How far apart a ceiling drips

The best-known instabilities of a moving fluid are fed by shear. A layer with a kink in it needs an inflection in the velocity profile; a vortex sheet is unstable at every wavelength at once; the channel that Rayleigh cleared is destabilised by viscosity itself. The oldest instability of all needs no motion to start. Put a heavy fluid on top of a light one and the interface between them cannot stay flat.

The everyday version hangs from a ceiling. Condensation in a bathroom, a coat of wet paint, a spill of honey on the underside of a shelf: each is a layer of liquid over air, held up by nothing but its grip on the surface above, and each develops a pattern of bulges that grow into drips. The spacing of those drips is a number that can be measured with a ruler, and the standard account gives it as one constant times the capillary length. The calculation below says the standard account is right for one kind of layer and wrong for the two that ceilings usually carry.

Heavy over light: gravity pays, surface tension charges #

Take the simplest version first: a deep layer of inviscid liquid above air. Displace the surface by a ripple of wavenumber $k$. Where the liquid bulges downward, the weight of the extra liquid pulls it further down; where it thins, the air pushes up into the gap. Gravity feeds every ripple. Surface tension resists them, because bending the surface costs energy in proportion to its curvature, and short ripples are sharply curved. The growth rate $s$ follows from balancing the two:

$$s^2 = gk - \frac{T k^3}{\rho}.$$

Every ripple longer than the cut-off $k = 1/\ell_c$ grows, where $\ell_c = \sqrt{T/\rho g}$ is the capillary length — 2.71 millimetres for water. The cut-off wavelength is $2\pi\ell_c$, seventeen millimetres. The growth rate is largest at $k = 1/(\sqrt3\,\ell_c)$, a wavelength of $2\pi\sqrt3\,\ell_c$ = 29.5 mm for water, and that is the number usually quoted for the spacing of the drips.

The contrast with the shear-driven instabilities is worth a sentence. A vortex sheet grows fastest at its shortest wavelength, which is why its linear theory has no answer to “what size?”. This instability has a band edge set by surface tension and a well-defined fastest wave inside the band. It is a selection problem with an answer, and the only question is whether the simplest version of the answer is the right one.

The layer a ceiling actually holds #

A real hanging layer differs from the textbook one in two ways at once. It is attached to a solid ceiling, where it cannot slip, so it has a finite depth $h$. And it is viscous, which matters because a bulge can only grow if liquid flows sideways to feed it.

The full linear problem is set up for a layer of depth $h$ under a no-slip ceiling, with air of negligible density below. Lengths are measured in capillary lengths, times in $\sqrt{\ell_c/g}$, and the viscosity appears as a single number, $N = \nu/\sqrt{g\ell_c^3}$ — about 0.002 for water, 3 for glycerol and 27 for honey. The vertical velocity of a disturbance is a combination of four exponentials: two that decay over the wavelength, as a potential flow’s would, and two that decay over the thickness of a viscous layer, $1/q$ with $q^2 = k^2 + s/\nu$. The ceiling imposes no flow through it and no slip along it; the free surface imposes no shear stress, and a balance of normal stress between the liquid’s pressure, its viscous stress, gravity on the displaced liquid and the surface tension. Four conditions on four coefficients make a four-by-four determinant, and the growth rate is the positive real $s$ at which it vanishes.

Two constants and a depth #

The determinant, solved across depths and viscosities, has three regimes, and the figure above shows one of each.

A deep inviscid layer reproduces the textbook answer: the fastest wave at $k = 1/\sqrt3$, recovered by the determinant to one part in ten million.

A thin film — any film much thinner than a capillary length, whatever its viscosity — grows fastest at $k = 1/\sqrt2$, a wavelength of $2\pi\sqrt2\,\ell_c$, which for water is 24.1 mm. The reason is the way a thin film has to thicken. A bulge in a deep layer is fed from below it, by liquid moving vertically over a depth comparable with the wavelength. A bulge in a thin film can only be fed by liquid squeezed sideways along the film from the thinning regions either side, through a gap of thickness $h$, and the flux that squeezing drives is $h^3/3\mu$ times the pressure gradient. The pressure gradient along the film brings in one more factor of $k$ than the deep layer’s vertical flow does, so the thin film’s growth rate is $s \propto k^2 - \ell_c^2k^4$ rather than $s^2 \propto k - \ell_c^2k^3$, and its maximum moves from $1/\sqrt3$ to $1/\sqrt2$. The drip spacing is shorter by the ratio $\sqrt{2/3}$, eighteen per cent.

