Illustrated Physics
physics, drawn
Concept

Reynolds number — where it appears

The ratio of inertial to viscous forces in a flow, its speed times a length divided by the kinematic viscosity. Below about one viscosity dominates and flow is reversible and smooth; at thousands it becomes turbulent, and two flows with the same value and shape are similar.

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

The flow past a thread takes thousands of widths to recover. The speed of the fluid beside a body, at right angles to the stream, as a fraction of the free stream, against distance from the body's axis in body radii, in slow viscous flow. Past a sphere, from Stokes's solution, the flow is back to 90 per cent of the stream by 7.5 radii whatever the Reynolds number. Past a cylinder, from Lamb's inner solution, the speed grows only as the logarithm of distance, with a slope set by the Reynolds number: it reaches 90 per cent of the stream at 29, 232, 1,843 and 1.5 × 10⁴ radii for Re = 10⁻¹, 10⁻², 10⁻³ and 10⁻⁴ in turn. Each cylinder curve is drawn out to where the inner solution reaches the stream speed, beyond which Oseen's outer flow takes over. Each tenfold drop in the Reynolds number pushes the recovery ten times further out, because what finally stops the disturbance is inertia, at a distance of order ν/U.

The drag a distant wall decides

A sphere moving slowly through a liquid has a drag that belongs to the sphere: six pi times the viscosity, the radius and the speed, and the liquid a few radii away hardly knows it is there. An infinitely long cylinder has no such drag. The equations of slow viscous flow have no solution past it at all — Stokes found this in 1851 — because a force spread over circles instead of spheres never finishes spreading. What rescues the cylinder is whatever distant thing supplies a largest length, and its drag carries the logarithm of that length for ever after.

fluids · Viscosity Three vessels emptying through the same hole. The water level against time in three vessels that each hold 25 litres to a depth of 35 cm and drain through the same 0.8 mm hole in the floor, the outflow following Torricelli's law. The cylinder empties in 16.0 hours, but its level falls fast at first and slowly at the end — a parabola in time — and it is half empty in depth after only 32% of the run. The bowl whose radius grows as the fourth root of the height lowers its level at one rate from top to bottom, a straight line, and empties in 12.9 hours. A straight-sided cone sloping outwards, the shape a potter can actually throw, comes between: half its depth goes in 49% of its run.

The vessel whose level falls evenly

Water leaves a hole in the floor of a vessel faster when the vessel is full, because the pressure at the hole is the depth of water above it. In a cylinder that makes the level fall quickly at first and slowly at the end, so hour marks on its wall crowd towards the bottom. A level that falls at one rate needs a vessel that widens as the outflow quickens: its radius must grow as the fourth root of the height. The Egyptians who made the oldest surviving water clock, about 1400 BC, gave it sloping sides, which halves the error of a cylinder. They also gave it a scale for every month, because the hours they wanted to count were not all the same length.

fluids · Hydrostatics

Named alongside it

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

Boundary conditionsDimensional analysisDischarge coefficientDragHydrostatic pressureIsochronismLogarithmPoiseuilleStokes flowTautochroneTorricellis lawViscosity

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