The teapot effect is a tension, not a pressure

The effect that explains nothing takes the name away from the mechanism: a jet following a curved wall is the normal momentum balance across curved streamlines, and naming it after Coandă adds nothing. The effect that is real, and where it stops gives the name back to the flow it belongs to. A wall jet running into fluid at rest makes all of its suction inside its own thickness, the wall seals the side its entrainment would empty, and the atmosphere presses it on.

That essay lists the demonstrations of the real effect, and one of them — a stream of water from a tap, bending round the back of a spoon — belongs to a different effect altogether. So does the most familiar case of all, the one the Coandă effect is most often invoked for in a kitchen: a slow pour running back under the spout of a teapot. Both are liquids in air, and a liquid in air cannot do what the wall jet does.

Why a liquid in air cannot borrow the atmosphere #

A wall jet of air on a convex surface stays attached because of what is on each side of it. On the outside is still air at atmospheric pressure. On the inside is the wall, and between the jet and the wall there is no way for the surrounding air to get in: whatever the jet entrains from that side is not replaced, so the pressure there falls, and the atmosphere on the outside pushes the jet onto the wall with the difference. The pressure the jet needs to be turned is small against what the atmosphere can lend it.

For an air jet a millimetre thick at fifty metres a second round a five-centimetre radius, the turning needs $\rho U^2t/R = 60$ pascals, and the atmosphere offers a hundred and one thousand. A margin of 1683, which is why an air wall jet’s real limit is not the pressure at all but a viscous separation, where the recovery it has to manage exceeds what its layer can take.

A water sheet flowing off a spout has air on its outside too. But once it leaves the wall at the lip’s edge, it has air on its inside as well — the same atmosphere, at the same pressure — and the air is eight hundred times lighter than the water, so nothing the water does to it produces a pressure difference worth counting. The atmosphere pushes on both faces of a liquid sheet in air equally, and lends it nothing — a push on both sides of a surface is no force on it, however large the push. What the sheet does have, that a jet of air in air does not, is two free surfaces, each pulling along itself with the surface tension. For a two-millimetre water sheet at 0.2 metres a second round a five-millimetre lip, turning needs 13.4 pascals and the two surfaces supply 24.9. A margin of 1.86 — and a margin that, unlike the air jet’s, runs out at a definite speed.

A sheet is a membrane #

The balance is clearest if the sheet is thought of as a membrane rather than as a stream.

A liquid sheet of thickness $h$ has two surfaces, and each pulls along itself with $\sigma$, so the sheet carries a tension $2\sigma$ per unit width — exactly like a stretched rubber membrane, except that its tension does not depend on how far it is stretched. It also carries momentum along itself, a flux of $\rho U^2h$ per unit width.

Bend the sheet round a radius $R$ and both of those give a force normal to it. A tension round a curve pulls the curve towards its centre with the tension over the radius, $2\sigma/R$ per unit area — which is the Young–Laplace relation for two surfaces. A momentum flux round a curve needs a force towards the centre of the flux over the radius, $\rho U^2h/R$, to be turned. So the net force pulling the sheet onto a convex lip is

$$\frac{2\sigma - \rho U^2h}{R},$$

an effective tension $2\sigma - \rho U^2 h$ times the curvature. Positive, and the sheet is pulled round the lip and follows it. Negative, and the momentum wins and the sheet goes straight.

The radius has cancelled. Both terms are forces in proportion to the curvature, so the threshold does not depend on how sharply the lip curves at all. It is a speed:

$$U^{*} = \sqrt{\frac{2\sigma}{\rho h}},$$

which for a two-millimetre sheet of water is 0.270 metres a second, for a one-millimetre sheet 0.382 and for a four-millimetre sheet 0.191.

The speed of waves on the sheet #

That speed has a name and a second meaning, and the second meaning is the surprising part.

It is the Taylor–Culick speed, known since 1959 as the speed at which a punctured soap film retracts: the rim of a hole in a sheet is pulled back by the tension $2\sigma$ and gathers up the sheet’s mass as it goes, and the balance of the two gives $\sqrt{2\sigma/\rho h}$. It is also exactly the speed of transverse waves on the sheet regarded as a membrane: a membrane of tension $T$ and mass $m$ per area carries waves at $\sqrt{T/m}$, and with $T = 2\sigma$ and $m = \rho h$ that is the same expression.

So the teapot’s threshold can be stated in one sentence. A sheet poured slower than the speed of waves along itself is pulled round the lip; a sheet poured faster goes straight.

The same statement for a string is older and more famous. A rope or chain moving along its own length at the speed of transverse waves on it has no preferred shape: the tension round any bend exactly supplies the force its momentum needs to be turned, so it can hold a loop in the air, which is the trick of a lasso and the physics of a chain pouring itself out of a beaker in an arch. A liquid sheet is that rope with its tension supplied by its surfaces. Poured slower than its wave speed it behaves like a rope with tension to spare, and wraps round whatever it is draped over.

