Illustrated Physics
physics, drawn
Concept

Neutral buoyancy — where it appears

The condition in which a submerged body's weight exactly equals its upthrust, so it neither rises nor sinks. Whether that depth is stable depends on a comparison of gradients: the surrounding fluid's density must rise with depth faster than the body's own does under compression.

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

Buoyancy that falls away as the body sinks. The net upward force on a body containing a little gas, against how deep it has been taken, for 3 gas fractions. The weight does not change with depth. The buoyancy does, because the gas obeys Boyle's law and the pressure rises by an atmosphere every ten metres, so a body that displaced its own weight at the surface displaces less at depth. Every curve therefore slopes downward, and that slope is the whole point: where a curve crosses zero the body is in equilibrium, and the crossing is always from above, which makes every one of these equilibria unstable. Push the body a little deeper and the force does not push back — it turns downward and grows. The crossings drawn are at 8.2 m, 10.0 m, 13.6 m, and a body sitting at one of them is balanced in the sense that a pencil is balanced on its point. This is why a diver at neutral buoyancy has to keep adjusting, why a submarine's depth is held by hydroplanes and not by ballast alone, and why a fish that loses the use of its swim bladder sinks rather than drifting.

The depth past which it must sink

A body carrying a pocket of gas can be trimmed to hang motionless in water at exactly one depth. Push it a little deeper and it does not come back — the gas compresses, the buoyancy falls, and the equilibrium turns out to have been balanced on its point.

fluids · Buoyancy A body at an interface, and the difference that holds it. The net upward force on a ten-centimetre cube straddling the boundary between two fluids, against how far its underside sits below that boundary — each curve scaled by its own largest value so that three very different cases fit on one axis. The displaced weight has to be counted twice, once with each density, so the force is linear in the displacement with a stiffness set by gravity, the cube's cross-section and the difference of the two densities — the difference, and neither of them alone. air over water holds the cube with its underside 60 per cent of the way through, at a stiffness of 97.8 newtons per metre; oil over water holds the cube with its underside 47 per cent of the way through, at a stiffness of 14.5 newtons per metre; water over mercury holds the cube with its underside 24 per cent of the way through, at a stiffness of 1229.4 newtons per metre. Both numbers are read off the drawn curve rather than substituted. The consequence is the one worth the figure: a body at an oil–water interface is held six times less stiffly than the same body at an air–water one, because the density difference is six times smaller, and the ordinary intuition that a denser fluid holds a body more firmly is exactly wrong — what matters is the contrast across the surface the body is sitting in.

The body that displaces two things

Every earlier argument has a body wetted by one fluid, so the displaced weight is a volume times a density. A body at an interface displaces two, and what holds it is the *difference* between them — so the same block is held six times less stiffly at an oil–water boundary than at an air–water one. In a continuously stratified column the neutral depth becomes stable, which is the exact opposite of the compressible case.

fluids · Buoyancy A plume rising into a stratified fluid, until it cannot. A turbulent plume from a small steady source of buoyancy, rising through a fluid whose density increases steadily downward, integrated from the Morton–Taylor–Turner equations with an entrainment coefficient of 0.1; height across, in units of F₀^(1/4)N^(−3/4), the only length the source's buoyancy flux and the stratification can make. Up: the buoyancy flux, as a fraction of the source's (solid); the momentum flux, as a fraction of its peak (dashed); and the plume's radius, as a fraction of its radius at the neutral level (dotted). The buoyancy flux falls from the start, because the fluid the plume entrains came from lower down and is denser than the fluid it is carried up into; it reaches zero at 4.37 — the level at which the plume will spread sideways — and goes negative. The momentum flux, built up by the buoyancy, peaks at 4.37 and is then spent carrying the now-heavy plume on to 5.75, where it stops. The radius grows almost in proportion to height until near the top, where the model's top-hat radius runs away as the momentum vanishes — the point at which the plume stops being a plume and becomes a spreading intrusion the equations do not describe.

The plume that stops below the surface

Hot smoke rises, oil and gas from a seabed leak rise, and in a fluid of uniform density they would go on rising for ever. Real oceans and atmospheres are stratified, denser lower down, and a rising plume carries its own downfall with it: it rises by drawing in the fluid around it, and the fluid it draws in near the bottom is denser than the fluid it is carried up into. Its buoyancy runs out, it overshoots, and it spreads sideways at a height set only by how buoyant its source is and how strongly the surroundings are layered — a quarter power of one and a minus three-quarters power of the other.

fluids · Stratification

Named alongside it

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

StratificationBuoyancyCompressibilityEquilibriumStabilityArchimedes principleBoyle lawBuoyancy frequencyDensityDimensional analysisDisplaced volumeEntrainment

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