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The thread: The exact theory is wrong — page 38

Page 38 of 38, continuing through the 335 essays this motif runs through.

335 essays carry this thread — page 38 of 38.

Surges merge down the chute until about four hundred time units, and then the train stops merging. The number of surges on a periodic chute 200 units long against time, after uniform flow at Froude number 3 is seeded with depth noise, at two resolutions, eddy viscosity 0.05. At the finer the count is 71, 55, 45, 45, 45 at times 100, 200, 400, 700 and 1000 — a spacing of 4.44; at the coarser, 63, 45, 32, 30, 30, a spacing of 6.67.

Roll waves stop merging at a spacing their fronts choose

On a steep chute the uniform flow breaks into a train of surges, and the train coarsens: a taller surge runs faster, catches the one ahead and swallows it, so the spacing grows down the chute. If the merging never stopped, the spacing in a photograph would only record how far the water had come. Marched down a long channel from noise, it stops. After a few hundred units of travel the surges stop catching each other and the train keeps its spacing for as long as it is followed. The spacing it keeps is not set by the chute or the noise but by the dissipation inside the surge fronts — which is the one part of the flow the shallow-water equations describe only roughly.

Pressed with one load, a steeper blade leaves a thinner film, and without a land a steep one cannot float at all. The film a loaded scraper leaves on the surface, in micrometres, against the angle its face makes with the surface, at a load of 100 N per metre of span on an ink of 0.1 Pa·s moving at 5 m/s: with a flat land 0.1 mm long under the edge, and with none. With the land the film is 110 µm at 10°, 1.69 at 30° and 0.268 at 60°. Without one it is the whole gap, 145 µm at 8°, and above 44° the gap the load needs is below a ten-thousandth of a nanometre: the edge is in contact.

Viscosity

A pressed blade floats on its land and leaves half the gap behind

A scraper or a doctor blade is not held at a gap; it is pressed on with a load, and it rises until the liquid it lets through carries that load. Without a flat land under its edge the liquid can carry it only through the corner's logarithm, so the gap falls exponentially with the load, and a blade steeper than about forty-four degrees cannot float at a working load at all — its edge is in contact. Give it a land a tenth of a millimetre long and the film becomes half the gap, falls as one over the load, and is a quarter of a micrometre at sixty degrees. The land is also why a blade's film grows as its edge wears.

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