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The thread: Taught wrongly, everywhere — page 26

Page 26 of 26, continuing through the 230 essays this motif runs through.

230 essays carry this thread — page 26 of 26.

Between about 4/3 of the lift and just over twice it, opening the bridge cuts the flow, by up to nearly half. Total flow against the head, with a lift of 1 in each route. With the bridge shut, flow starts at H = 1 and rises by nearly 2 per unit head. With a bridge of resistance 0.001, the two small-bore pipes in series carry H/2 from zero head. Just below H = 2 the flows are 1.82 and 0.995.

An extra pipe can cut the flow

Connect a new pipe between two points in a network and more paths exist, so more water should flow. In a network of identical resistors that is a theorem. Mix a pipe whose loss grows with the flow and an element whose loss does not, and it is false: opening a bridge can nearly halve the flow, at the same head, with the same pipes. It's Braess's paradox from road traffic, in a form that can be built on a bench. The reason is one short argument about what a network minimises, and what it does not.

One formula draws a smooth layer, a row of cat's eyes, and in the limit a row of point vortices. Streamlines of ψ = ln(cosh y + A cos x) at A = 0.5, over two periods, in the frame of the eyes. The separatrix encloses eyes of half-height 1.32; the peak vorticity at their centres is 3 times the layer's, and on the braids 0.333.

Ideal flow

A shear layer and a row of vortices are one formula, and not one flow

There is an exact steady solution of Euler's equations with one parameter in it, written in a line by Stuart in 1967. At one end it is the smooth shear layer every instability calculation starts from, and at the other it is a row of point vortices, with cat's-eye streamlines between. It looks exactly like the history of a layer rolling up, and textbooks draw it next to one. It cannot be. Each member has a different area of fluid at each vorticity and a different energy, and those are the two things an inviscid flow cannot change.

A cart that leaks straight down keeps its speed, and the rule that says it gains one has lost track of what left. A frictionless cart losing three quarters of its mass at a steady rate, moving at speed 1. With its leak leaving backward at 0.5 of that speed relative to the cart it ends at 1.69. The rule F = d(mv)/dt, read as d(mv)/dt = 0, says the cart reaches 4 whatever the leak does.

Flows and fields

Mass leaving a cart pushes nothing unless it leaves with a speed of its own

A cart leaking straight down keeps its speed, though its mass falls and every first course says force is the rate of change of mass times velocity. Apply that form to the leak and the cart speeds up fourfold, at a rate that depends on who is watching, which no force may. The transport theorem has the right rule in a line: only what crosses the boundary at a speed relative to it pushes. The rule then does four more things, from a cart that slows with a constant mass to a tank whose thrust is exactly twice the missing wall's.

The friction a coil adds is one curve in one number, and it is near a square root at large Dean number. Friction over a straight pipe's, at the same flow, against De = Re √(a/R): 1.01, 1.13, 1.4, 1.79, 2.36 at De = 11.6, 30, 70, 150, 300, beside White's fit to measurements and a large-Dean boundary-layer law. Below about 11.6 the ratio is one within a per cent.

Fluids at work

A coil's friction is one number, and the coil holds its flow laminar

Bend a pipe into a loose coil and the flow inside is not the same flow bent. The fast core is thrown outward, slow fluid returns round the walls, and a pair of counter-rotating cells appears, about a tenth of the mean speed in an ordinary coil. Dean showed in 1927 that one number governs all of it: Reynolds times the square root of the tightness. Below about 12 nothing changes. Above it the friction rises by a curve the coil's geometry never enters again. And the same secondary flow does something unexpected to transition. A coil stays laminar at four or five times the Reynolds number that breaks a straight pipe.

Two equal balloons below the peak even out, past the peak split into a small one and a large one, and at great stretch even out again. Where two balloons that start one hundredth apart in stretch finish, against that starting stretch. From 1.2 they settle at 1.2 and 1.2; from 2, at 1.15 and 2.44; from 3, at 1.11 and 3.75; from 5.5, at 5.5 and 5.5. Two balloons of different size can then sit at one pressure, and no flow in the tube changes that.

What is taught wrongly

Two connected balloons part over a range of stretch, and even out on either side

Air flows from high pressure to low until the pressures match, and everyone concludes that two connected balloons end up the same size. Soap bubbles famously do not. Rubber does something stranger than either: a pair evens out at low stretch, parts into one large and one small balloon over a wide middle range, and evens out again at high stretch. Equal pressure is all the air ever asks for, and one rubber curve lets the same pressure be held at three different sizes. What the pair does also depends on how it was filled.

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