The radius that drops out #

The cancellation of the radius was a statement about a thin sheet, and a spout’s sheet is not always thin against its lip. The balance can be written without that approximation: the inner surface is concave towards the liquid and the outer convex, so the two tensions give $\sigma(1/R + 1/(R+h))$ across the sheet, and a plug flow turning round the lip needs $\rho U^2 \ln(1 + h/R)$.

On a twenty-millimetre lip the critical speed is 0.08 per cent above the thin-sheet value; on a five-millimetre lip, 0.94 per cent; on a two-millimetre lip, as thick as the sheet, 4.02 per cent. The deviation falls as the square of the thickness over the radius — measured over a halving of the ratio, the exponent is 1.93 — because expanding both sides of the balance, the first-order terms in $h/R$ cancel as well. The radius drops out of the teapot’s threshold to second order, not merely to first.

That has a practical corollary that runs against intuition. A rounded spout and a moderately sharp one dribble at the same speed. Only when the lip is smaller than the sheet is thick does the geometry start to matter, and by then the membrane picture itself is failing: at a truly sharp edge the sheet does not bend round a radius at all, and what decides whether it follows is where its contact line on the solid can sit — a question of wetting rather than of tension and momentum.

Where a pour dribbles #

On the plane of sheet thickness and pour speed, the threshold is a straight line of slope minus a half on logarithmic axes, and it separates a pour that runs back under the spout from one that leaves cleanly. A slow pour — a two-millimetre sheet at fifteen centimetres a second — sits below the water line and dribbles. The same sheet poured at sixty centimetres a second sits above it and does not.

A thinner stream dribbles at a higher speed. A half-millimetre trickle has a threshold of 0.54 metres a second, twice the two-millimetre sheet’s, so a gentle pour from a nearly full pot — where only a thin sheet crosses the lip — is the case that runs down the outside, even at a speed that would pour cleanly as a thicker stream.

The other liquids move the line in the direction the formula says. Soap halves the surface tension and so lowers the threshold by a factor of 1.44: soapy water pours more cleanly, which is the reason a little detergent is the old cure for a dribbling spout. A light oil has less than half water’s tension and slightly less density, and mercury nearly seven times the tension and thirteen and a half times the density; both lines sit about 30 per cent below water’s.

A slow pour, in litres a minute #

The threshold can also be read as a flow rate. A sheet of thickness $h$ at its critical speed carries $U^*h = \sqrt{2\sigma h/\rho}$ per unit width of lip, and for water that is 0.324 litres a minute over each centimetre of spout for a two-millimetre sheet, 0.229 for one millimetre and 0.458 for four.

The flow grows only as the square root of the thickness, so across every plausible sheet the threshold sits at a few tenths of a litre a minute per centimetre. The teapot effect is a property of a slow pour, and a pour is slow in litres a minute, not in how steeply the pot is tilted. A spout a centimetre wide pouring a third of a litre a minute is at the edge of dribbling whatever its sheet happens to be, and a spout pouring a litre a minute through the same lip is well clear of it.

What the spoon under the tap demonstrates #

The demonstration that accompanies almost every telling of the Coandă effect is a stream from a tap touching the back of a hanging spoon. The stream bends round the spoon, the spoon is drawn into the stream, and the audience is told it has seen a flow follow a curved surface.

It has, and the demonstration is honest about that. What it is not is a demonstration of the mechanism a wall jet of air uses, because the stream is water in air: nothing seals a region of low pressure between the water and the spoon that the atmosphere could push against, and the air’s inertia is negligible. Whatever holds the water to the spoon is carried by the water — its surface tension, its wetting of the metal — and the spoon is drawn in by the reaction to the water’s momentum being turned.

There is a quick test that separates the two, and it is one a kitchen can run. The effect this essay describes depends on the water wetting the surface; a wall jet of air does not care what the wall is made of. A clean spoon and a greasy one hold a stream very differently, and anybody who has tried to demonstrate the “Coandă effect” with a spoon that has been through a dishwasher with too little rinse aid has run the experiment. The effect that is real remains real for air; it is simply not what the spoon shows.

Why the cure is a coating rather than a curve #

The arithmetic says the lip’s radius hardly matters, and the history of the problem agrees in the way it has been solved. Spouts have been reshaped for as long as there have been teapots, and the cures that have lasted are not gentler curves or sharper ones. They are changes to what happens where the liquid meets the solid at the edge: a ridge under the spout, a groove cut into the underside, a thin downturned lip, and most recently a coating the liquid will not wet.

