Illustrated Physics
physics, drawn
Concept

Bernoulli equation — where it appears

The statement that along a streamline of steady, frictionless flow the sum of pressure, kinetic energy per volume and gravitational energy per volume stays constant. It is why faster-moving fluid stands lower or presses less, from a carburettor's throat to the water beside a moving boat.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Every steady flow of gas onto a point mass. The Mach number of steady, spherical, isothermal flow round a point mass against distance from it, both on logarithmic axes, distance in units of the sonic radius, where the inflow reaches the speed of sound. Every curve is a contour of Bondi's conserved integral, which ties the Mach number to the distance. The faint curves are flows that never pass the speed of sound, or that exist only close in or only far out. Two curves cross at the saddle (1, 1): the solid one is accretion, starting from rest far away, reaching the speed of sound exactly at the sonic radius and falling supersonically within it; the dashed one is its time-reverse, a wind. Only the crossing flow joins a gas at rest to a hole that swallows it, and it fixes the rate: λ = exp(3/2)/4 = 1.120 in Ṁ = 4πλ(GM)²ρ∞/c³.

The gas a hole catches from far away

A particle drifting past a black hole is captured only if its aim is almost perfect; miss by a little and it swings round and leaves. Gas is different. Streams passing on opposite sides collide, lose their sideways motion as heat, and fall in, so a hole in warm gas draws from the whole sphere inside which its gravity beats the gas's thermal motion. Bondi worked out the flow in 1952: one steady solution passes smoothly from rest to free fall, it crosses the speed of sound at a single radius, and the rate it fixes grows as the square of the hole's mass — fast enough, left alone, to run away in a finite time.

astrophysics · Horizons A bridge that never feels the boat crossing it. The weight of water an aqueduct 300 m long, 6.5 m wide and 2.7 m deep carries, against the position of an 80-tonne boat crossing it, with ten kilometres of canal joined to it. The boat pushes its own weight of water out of the aqueduct, and the water spreads along the whole reach, raising it by 1.2 millimetres. The load stays at 5265 tonnes whether the boat is on the aqueduct or not; a road bridge would gain the vehicle's whole weight. The aqueduct is designed for its water alone, and a boat crossing it is invisible to the structure.

The bridge that does not feel the boat

An aqueduct carries a canal across a valley, and a barge of eighty tonnes crosses it. The aqueduct carries no more weight with the barge on it than without, because a floating barge pushes out exactly its own weight of water, and the water goes elsewhere. Boat lifts are built on the same fact: a caisson of water weighs the same whatever floats in it, so two caissons balance whatever boats they carry, and the only thing that needs measuring is the water level.

fluids · Buoyancy

Named alongside it

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

AccretionAngular momentumArchimedes principleBlack holeBoat liftBondi radiusBuoyancyContinuityEddington limitHydrostatic pressureMach numberSound speed

All concepts