Surface at fixed volume
At its defaults it draws surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.
surface-energy is one function in lib/figures/fluids.js —
matter that will not hold a shape, and the forces in it. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.
Surface at fixed volume
The options are the ones How high water will climb passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.
Surface at fixed volume
The options are the ones The angle a liquid makes with what it sits on passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.
Two drops, then one
The options are the ones The angle a liquid makes with what it sits on passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two water drops of radius 1.0 mm merging into one of the same total volume, whose radius is the cube root of two times larger. The surface falls by 20.6 per cent — a figure that depends on nothing but the cube root of two — and the 0.38 µJ of surface energy that went with it has to go somewhere. It goes into warming the drop and into the ringing that follows a merge.
Two angles a film is not allowed to depart from
The options are the ones The angles a film has no choice about passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The two junctions Plateau's laws permit, drawn at the angles a balance of equal tensions requires. A soap film pulls equally in every direction along itself, so where films meet the pulls must sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films can only meet along a line, at 120.0000° to one another, because three equal coplanar vectors sum to zero at 120° and at no other angle. Four such lines can only meet at a point, at 109.4712° — arccos(−1/3), the tetrahedral angle — for the same reason in three dimensions. Both numbers are found here by solving the balance rather than by drawing what is expected, and neither depends on the liquid, the temperature or the size of the foam. A junction of four films along a line, or of three lines at a point, is not merely unusual: the tensions cannot balance there, so it rearranges within milliseconds into the two arrangements drawn.
The film between two rings, and where it stops existing
The options are the ones The angles a film has no choice about passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The area of the soap film spanning two coaxial rings, against how far apart they are in ring radii, beside the area of the two flat discs that are the alternative. The film is a catenoid, and the equation fixing it has two solutions, one, or none: the lower curve is the stable catenoid, the upper one the unstable solution it merges with, and past a half-separation of 0.6627 radii there is no catenoid at all and the film snaps. What is worth reading carefully is that the two events are not the same event. The catenoid's area exceeds the two discs' at 0.5293, so between there and 0.6627 the film is no longer the least-area solution and survives anyway, held in a local minimum. Pulling the rings apart slowly therefore does not break the film where the arithmetic says the discs win; it breaks it where the catenoid ceases to be available, which is 25 per cent further on.
What checks it
physicscheck asserts something about surface-energy that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
How high water will climb
Water rises up a narrow tube against gravity, and the narrower the tube the higher it goes. The height is set by a curved surface pulling on a circumference while gravity pulls on an area, and the two scale differently — which is the whole of it.
FluidsThe angle a liquid makes with what it sits on
Everything capillarity does — climbing, beading, wicking, waterproofing — is the sign and size of one cosine, and that cosine belongs to three interfaces at once rather than to the water. Change the solid and nothing about the water has changed, yet the same five microlitres goes from a footprint 2.61 mm across to one of 0.69 mm.
FluidsThe angles a film has no choice about
A soap film pulls equally hard in every direction along itself, so wherever films meet the pulls have to sum to zero — and equal vectors summing to zero fixes the geometry completely. Three films meet at a hundred and twenty degrees and four edges at a hundred and nine point four seven, in every foam, of every liquid, at every scale, and nothing about the material appears in either number.
ThermodynamicsThe barrier a new phase has to climb
Water vapour three times supersaturated is thermodynamically desperate to condense and will sit there indefinitely if it is clean enough. The obstacle is that a droplet has to start small, and a small droplet is nearly all surface — so the first nanometre of every phase transition costs energy rather than releasing it, and what decides whether anything happens is the height of that cost divided by kT.
FluidsThe film that goes black before it bursts
A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.
ElectromagnetismThe pressure a charge puts on its own metal
Charge on a conductor sits on the surface and tries to leave. The outward pull is half epsilon-nought E squared, the half is because a charge exerts no force on itself, and setting that pull against surface tension gives the largest a charged drop is allowed to be — a number Rayleigh wrote down in 1882 and an industry now depends on.
FluidsThe ring the drop leaves behind
A drop of coffee dries into a ring rather than a disc, and nothing about coffee is responsible. The pattern is produced by a boundary condition — an edge that cannot move — and it survives replacing the coffee with anything else that will stay suspended.
FluidsThe skin that is not a skin
A drop of water behaves as though it were wrapped in a stretched membrane, and there is no membrane. What there is instead is an energy cost per unit of surface, and almost everything the apparent skin does follows from a liquid trying to have less of one.
FluidsThe small bubble blows up the big one
Connect two soap bubbles of different size and the small one empties into the large one. Everybody expects the opposite, and the reason it happens is one equation with a radius in the denominator — which also means the process runs away rather than settling.
FluidsThe surface that pulls toward the stronger side
Surface tension is usually treated as a constant of a liquid, and it is not — it depends on temperature and on what is dissolved, and a difference in it along a surface is a force along that surface — which drags the liquid underneath and needs no pressure difference at all.
FluidsThe thread that cannot stay a thread
A stream of water from a tap breaks into drops, and it does so at a spacing that is always about four and a half diameters. Nothing chooses that number — it is the wavelength that grows fastest out of a competition between all of them, and it can be computed before any water is poured.