Concept

Capillary length — where it appears

The length at which surface tension and weight balance, the square root of the surface tension over the density times gravity. It is 2.7 mm for water, and it is the only length in problems that have no apparatus size in them.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

How big before gravity shows. A drop's height over its width against the Bond number, which is the ratio of its weight to the force its own skin can supply. The number is one where those two are equal, and by then the drop is a bun: it is one per cent from a ball at Bo = 0.0079, five per cent at 0.054 and ten at 0.13. Every one of those is below one, and the first is below it by a factor of a hundred and twenty-six.

The size a drop is allowed

The Bond number sets a drop's weight against the force its own skin can supply, and it is one when they are equal. By then the drop is a bun — it is a per cent from being a ball at Bond number 0.0079, which is a water drop half a millimetre across.

regimes · Bond

Twice as fast is not twice as thick. The film a plate carries out of water, against the speed it is withdrawn at, both logarithmic. The slope is exactly two-thirds, so doubling the speed thickens the film by 58.7 per cent and never by more. The open marks are outside the range the derivation holds in — above a capillary number of about a hundredth the film is no longer thin against the capillary length, and the measured thickness leaves this line.

What a plate takes with it

Pull a plate out of a bath and it comes out wet. How wet is not set by the plate, the bath or how much liquid there is, but by a competition in a region a fraction of a millimetre long that nobody looking at the plate can see — and the film goes as the two-thirds power of the speed, never as the first.

viscous · Coating

A drop fed into a tube, to scale, up to the largest one it can hold. Six members of the family of static drops a tube of radius half a capillary length holds, drawn to scale and hung from the rim, with their volumes in cubic capillary lengths. The drop grows from a shallow cap through a hemisphere to a bulb with a neck. The last is the fold: no static drop on this tube holds more, and it holds 76 per cent of what Tate's law says the tube can carry.

The drop falls at a fold

Tate's law says a drop leaves a tube when its weight equals the tension round the rim. No force balance decides it. A tube holds a family of static drops, the family has a largest member, and the drop falls because there is no static shape with more liquid in it — which Tate's balance overestimates by a quarter on a millimetre tube and underestimates on a wide one.

regimes · Bond

Four tubes to scale, and the level Jurin's law gives each. Water rising in tubes of radius half a capillary length to four, drawn to one scale, with the level of the liquid far outside the tubes at the bottom. The red line in each tube is Jurin's height, 2ℓ²/R. In the narrow tube the surface is nearly flat and sits on it. In the wide ones the liquid gathers in a rim at the wall, the centre hardly rises, and Jurin's level runs through the middle of a surface that is nowhere near it — while still being its exact average.

A law that is exact as an average

Jurin's law gives the height water climbs in a tube as twice the square of the capillary length over the radius. As a statement about the height it is an approximation for narrow tubes. As a statement about the mean height of everything lifted it is exact for every tube — and a wide tube shows what the average was hiding: a rim of water at the wall and almost nothing in the middle.

regimes · Bond

The drip spacing is set by the depth of the layer. The spacing of the fastest-growing wave, in units of 2π capillary lengths, against the depth of the hanging layer in capillary lengths, for water, glycerol and honey. A thin film of any of them drips at √2. Deep water drips at √3. Deep glycerol and honey keep growing past √3, their fastest wave as long as the layer is deep, because viscosity slows short waves more than long ones.

How far apart a ceiling drips

A layer of liquid hanging from a ceiling is heavy fluid over light, and every ripple on it longer than about seventeen millimetres grows. Which ripple grows fastest, and so how far apart the drips form, is usually given as one number. It is at least three, and what chooses between them is not the liquid's surface tension but the depth of the layer.

turbulence · Instability

Whether a sloping ceiling drips in place is one ray's growth. How fast a disturbance grows as seen by an observer moving along the ceiling at speed v, for four speeds at which the film carries its disturbances downhill, in the film's own units. On a flat ceiling (V = 0) the growth peaks at a quarter for the observer standing still and falls to zero for one running at ±1.622. Tilting the ceiling slides the whole curve downhill. While the observer at the point disturbed, v = 0, still sees growth, the ceiling drips where the disturbance began; at V = 1.622 that observer sees none, and above it the disturbance grows only while it is carried away.

A sloping ceiling drips downhill, or not at all

A film hanging from a level ceiling drips where its ripples form. Tilt the ceiling and the film flows downhill, carrying its ripples with it, and at some slope they are carried away faster than they spread and the ceiling stops dripping in place. That slope is set by one number, 1.622 — the speed at which a level ceiling's disturbance spreads — and for water it is tiny: a degree for a tenth of a millimetre of film, six for half a millimetre. Past it the ripples still grow, and still drip, but downhill, at a distance that grows with the slope: a ceiling shorter than that delivers its water to the edge.

turbulence · Instability

What a tilted pane holds is a difference of two cosines. The largest ridge a tilted plate holds, as a cross-section in square capillary lengths, against the tilt, for clean glass (advancing 30°, receding 10°), a plastic (90°, 70°) and a water-repellent coating (115°, 95°): the force balance (cos θᵣ − cos θₐ) ÷ sin α, and, as points, the areas of drops shot from Young–Laplace, which do not use it. Clean glass holds least, not because water sticks to it less but because on a surface it wets well the two cosines are nearly equal.

The force a contact line holds is a range

Capillary rise and the drop on a window are usually drawn with one contact angle, and a contact line with one angle makes a force that is a single number. A real contact line pins, and stops anywhere between a receding and an advancing angle. The force it holds is then a range, as static friction is, and its width is surface tension times the difference of two cosines. A tube holds its column at any height in the range, so which way the meniscus last moved matters more than how patchy the wall is — and a tilted pane holds a drop only as large as that difference allows.

regimes · Bond

Below the capillary length, the coat is set by the fibre, not the bath. The coat's thickness in units of ℓ_c Ca^(2/3) against the fibre's radius in capillary lengths, on logarithmic axes: the dynamic meniscus's 1.3376 over the curvature the static meniscus asks of it, 1/b + Z/ℓ_c². Thin fibres take Quéré's 1.3376 b Ca^(2/3), set by their own radius; thick ones take Landau–Levich's 0.9458 ℓ_c Ca^(2/3). The two laws cross at b = 0.71 ℓ_c, where the coat is 0.61 of either; at b = 0.1 ℓ_c it is 0.137 of what the plate's law would give.

A thin fibre coats by its own radius, and beads by it

A plate drawn out of a bath carries a film set by the capillary length, the size at which surface tension and gravity balance. A fibre thinner than that length carries a film set by its own radius instead, often a tenth of what the plate's law promises — and the same radius then decides how quickly that film gathers into beads. The faster the fibre is drawn, the thicker its coat and the shorter the length of it that comes out smooth.

viscous · Coating

Named alongside it

The objects these essays reach for when they reach for this one.

Surface tension Bond number Model limit Lubrication film Young laplace Contact angle Drop Measurement Threshold Viscosity Buoyancy Capillary number

All concepts