The siphon that does not need the air

A siphon lifts liquid over a barrier and delivers it below the level it started from, with nothing driving it but the arrangement of the tube. It is one of the oldest devices in engineering and one of the most confidently misexplained.

The standard account is that atmospheric pressure pushes the liquid up the short leg and over the crown. It has a testable consequence, and the test is easy: if the atmosphere is doing the driving, raising the hump should make the siphon work harder and run slower.

It does not. The height of the hump is not in the flow rate at all.

The flow rate, and what is missing from it #

Apply Bernoulli from the free surface of the source to the outlet — a steady streamline in a fluid whose viscosity is being neglected, which is where the equation holds. The pressure is atmospheric at both ends, because both are open to the air. What is left is

$$V = \sqrt{2g\,\Delta z}$$

with Δz the drop from the source surface to the outlet. That is Torricelli’s expression, the same one a hole in the side of a tank obeys, and the crown’s height appears nowhere in it.

A siphon over a garden wall and a siphon over a kerb, draining to the same place from the same source, run at exactly the same rate. If the atmosphere were doing the pushing, the taller one would be doing more work against gravity on the way up — and it is not, because whatever is spent going up is recovered coming down. The two legs cancel, and what is left is the difference in level between the ends.

What drives a siphon is the drop, and only the drop. The same statement in energy terms: the liquid is falling from the source surface to the outlet, and everything in between is a path.

What the atmosphere is actually for #

None of that means the atmosphere is irrelevant, and the second job is where the ten-metre limit comes from.

The pressure at the crown is below atmospheric — it must be, since the liquid there is above the source and moving. Bernoulli gives it as

$$p_{\text{crown}} = p_{\text{atm}} - \rho g h_{\text{rise}} - \tfrac{1}{2}\rho V^2$$

and it falls one atmosphere for every 10.3 metres of rise. The atmosphere is not pushing the column along; it is preventing it from coming apart, by keeping the pressure everywhere in the tube above the point at which the liquid boils.

That symmetry is worth pausing on. If the atmosphere were driving the flow, there would be an asymmetry somewhere: a net force in the direction of travel. There is not. The pressure profile is a function of height alone, and the tube’s two legs mirror each other about the crown. What is asymmetric is the geometry — one end is lower than the other — and that is where the driving comes from.

Where the ten metres comes from, and the coupling nobody expects #

The ceiling on the crown’s height is where the crown pressure reaches the vapour pressure:

$$h_{\max} = \frac{p_{\text{atm}} - p_v - \tfrac{1}{2}\rho V^2}{\rho g}$$

which for water at twenty degrees, at rest, is 10.11 metres — the barometric height less the vapour head.

The velocity term in that expression is the interesting part, because it produces a coupling a reader would not guess: running the siphon faster makes it break sooner.

One metre of drop costs one metre of hump, exactly — which follows from the arithmetic and is not obvious from the description. A siphon designed at its height limit will fail when the outlet is lowered, which is the opposite of what an intuition about “more driving force” would predict.

The experiment the standard account forbids #

If the atmosphere holds the column together by pushing on it, then a siphon over more than ten metres of water is impossible, and a siphon in a vacuum is impossible.

Both have been done.

Water that has been carefully degassed can sustain tension — a negative absolute pressure — because there is nothing in it for a cavity to nucleate on. Under those conditions the liquid column behaves like a chain: the descending leg pulls the ascending one over the crown, and the atmosphere’s role disappears entirely. Siphons have been run at fifteen metres this way, and in a vacuum chamber with degassed liquid the arrangement works with no gas anywhere.

That does not make the atmospheric account useless; ordinary water is full of dissolved gas and nucleation sites, so for any practical siphon the ten-metre limit is real and the atmosphere is what enforces it. It makes the account incomplete — a statement about a limit, mistaken for a statement about a mechanism.

The parallel with the site’s cavitation rung is exact. There a body moving through water drops the pressure below vapour and the liquid tears; here a column of water lifted too high does the same thing; and in both cases what decides the threshold is whether the liquid has anything in it for a bubble to start on. The tensile strength of pure water is enormous — tens of megapascals in careful experiments — and the tensile strength of tap water is essentially zero.