Each of those works on the contact line rather than on the sheet. A ridge or a groove forces a sheet that has started to wrap round the lip to cross a second edge, where it has to wet a fresh surface at an awkward angle and gives up. A water-repellent coating removes the wetting that let the sheet follow the solid in the first place, so the balance computed here never gets to act: the sheet reaches the edge, meets a surface it will not spread on, and leaves. Experiments comparing spouts of the same shape with different wettability found the flow separating at speeds that depended strongly on the wetting, which is the direction the argument predicts.

That is the membrane argument’s own boundary, and it is a useful one to know precisely. Where the balance of tension and momentum applies, the radius does not matter; where the radius would matter, the balance has already stopped applying, because a lip sharper than the sheet is thick is a contact-line problem and not a curvature problem. The design space for a spout therefore has two regions and not a continuum between them: pour fast enough to be above the sheet’s wave speed, or make the edge a place the liquid cannot wet.

The same division runs through the rest of the subject wherever a liquid meets a solid with a free surface nearby. A film drawn up by a moving plate has a thickness set by viscosity against a meniscus’s tension, and a surface loaded with surfactant stops moving freely with the flow; in both, the mechanism is in the bulk and the control is at the surface.

What the membrane leaves out #

The lip is perfectly wetted. Every figure assumes the liquid wets the spout completely, so that nothing holds the sheet back from following the surface except its own balance. Real spouts are partly wetted, and the angle the liquid makes with the solid at the line where it leaves is what decides separation at a sharp edge. That line is where the effect is controlled in practice.

The sheet is uniform and inviscid. A real sheet leaving a spout has a velocity profile shaped by the flow inside the pot, a boundary layer on the lip, and a thickness that changes as it goes. The plug-flow sheet is the leading term of that and not its structure.

There is no gravity. Once the sheet has wrapped round the lip it runs down the outside of the spout under its weight, which is what turns a sheet that follows the lip into a dribble that reaches the table. The threshold is about whether it follows; how far it then runs is a separate question of drainage.

The surface is clean. Surfactants change the tension, and a moving sheet can stretch its surface faster than surfactant can diffuse to it, so the tension of a flowing soapy sheet is not the tension of the same liquid at rest. The line drawn for soapy water uses the static value.

And the sheet is a sheet. A round stream is not a membrane of one thickness, and its balance round a lip has an extra curvature in the other direction; the essay’s arithmetic is for a stream flattened over a lip, which is what a pour usually is.

Each claim is computed a second way. The Taylor–Culick speed equals the membrane wave speed to machine precision for every liquid and thickness tried, the momentum flux at that speed equals $2\sigma$ exactly, the finite-thickness critical speed on a lip ten thousand times the sheet’s thickness agrees with the thin-sheet value to a part in a billion, and its deviation on thicker sheets falls with an exponent of 1.93 against the two the expansion predicts.

A kitchen problem that took thirty years #

The teapot effect was put to physicists as a named problem by Markus Reiner in 1956, in a short article that asked why tea runs down the spout, and answered by Joseph Keller the following year with a potential-flow analysis in which the stream’s own pressure field, not the atmosphere, holds it to the lip. G. I. Taylor and F. E. C. Culick worked out the retraction speed of a punctured sheet in 1959 and 1960, without any connection to teapots. Kistler and Scriven treated sheet-forming flows with wetting and hysteresis in 1994, and in 2010 Duez and colleagues showed experimentally that the wettability of the spout controls whether the flow separates — that a spout made hydrophobic stops dribbling.

The name Coandă arrived by a different route, through patents in the 1930s on jets of gas deflected by surfaces, and it was attached to the teapot afterwards because both show a flow following a curve. Thomas Young had described a stream of air following a curved surface in 1800, and the attachment of one mechanism’s name to the other’s demonstration is the pattern this field keeps finding — a real observation, a real mechanism, and a word that crossed from one to the other because they looked alike.

Still open: what the lip’s wetting decides #

The membrane argument decides whether a sheet leaving a rounded lip is pulled back round it, and it deliberately stops short of the edge itself. At a sharp edge, the liquid’s contact line on the solid sits at a definite place, it resists moving by an amount that depends on the surface’s chemistry and roughness, and a sheet that has once wrapped round an edge can stay wrapped at speeds that would not have wrapped it — a hysteresis between starting a pour and stopping one.

That is the question the arithmetic here does not reach: how a liquid meets a solid at a moving line, and how much of the teapot effect is decided there rather than in the sheet. Beside it is the other thing a fast sheet does once it has left the lip cleanly, which is to thin, flap and break into drops — where a sheet or a jet stops being one, at a Weber number rather than at the wave speed — and the number that cannot break a drop is the same competition between inertia and tension, asked of the pieces.