A deep viscous layer has no fastest wave of its own. In a deep layer of honey, a ripple of wavelength $\lambda$ sets in motion a region about $\lambda$ deep, the viscous stress resisting it scales as $\mu k$, and the growth rate is $(\rho g/2\mu k)(1 - \ell_c^2k^2)$ — largest for the longest ripple. Nothing inside the liquid stops the preferred wavelength growing; only the ceiling does, once the ripple is long enough to feel it, at $kh$ of order one. The fastest wave in a viscous layer is as long as the layer is deep.

The figure puts the three together. Water, which is nearly inviscid on the capillary scale, moves from $\sqrt{2}$ for a film to $\sqrt{3}$ for a pool as its depth passes a capillary length, and stops there. Glycerol and honey start at the same $\sqrt{2}$ when thin, pass through $\sqrt{3}$ around a capillary length deep, and carry on climbing: at twenty capillary lengths deep, a honey layer’s fastest wave is several times longer than the textbook’s. The drip spacing is the depth’s, not the liquid’s. Surface tension sets the cut-off and the unit; the depth decides which multiple of the unit.

The determinant, checked against what it was not told #

The determinant is built from boundary conditions, not from any of the three formulas above, and each formula is a limit it must reach on its own.

At vanishing viscosity and a depth of two capillary lengths it gives $s^2 = (k - k^3)\tanh kh$ to one part in ten thousand; the $\tanh$ is the ceiling, felt by an inviscid layer as a depth limit on how far below the surface the motion reaches. For a film a twentieth of a capillary length thick it gives the lubrication result $h^3(k^2 - k^4)/3N$ to within three parts in a thousand, the difference being the film’s small inertia. For a layer forty capillary lengths deep and five hundred times more viscous than honey it gives $(1 - k^2)/2Nk$ to the same accuracy. The worst of nine comparisons is $2.9 \times 10^{-3}$, and the two constants come out at $\sqrt3\,k = 1.0000$ and $\sqrt2\,k = 0.99944$.

What holds the liquid up, and why a narrow tube never drips #

Before any ripple grows there is a static question: why a layer of liquid can hang from a ceiling at all. Nothing pulls it up. The air below pushes it up, and it can because the pressure inside the layer is below atmospheric — by $\rho g h$ at the ceiling, a hundred pascals for a centimetre of water, a thousandth of an atmosphere. The ceiling carries that deficit as a stress on its surface, and a flat layer is in perfect equilibrium. The equilibrium is unstable, which is the whole of this essay, but it exists.

The cut-off turns that into a practical rule. A ripple can only grow if it fits, so a layer whose horizontal extent is shorter than the cut-off wavelength cannot develop one at all. The familiar case is an inverted tube with its top closed, a finger over a straw. The air trapped above stops the liquid from simply falling out as a plug, so the only way out is for one side of the surface to bulge down while the other rises — the tube’s lowest sloshing mode, whose wavenumber is $1.841/R$ for a tube of radius $R$. That mode grows only if its wavenumber is below the cut-off, $1/\ell_c$, so water stays in an inverted tube narrower than $2 \times 1.841\,\ell_c$ = 10 mm across and runs out of a wider one. A drinking straw, a few millimetres across, holds its water; a tube wider than about a centimetre lets it go, air bubbling up one side as the water runs down the other. The number and the drip spacings share one unit.

How long a ceiling takes to drip #

The spacing is one number a hanging layer has; the time it takes to show it is another, and the time is far more sensitive to the depth.

A thin film is held to the ceiling by its own viscosity: the liquid that must move sideways to feed a bulge moves through a gap $h$ thick, and the flux falls as $h^3$. A water film a tenth of a millimetre thick grows by a factor of $e$ in nine seconds, one a millimetre thick in 64 milliseconds, and a deep pool in 27. Glycerol, a thousand times more viscous, takes nearly two hours per e-fold as a tenth-of-a-millimetre film and eight seconds as a millimetre. Honey as a tenth-of-a-millimetre film takes more than eight hours.

That is why a ceiling covered in condensation drips at all, and why a painted one usually does not. A condensation film grows thicker as it collects water, its growth time falls as the cube of the thickness, and at some thickness it starts to drip within seconds. Paint is applied at about a tenth of a millimetre, dries in an hour or two, and at its viscosity would need a comparable time per e-fold — and paint is formulated with a yield stress besides, which stops a film from flowing at all while $\rho g h$ is less than the yield stress. A paint’s resistance to dripping is designed, and the arithmetic says what it has to beat.

