A sloping ceiling drips downhill, or not at all

How far apart a ceiling drips solves the Rayleigh–Taylor instability of a liquid layer hanging from a level ceiling, and finds that a film thinner than the capillary length — the kind that condenses on a bathroom ceiling or the underside of a cold roof — grows fastest at a spacing of $2\pi\sqrt2$ capillary lengths, 24 millimetres for water. It closes on the ceiling that is not level. A film on a sloping underside flows downhill, and carries its ripples with it. A ripple carried away faster than it grows never drips where it formed; the question is at what slope that happens, and what it means for where the water goes.

The question has a name. An instability whose disturbances grow at the place they started is absolute; one whose disturbances grow only while being carried away, leaving the place they started quiet again, is convective. The two can have exactly the same growth rate in a frame that moves with the disturbance, and entirely different consequences for anything fixed to the ceiling. Every wavelength at once met the distinction in a vortex sheet. Here it can be computed in closed form, and checked against the exact response to a single disturbance.

The level ceiling, carried #

In the thin-film limit of the earlier essay, a small change $\eta$ in the thickness of a film $h_0$ thick, hanging under a plane tilted at $\theta$, obeys

$$\eta_t + V\eta_x + \eta_{xx} + \eta_{xxxx} = 0, \qquad V = \frac{3\tan\theta\,\ell}{h_0},$$

in units of the capillary length $\ell = \sqrt{\sigma/\rho g\cos\theta}$ and the time $3\mu\ell^4/\sigma h_0^3$. The two derivatives on the right are the level ceiling’s: the normal part of gravity pulls long waves down and surface tension flattens short ones. The new term, $V\eta_x$, is the downhill part of gravity, and it does only one thing — it carries the pattern downhill at speed $V$, the speed at which a thickness wave travels on a film draining down a slope, three times the film’s own mean speed.

A wave $e^{ikx + st}$ grows at

$$s = k^2 - k^4 - iVk .$$

The real part is exactly the level ceiling’s: the fastest wave is still $k = 1/\sqrt2$, the earlier essay’s $2\pi\sqrt2$ capillary lengths, growing at a quarter. Tilting the ceiling changes no wave’s growth. It changes only where the growing wave is.

That is worth pausing on, because every stability criterion used on these shear layers and films so far is a statement about growth in time at a fixed wavenumber. A layer with a kink in it asks whether any wave can grow at all; every unstable wave is inside one circle bounds how fast one can; the profile Rayleigh cleared and viscosity did not finds the Reynolds number above which one does. All of them would give the sloping ceiling the level ceiling’s answer — unstable, fastest at $2\pi\sqrt2$, growing at a quarter — because a uniform drift cannot change a growth rate measured in a frame that drifts with it. The question the slope raises is not in that frame. It is about growth at a fixed place, and a fixed place sees a mixture of every wavenumber the packet contains, each arriving and leaving at its own speed.

Seen from a moving observer #

A single disturbance on a level ceiling grows into a packet of ripples that spreads both ways from where it started. An observer moving along the ceiling at speed $v$ sees it grow at a rate that is largest for an observer standing still and falls to zero for one moving at the packet’s edges. The first figure is that growth, for four carrying speeds. On the level ceiling it peaks at a quarter at $v = 0$ and falls to zero at $v = \pm1.622$: the packet’s edges spread at that speed.

The rate along each ray is computed at the saddle point of $s + ivk$ in the complex wavenumber plane — the wave that dominates what an observer moving at $v$ sees at long times. Of the three saddles, the one that pinches the integration path is the one that continues from the level ceiling’s fastest wave, and the check below shows it is the right one.

Tilting the ceiling slides the whole curve downhill by $V$. That is all it does. The packet grows at a quarter in its middle, which now moves at $V$, and spreads at 1.622 either side of it. The observer at the point disturbed, $v = 0$, sees growth as long as the packet’s trailing edge is still moving uphill — as long as $V < 1.622$. At $V = 1.62208$ the trailing edge stands still over the point disturbed; above it, it moves downhill, and the point disturbed is left behind the packet, flat again.

One disturbance, followed #

The second figure is the same thing without the complex plane. Because the equation is linear, the film’s response to a single small disturbance can be written down exactly as a Fourier integral, and it is drawn here at three times, each scaled to its own largest value so that its shape can be compared as it grows. At $V = 1$ the packet is carried downhill, but it spreads uphill faster than it is carried, and the point where it began stays inside it. At $V = 2.5$ it grows just as fast — in the frame moving with it, nothing has changed — but its trailing edge has left the origin, and the ceiling behind it is flat.

