres a second, and it delivers that rate, unchanged to the second decimal place, for four and a half minutes while its level falls from 30 to 10 cm. Then the level passes the tip. The pocket’s suction drops to zero, the tiny surcharge the bubble had been taking is released — the small step up in the curve — and from there the bottle drains as an open one would from a 10 cm fill.

The steadiness is bought with the head. The sealed bottle runs at the pressure of 9.3 cm of water for its whole working life, while the open bottle starts at 30 cm and takes most of its time at depths above 10. So the open bottle empties first, in 6.9 minutes against 8.2. What the sealed bottle offers is not speed but a flow that is a property of where the tube was set, chosen once and held while the reservoir runs down.

The tube decides and the bottle does not #

The claim can be put as a test. Set the tube’s tip at different heights, fill the bottle to different levels, and see what the steady flow depends on.

The two curves lie exactly on each other as long as the tip is under water. The flow rises as the square root of the tip’s height, so that four times the height gives twice the flow, and the level of the water in the bottle is not in it anywhere. Raise the tip above the surface and the tube no longer touches the water. The pocket is then connected straight to the outside, the bottle is open, and the flow is set by its level; the curves separate there and run flat, at each fill’s own open-bottle flow.

The same independence holds for everything else about the bottle. A wider bottle holds more water and runs for longer at the same rate. A taller one, filled higher, has a stronger partial vacuum in its pocket and the same rate. The bottle’s shape does not appear, which is the hydrostatic paradox of the pressure that only knows depth applied twice: once above the tip, where the shape of the water does not change the suction the pocket needs, and once below it, where it does not change the pressure at the outlet.

The arrangement is in daily use in a degenerate form. An upturned bottle standing in a dish — the drinking fountains sold for poultry and pets, and the reservoirs that fed the oil lamps of the early nineteenth century — holds the level in the dish at the bottle’s mouth: the mouth is the tube’s tip, and air enters only when the dish’s level falls below it. Laboratory chemists use Mariotte bottles to feed columns at a fixed pressure, and soil scientists measure how fast water soaks into the ground with infiltrometers and permeameters whose reservoirs are Mariotte bottles, so that the water is supplied at a constant pressure while the reservoir drains, and the rate at which the reservoir falls is the rate at which the soil takes water.

The toll a bubble takes #

The one correction is at the tube’s mouth. Air does not leave the tube as soon as the water there drops below atmospheric. It has to form a bubble first, and a bubble is a curved surface with a pressure step across it. The small bubble blows up the big one found that step for a soap film; for a single surface between air and water it is $2\sigma/R$, where $\sigma$ is the surface tension and $R$ the radius of curvature. As a bubble grows at the end of a tube of inner radius $r$, its radius of curvature first falls, to a minimum of $r$ when the bubble is a hemisphere, and then rises again. So the air must be pushed in with an overpressure of at least $2\sigma/r$ to get past the hemisphere, and the pocket must be that much further below atmospheric before a bubble can leave.

The pinned pressure at the tip is therefore not atmospheric but atmospheric less $2\sigma/r$, and the head the outlet feels is the tip’s height less $2\sigma/(\rho g r)$. For water, with a surface tension of 0.073 newtons per metre, that is 7.4 mm for a tube 2 mm in radius — the 7.4 mm the earlier figures have carried — and 29.7 mm for a fine tube half a millimetre across. A fine tube costs a sixth of the flow from a 10 cm head, which is the same capillary effect that how high water will climb found lifting water up a narrow tube, here acting in reverse to hold air out of one.

Wider tubes cost less, but only up to a point. Once the tube’s radius approaches the capillary length, $\sqrt{\sigma/\rho g}$, 2.7 mm for water, the bubble at its mouth is large enough for its own buoyancy to matter: it no longer swells as a hemisphere and detaches, but sags and breaks away earlier, and the simple formula overstates the toll. The constant head of a well-made Mariotte bottle is therefore set by a tube a few millimetres across, wide enough that the bubble’s surcharge is small and steady, and the surcharge is measured once and subtracted rather than ignored.

What the figures leave out #

The figures treat the outflow with a fixed discharge coefficient, which is accurate for a sharp-edged hole at these speeds but not for a long outlet tube, where viscous losses grow with the flow and lower it further. They treat the bubbling as smooth. In fact air enters as discrete bubbles, each bubble restoring the pocket’s pressure in a step, so the pressure at the outlet carries a small sawtooth of the size of the bubble’s surcharge, at the bubbling frequency; at the flows drawn here, with each bubble from a 2 mm tube holding about a tenth of a millilitre, that is some sixty bubbles a second, and a ripple of a few millimetres of head.

They leave out temperature. While the bottle bubbles, the pocket’s pressure is fixed by the water, so a pocket that warms expands at that fixed pressure and pushes out extra water. A litre of air warming by one kelvin swells by about 3.4 millilitres, which is a second’s worth of the flow drawn here, delivered over however long the warming takes. A bottle in sunlight or in a draught runs fast by day and slow by night, and field instruments built on the principle shade or insulate the reservoir for that reason. And a leak anywhere above the water undoes the whole arrangement: the pocket comes to atmospheric pressure, the bottle becomes open, and the flow jumps up to the open-bottle value for the level — a test for a bad seal that needs no instrument.

Still open: the rhythm of the bubbles #

How the air actually enters is less settled than what it accomplishes. A gas fed through a submerged nozzle from a chamber forms bubbles at a rate set by the gas flow, the nozzle’s size, the liquid’s properties and — importantly here — the volume of gas the chamber holds between the supply and the nozzle, because a large chamber acts as a spring that lets the pressure at the nozzle swing within each bubble’s growth. In the bottle that chamber is the tube itself. Experiments on bubbling from a nozzle find a regular rhythm at low flows that can give way, as the flow rises, to bubbles that pair up, alternate in size, and eventually follow no repeating pattern, a sequence with the same structure as the map a dripping tap turns out to be. Which regime a given Mariotte bottle is in, and how far its unsteady bubbling limits the steadiness of a very slow, very precise constant-head supply, is decided case by case rather than predicted, and models that couple the bubble’s growth to the gas column feeding it are still being refined.

The hydrostatic statement is short. In a sealed bottle bubbling through a tube, the pressure is pinned at atmospheric — less a bubble’s $2\sigma/r$, 7.4 mm of water for a 2 mm tube — at the height of the tube’s tip, so the outlet works against the tip’s height whatever the level: a bottle filled to 30 cm with its tip at 10 delivers 5.81 mL/s, unchanged for four and a half minutes, while its air pocket falls from 2.0 kPa below atmospheric to 73 Pa holding up the water above the tip. A still fluid knows only depth, and Mariotte chose the point the depth is measured from.