A wet ceiling, a pot of honey #

A bathroom ceiling carries a film of the order of a tenth of a millimetre and drips at about 24 mm. The underside of a lid over a pan of boiling water carries a similar film and a similar spacing. A tray of water turned upside down, if it could be done without the water simply falling out as a sheet, would start to break up at about 30 mm. A glycerol film drips at 20 mm, because glycerol’s capillary length is shorter than water’s, and a 3-centimetre layer of glycerol hanging from a ceiling would drip at 64 mm. Honey 3 centimetres deep: 88 mm. The capillary length is between 1.9 and 2.7 millimetres for all three liquids, and the spacings range over a factor of five.

On a real ceiling the drips form a two-dimensional pattern rather than a row, and the linear calculation fixes only the distance between neighbours, not their arrangement. Experiments on hanging films of silicone oil find the bulges arranging themselves into hexagons at the spacing the thin-film theory predicts; the choice of hexagons over squares or stripes is made by the nonlinear terms, which are not computed here. Once a bulge has grown to about a capillary length it stops being a ripple and becomes a pendant drop, and from there it detaches at a fold of its family of hanging shapes rather than by any continued instability of the film.

From ripples to a mixing layer #

Everything so far is the beginning of the instability. When two fluids of comparable density are forced into this configuration and kept there — a heavy gas above a light one, the interface of a supernova’s ejecta, the fuel pellet of an inertial-fusion capsule — the bulges become fingers of heavy fluid falling and bubbles of light fluid rising, the fingers roll up at their sides by the shear-driven instability of a layer with a kink in it, and the interface becomes a turbulent mixing zone. That zone is found to grow as $\alpha A g t^2$, with $A$ the Atwood number and a growth constant measured anywhere between about 0.02 and 0.07 depending on how the initial ripples were prepared.

That spread is the turbulent descendant of this essay’s selection problem. Once the flow has forgotten the fastest linear wave the zone should be self-similar and $\alpha$ universal, and the fact that it is not — that the constant remembers the initial conditions — is the same kind of memory as a parcel’s dispersion remembering its start, and is not computed here. What the linear theory does fix is where the memory comes from: the band of growing ripples, its cut-off and its peak, which is what an experiment’s preparation either matches or overrides.

What the picture cannot show #

Linear growth only. Every number is the growth of an infinitesimal ripple. The spacing it selects is the spacing of the first bulges; the drips they become may merge or coarsen.

Air of no density. The Atwood number is taken as one. Two liquids of similar density, the laboratory version, have a slower instability and a different viscous regime.

A perfectly wetting, smooth ceiling. A real ceiling has roughness, contact lines and patches where the film is pinned, all of which seed particular wavelengths and can override the fastest one.

A Newtonian liquid. Paint and honey are not quite Newtonian, and paint’s yield stress changes the problem from “how fast” to “whether at all”.

The convention the numbers depend on #

The capillary length is $\sqrt{T/\rho g}$. The wavenumber is in units of $1/\ell_c$ and the spacing is the wavelength of the fastest-growing ripple, $2\pi/k$. Growth rates are in units of $\sqrt{g/\ell_c}$, and the viscosity enters as $N = \nu/\sqrt{g\ell_c^3}$. The layer’s depth is measured from the ceiling to the undisturbed free surface. The liquid properties are water at 20 °C, glycerol at 20 °C and a typical honey, the last being the least certain by a factor of two in viscosity.

Who found it, and when #

Rayleigh derived the instability of heavy fluid over light in 1883, and G. I. Taylor gave the accelerating-interface version in 1950, which is why it carries both names. Bellman and Pennington added surface tension and viscosity in 1954, and Chandrasekhar’s 1961 treatise gave the full viscous dispersion relation for two semi-infinite fluids. The thin-film version under a ceiling, with its $\sqrt{2}$ and its hexagons, was worked out experimentally and theoretically by Fermigier, Limat, Wesfreid and colleagues in the early 1990s.

It belongs with the size a drop is allowed and what a plate takes with it as a problem whose unit is the capillary length, and with the number that stops the mixing as its mirror image: a stable density difference suppresses exactly the motion that an unstable one amplifies.

Still open: a ceiling that slopes #

Every layer here hangs from a horizontal ceiling, so every bulge grows where it started. Tilt the ceiling and the film flows downhill along it, carrying the bulges with it. A bulge that is carried away faster than it grows never drips at the place it formed; if the tilt is steep enough, it may never drip at all before it reaches the edge. The instability is then said to be convective rather than absolute, and the tilt at which one becomes the other is a calculation of the same determinant with a mean flow added and the growth rate made complex. It would say at what slope a wet underside stops raining on what is below it and starts delivering its water to the edge instead, which is the question behind every drip edge on a building.