A ceiling carrying its film faster than 1.622 is still unstable. Every disturbance still grows by the same factor. What it no longer does is grow in the place it started, and for anything fixed to the ceiling — a drip, a stain, the floor below — that is the difference that matters.

The saddle, checked against the exact response #

The third figure checks the saddle against the exact integral. It takes the response at the point disturbed, multiplied by $\sqrt t$ to remove the spreading of the packet, and follows it in time. At a carrying speed of 1.2 it oscillates and grows; at 1.622 its envelope is flat; at 2 it oscillates and dies away. Fitted over the latest times the integral can still be computed accurately — its own cancellation, a growing $e^{t/4}$ summed to a decaying answer, sets the limit — the growth rates are 0.2214, 0.1020 and −0.1053 at carrying speeds of 0.5, 1.2 and 2, against the saddle’s 0.2203, 0.1015 and −0.1054. The worst disagreement is 0.0011, which is the saddle’s own next correction at those times.

Across the transition the fitted growth at the point disturbed changes sign: 0.042 at nine-tenths of the critical speed, −0.044 at eleven-tenths.

One number #

The fourth figure is the growth at the point disturbed against the carrying speed. It is a quarter on the level ceiling, falls as the ceiling tilts, and crosses zero at

$$V_c = 1.62208 .$$

The number is a property of $s = k^2 - k^4$ alone: it is the speed at which a level ceiling’s disturbance spreads. The same operator is the linear part of the Kuramoto–Sivashinsky equation, and $V_c$ is that equation’s linear spreading speed — the speed at which a front of pattern invades a flat state. It has appeared before in another form: a wrinkled flame obeys an equation whose growth is $|k| - \nu k^2$ rather than $k^2 - k^4$, but it too is an instability whose linear part decides how fast disturbances spread and whose nonlinear part decides where they stop.

The same division, absolute against convective, is why a cylinder’s wake sheds vortices at a frequency of its own. Close behind the body the reversed flow makes the wake absolutely unstable: a region where disturbances grow in place and impose their frequency on everything downstream, which is the regularity the frequency a wake chooses turns into a Strouhal number. A jet, carried away from its nozzle, is usually convectively unstable instead, and amplifies whatever noise it is fed. A sloping ceiling is a wake without the reversed flow. Tilted past its transition it has no region that can choose a frequency or a place, and its drips form wherever the amplified noise happens to reach the film’s own thickness.

The slope, in degrees #

Written in the ceiling’s own terms, the transition is

$$\tan\theta_c = \frac{V_c\,h_0}{3\,\ell},$$

a slope in proportion to the film’s thickness over the capillary length. The fifth figure draws it for water, whose capillary length is 2.7 millimetres, and for a wet paint, whose surface tension is half water’s and whose capillary length is 1.7. A tenth of a millimetre of water stops dripping in place at 1.1 degrees of slope; half a millimetre at 5.7; two millimetres at 21. A film of paint of the same thickness needs more, because its capillary length is shorter and gravity carries it less far per unit of the instability’s own length.

The viscosity is not in it. Viscosity slows the draining and the growth by the same factor, and sets only the clock: a tenth of a millimetre of water grows by a factor of $e$ in about nine seconds, half a millimetre in 72 milliseconds. A thick honey film, draining and growing a few thousand times more slowly, has exactly the same critical slope as water of the same thickness and capillary length.

The slopes are small because condensation films are thin. A level ceiling’s film grows fastest at a wavelength of a few capillary lengths, and its packet spreads at one and a half capillary lengths per unit of its own time. The film on even a gently sloping ceiling drains at a speed that, measured in those units, is three times the slope’s tangent times the ratio of the capillary length to the film’s thickness — a large multiplier for any film much thinner than the capillary length.

The same transition, in millimetres and seconds #

The transition has a plain physical reading that does not need the complex plane. At the critical slope, the film carries a ripple downhill by $4V_c = 6.5$ capillary lengths in the time the ripple takes to grow by a factor of $e$ — four of the film’s own time units, since its middle grows at a quarter. For water that is eighteen millimetres, about three-quarters of the level ceiling’s drip spacing, and it is the same for every thickness of film: a thicker film grows faster and drains faster by exactly compensating factors, which is why the transition is a slope and not a speed.

