Buoyancy — where it appears
Named by 15 essays across 5 fields — each of them below, with the objects they name alongside it.
A threshold with a closed form
A layer of fluid heated from below sits still until buoyancy overcomes both diffusions at once, and then it convects. Unlike every other threshold in this field, that one is an eigenvalue with an exact answer, and the answer is 27π⁴/4.
The tracer that is not one
Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.
The bed that weighs itself
Blow hard enough through a pile of sand and the pressure drop stops rising. It cannot rise — a control volume round the bed says the drop can never exceed the buoyant weight of the solid in it, and at the velocity where the two meet the bed stops being a structure and starts being a fluid.
A speed nobody imposed
Every regime number in this collection contains a velocity somebody chose. Natural convection has none: a warm plate makes its own flow, and the Grashof number is what is left when the speed is taken out. The Reynolds number of the result — six thousand, on an ordinary radiator — is an output of the solution rather than a setting on an apparatus.
Sufficient, and not necessary
A stratified shear layer whose Richardson number exceeds a quarter everywhere cannot go unstable. That is a theorem with an exact number in it. What it does not say — and what it is constantly read as saying — is that a layer below a quarter will.
Hexagons remember how the heat was turned up
A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.
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.
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.
A thin interface keeps its waves past a quarter
Miles' quarter rules a shear layer whose density changes over the same depth as its velocity. Make the density interface three times thinner and the stationary billow dies early, but a pair of travelling waves takes its place and is still growing at five times the quarter. No theorem is broken: at the edges of the shear, where the waves draw their energy, the local Richardson number has fallen to nothing.
The warm side pulls a bubble with no force on it
A bubble in a liquid whose temperature varies from place to place moves towards the warm side, with no gravity and nothing pushing it. The surface tension is lower where the liquid is warmer, the surface is pulled towards the cold pole, and the bubble goes the other way. Young, Goldstein and Block's speed comes out of four interface conditions and one more: that the total force on the bubble is zero. That one condition removes the point force a sinking or rising body always carries, so the bubble's disturbance dies a hundred times faster with distance, and the radius at which it balances its own buoyancy has no viscosity in it.
Over a plate in mercury the warm layer flows as if inviscid
The exact solution for a warm vertical plate has been tabulated since 1953 and is usually shot from the wall, which fails in a liquid metal unless the shot starts within a thousandth of the answer. Posed instead as one algebraic system, it solves from a Prandtl number of a hundred-thousandth to a hundred thousand — and shows that in mercury the layer's speed is set without viscosity, that the integral method runs low there rather than high, and that air sits where neither limiting law is within a fifth.
Stable air takes back the momentum a wing pushes down
Behind a wing in neutral air the trailing vortices carry the downward momentum the wing gave the air for ever, and a face across the wake keeps its share of the weight however far back it is drawn. In stably stratified air the column the pair carries down is lighter than the air around it, and buoyancy pushes back. The pair stops a few hundred metres down, its momentum returned, after a time that is the same for every aeroplane — 2.33 divided by the buoyancy frequency, about four minutes — and from there on the whole weight crosses the box's upper and lower faces as pressure.
Laid flat, a warm plate is pushed by the pressure of its own layer
A warm vertical plate drives its layer with buoyancy acting along it. Lay the plate flat and buoyancy acts across the layer instead, so the only push along it is the pressure deficit its own warm fluid leaves under the cold — deeper where the layer is thicker. That push gives a heat transfer growing as the fifth root of the Grashof number, not the fourth, works only on the plate's upper face, and survives a tilt of a degree for a third of a metre.
Above a warm wire the plume speeds up as it rises
A plume above a line of heat has no wall, no imposed speed and one conserved quantity, the heat it carries. That alone fixes a centreline speed growing as the fifth root of height — the only one of the four common plumes that accelerates — a temperature excess falling as the three-fifths power, and a warm core that is narrower than the moving sheet around it in any fluid more viscous than five ninths of its own diffusivity. At that Prandtl number the equations have a closed form, and the solver lands on it to three parts in a million.
A wake that sheds its fluid sinks further, and past a point does not stop
In stable air the pair of trailing vortices behind a wing carries a column of lighter air down with it, and buoyancy pushes the column back until the pair has given up its momentum and stops. That picture keeps every parcel of the column. A real pair sheds the light air as it sinks, and shedding does two things at once: it takes away part of the momentum the buoyancy had to return, and it takes away the buoyancy that was returning it. The second wins. A pair that sheds a little stops later and deeper; one that sheds faster than a threshold loses its column before buoyancy can stop it. The shed air keeps a quarter of the momentum swinging up and down behind the aeroplane.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitLinear stabilityEigenvalueHeat transferMeasurementPrandtl numberSimilarity solutionStratificationSurface tensionBoundary layerRegimeThreshold