What is holding the column up, if not the air #

The account so far has been negative — the atmosphere is not driving the flow — and it is worth stating the positive version, because “gravity does it” is too quick.

Consider the liquid in the upper leg, above the source’s level. It is being held there against gravity, and something is holding it. In an ordinary siphon that something is the pressure difference between the crown and the ends: the pressure falls with height in the tube exactly as it does in a static column, so every element of fluid has a slightly higher pressure below it than above it, and the difference supports its weight. That pressure field exists because the atmosphere sets the value at both ends, and this is the sense in which the atmosphere is doing something real.

In a degassed siphon above the barometric height the same job is done by tension. The liquid’s own cohesion carries the load: the pressure is negative, the column is being stretched rather than squeezed, and the descending leg pulls the ascending one over the crown like a chain over a pulley.

Both are correct descriptions of their own case, and the argument in the literature is largely two groups describing different liquids. What neither can be is the driver, because a support force does no net work on a steady flow: the liquid rises through the same height it later falls through, and the pressure field that holds it up returns everything it took.

Why raising the hump does slow a real siphon #

There is an observation that keeps the atmospheric account alive, and it deserves to be met head on, because it is correct: raise the hump on an actual garden hose and the siphon really does run more slowly. Anybody who has done it has seen it.

The reason is friction, and the frictionless expression above has none. With it, the exit speed becomes

$$V = \sqrt{\frac{2g\,\Delta z}{1 + fL/D + \sum K}},$$

and the crown height enters through $L$ — because a taller hump needs more tube. It does not enter the physics; it enters the plumbing.

The size of that term is the surprise. A garden hose of thirteen millimetres bore, five metres long, with a friction factor near 0.03, has $fL/D \approx 12$ — twelve times the unity it sits beside, so the frictionless answer is not a small overestimate but a factor of three and a half. Add two metres of hump, which means about four more metres of hose, and $fL/D$ rises to about 21 and the delivery falls by roughly a quarter.

So the observation is real, the size of it is right, and the cause is not the atmosphere. A viscous siphon is dominated by its own pipe loss, and everything about the arrangement that lengthens the pipe costs flow.

Which supplies the experiment that separates the two accounts. Take a hose long enough to have slack, and change the crown height without changing the tube length — raise the middle and let the slack out of the legs, keeping source and outlet fixed. The atmospheric account predicts a slower flow, since the liquid is being pushed higher. The correct account predicts no change whatever, since neither $\Delta z$ nor $L$ has moved.

Nothing changes. That is the whole refutation, performed with a hose and a bucket, and it is the version worth carrying because it removes the confounding variable that has been doing the persuading for three centuries.

There is a second consequence worth noting, because it goes the other way. Friction reduces the velocity at the crown, so it reduces the velocity head there, so it raises the height limit — a long thin siphon can go higher before it breaks than a short fat one delivering more. The ceiling figure above is drawn without friction and is therefore conservative in that respect and optimistic in another: it also omits the pressure lost to friction along the ascending leg, which lowers the crown pressure directly. The two corrections have opposite signs and neither is computed here.

Which parts of the usual account survive #

It is worth separating them, because the popular explanation is not simply wrong.

True, and the mechanism. The level difference drives the flow, and the flow rate is √(2gΔz).

True, and a limit rather than a mechanism. For ordinary liquids, atmospheric pressure sets how high the crown may be. Remove the atmosphere and an ordinary siphon fails — not because the driving force went away, but because the liquid boiled.

False. That the atmosphere pushes the liquid over the hump. If it did, the hump’s height would be in the flow rate, and it is not.

The failure mode is the field’s standing one: the same shape as the Coandă explanation and the bathtub’s, where something real is given a job it does not do. What makes this one durable is that the atmospheric account gets one prediction spectacularly right — the ten-metre limit — and being right about a consequence is easily mistaken for being right about a cause.

Two siphons that are not tubes #

The mechanism generalises past the tube, and two other arrangements are worth naming because they sharpen what the essential ingredient is.

A siphon over a weir. Large reservoir spillways are sometimes built as siphons: a hooded passage whose crest is above the water level and whose outlet is far below. Once primed it runs full and delivers far more than a free weir of the same width, because the driving head is the whole drop to the outlet rather than the depth over the crest. It is the same arithmetic as the specific-energy argument a weir obeys with the critical section removed, and its failure mode is priming: a siphon spillway that breaks its seal reverts to a weir, and its capacity collapses without warning.