The numbers it corresponds to are small. A tenth of a millimetre of water on a slope of 1.1 degrees drains at 0.6 millimetres a second on average, carries its thickness waves at three times that, and grows by $e$ in nine seconds: in that time a ripple has moved the eighteen millimetres. Half a millimetre of water on 5.7 degrees both drains and grows 125 times faster, and the ripple moves the same eighteen millimetres in 72 milliseconds. Nothing about the transition looks dramatic from underneath. A film just past it looks, if anything, calmer than one just short of it, because the ripples that form over any fixed point are always on their way somewhere else.

Where the drips form instead #

A convective film still drips. Its disturbances grow as they travel, and once one has grown from a ripple a thousandth of the film’s thickness to the film’s own thickness — seven factors of $e$ — it drips wherever it has got to. The sixth figure is how far that is. The spatial growth rate is the largest growth per unit distance along any ray, and far above the transition it approaches the packet’s middle, a quarter per unit time at speed $V$, so the distance is about $28V$ capillary lengths.

For a film of 0.2 millimetres of water the drips form half a metre downhill of their origin on a ten-degree slope, 1.2 metres on a twenty-degree one, and more than two on thirty. For a millimetre-thick film, which reaches the transition only at eleven degrees, they form within a quarter of a metre at twenty degrees.

This is the answer to the earlier essay’s question about drip edges. A sloping soffit shorter than the distance a disturbance travels before it drips delivers its water to the edge; a longer one rains on what is under it, though not where the water first gathered. The film on a wet overhang is thin and its slope usually modest, so the distance is tens of centimetres to a metre: the scale of the overhang itself. That is why a drip groove — a notch that interrupts the film and forces it to fall — works where it is placed and not a little further along.

The drop that finally falls is an old object again: a pendant bulge that has outgrown what its surface tension can hold, at a size set by the capillary length and nothing else, as the size a drop is allowed found for a drop resting the other way up. The slope decides where along the ceiling that happens, and the capillary length decides how big it is when it does. It turns a convective film, whose drips are spread over a length set by the slope, into one that drips at a line.

What the picture cannot show #

Linear growth only. The distance to drip is the distance to reach the film’s own thickness at the linear rate, which is where the linear theory stops. Nonlinear films on inclined undersides form travelling rivulets and drops that can slide a long way before detaching, so the distance here is a lower bound on where drips form.

A thin film. The lubrication limit holds while the film is thinner than the capillary length, which condensation films are. The earlier essay’s full determinant would carry the mean flow into the thick-layer case too.

A uniform film. The film is taken as uniformly thick, fed at the rate it drains. A film that thins downhill carries its disturbances more slowly as it goes, and its transition moves along the ceiling with it.

A film that is not being fed. A condensing film thickens as it drains, and an evaporating one thins; either moves the film’s thickness, and with it the transition, on the time scale of the condensation. Where condensation is fast compared with the drainage — a cold roof on a humid night — the film under a gentle slope can thicken through its own transition, and a ceiling that carried its water to the edge in the evening can start to rain in place by morning.

One dimension. Disturbances across the slope are not carried and grow as on a level ceiling. The downhill argument is about where the drips form along the slope; across it, they still form at the level ceiling’s spacing.

The convention the numbers depend on #

Lengths are in the capillary length $\ell = \sqrt{\sigma/\rho g\cos\theta}$, times in $3\mu\ell^4/\sigma h_0^3$. The carrying speed is $V = 3\tan\theta\,\ell/h_0$ in those units. A disturbance is taken to drip once it has grown by $e^7$, about a thousandfold.

Who found it, and when #

The distinction between absolute and convective instability comes from plasma physics — Briggs’s and Bers’s criteria of the 1960s — and was brought into fluid mechanics by Huerre and Monkewitz in the 1980s, where it explained why some wakes and jets oscillate at their own frequency and others merely amplify noise. The Rayleigh–Taylor instability under an inclined plane was studied this way by Brun, Damiano, Rieu, Balestra and Gallaire in 2015, who located the transition between the two, in the thin-film limit and in experiments with silicone oil. The spreading speed of the Kuramoto–Sivashinsky operator is tabulated in van Saarloos’s review of fronts propagating into unstable states.

Still open: rivulets that carry the water past the edge #

Past its linear stage a convective film on a sloping underside does not simply drip: it breaks into rivulets aligned down the slope, which carry the water faster than the film did and can reach the edge before any drop detaches. The next calculation follows the thin-film equation past the linear stage, with the cross-slope direction included, and asks at what slope the rivulets outrun their own dripping — whether there is a second critical slope, above the one here, beyond which a sloping ceiling delivers all of its water to the edge however long it is.