A chain siphon, with no liquid at all. A chain hanging over a pulley with more of it on one side runs exactly as a siphon does, driven by the difference in the two hanging lengths and needing no atmosphere whatever. The parallel is instructive rather than decorative: it is the mechanical analogue the degassed-water experiments realise, and it makes clear that what is needed is continuity — something that transmits tension or pressure along the path — rather than air.

What both share with the tube is that the driving is a difference in level at the two ends and the path between them is irrelevant. That statement is the essay’s whole content, and it survives being detached from any particular liquid, tube or atmosphere.

What is not computed here #

No friction. The tube is treated as frictionless, so the flow rate is an upper bound; a long thin siphon delivers considerably less, and the friction factor is what would say how much less. Adding it changes the flow rate and changes nothing about which height appears in it.

No entry loss and no vena contracta. The inlet is assumed smooth. A sharp-edged one contracts the flow exactly as an orifice does, which reduces the delivery by a factor this site computes elsewhere and does not apply here.

No starting problem. Everything above concerns a siphon already running. Getting one started — filling the tube, or drawing the liquid over — is a separate question, and it is the one place where the atmosphere genuinely does push something.

Nothing unsteady. A siphon whose source level is falling is unsteady, its flow rate declines as the drop shrinks, and the quasi-steady treatment used here is an approximation whose error nobody has computed on this site.

No flow field anywhere. A siphon is a one-dimensional duct and Bernoulli along it is exact; drawing streamlines inside the tube would be decoration of the kind this site’s first invariant forbids.

A sixth figure, and what a reader should take from the set #

Set beside the earlier profile, that figure is the whole argument in two pictures. The shape of the pressure curve changed completely; the flow rate did not change at all.

Who found it, and when #

Siphons appear in Egyptian reliefs from about 1500 BC and in Hero of Alexandria’s writings. The atmospheric explanation dates from the seventeenth century — from Torricelli, Pascal and Boyle, who had just discovered that the atmosphere had a weight and reasonably enough used it to explain everything.

The argument has never entirely stopped. In 2010 a physicist noticed that the Oxford English Dictionary defined a siphon in terms of atmospheric pressure and had it changed; the same year a paper demonstrated a working siphon at fifteen metres in degassed water; and there is a small, courteous literature of papers and rebuttals about which account is right, some of it arguing that gravity and molecular cohesion do all the work and some of it insisting the atmosphere is essential.

The site’s own reading is the division above. The flow rate contains the drop and not the hump; the ten-metre limit is real for ordinary water and is a limit on holding the column together; and both statements come out of the same two lines of Bernoulli, which was written down in 1738 and settles the question completely.

The refutation index, at ten entries #

This is the tenth explanation this collection has stated fairly and then tested, and with the field at ten the shapes have sorted themselves into three kinds.

A statement that is simply false. Equal transit time is the type specimen: parcels do not meet at the trailing edge, the site’s own solver shows they do not, and nothing survives.

A statement that is true and given a job it cannot do. The suction at a jet pump’s nozzle, the Coandă effect on a wing, the Coriolis term in a bath: all real, all present, none of them the reason for the outcome they are cited for.

A statement that is right about a limit and wrong about a mechanism. This one, and the airspeed indicator’s. The atmospheric account predicts the ten-metre ceiling correctly and would be very hard to dislodge on evidence alone, because its one testable consequence is right.

The third kind is the hardest to correct and the most interesting to hold, because being right about a consequence is genuine evidence — and the only thing that separates it from being right about a cause is finding a second consequence the two accounts disagree about. Here that second consequence was the flow rate’s independence of the crown height, and it was available all along in an expression anybody could write down.

Where this field goes next #

Ten misconceptions are now tested rather than asserted against, and the pattern has become clear enough to name. Every one of them takes something true — a low pressure, a curvature, a rotation, an atmosphere, a photograph — and promotes it from a participant to the cause. The repair is never rhetorical: it is a comparison of magnitudes, a control volume drawn where the faces are known, or an expression examined for what is missing from it.