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The thread: Everything happens in a thin layer — page 9

Page 9 of 10, continuing through the 84 essays this motif runs through.

84 essays carry this thread — page 9 of 10.

$How long a fluid takes to forget it was not rotating. The fraction of solid-body rotation a container's interior has reached, against time, by the two available routes. The Ekman layers on the end walls pump fluid radially and carry angular momentum inwards in a hundred seconds; diffusion alone would need ten thousand.$ Viscosity

How long a fluid takes to forget it was not rotating

Spin a container of water and the fluid inside reaches solid-body rotation in a hundred seconds rather than the three hours diffusion would need. The shortcut is the thin layers on the end walls, and the advantage they give is exactly the reciprocal of the square root of the Ekman number.

$A particle is a low-pass filter. The fraction of a fluctuating flow's velocity that a particle follows, and the phase by which it lags, against the Stokes number. At one the particle follows 71 per cent of the motion and lags by 45 degrees — which is the corner frequency of a first-order filter, arrived at from mechanics rather than from electronics.$ Regimes and numbers

A particle is a low-pass filter

A seeding particle does not report the flow; it reports the flow through its own transfer function. At a Stokes number of one it follows 71 per cent of the motion and lags it by 45 degrees, and a fifty-micron droplet at a kilohertz is following two per cent of what it is supposed to be measuring.

A band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached.

Transition and turbulence

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

A vortex sheet rolling up, at five stages. One period of an initially flat vortex sheet with a small perturbation on it, drawn at equal intervals. The perturbation grows, the sheet steepens, and the ends wind into a spiral. Nothing is added to the sheet after the first instant.

Ideal flow

A spiral is a legible record

When a vortex sheet rolls up, the fluid in it can never change places: two points on a sheet cannot pass one another. So the arms of the spiral are a map of the initial sheet, in order, and the picture is a record of the roll-up rather than a snapshot of it.

Viscosity

A duct that forgets everything but one number

Whatever is fed into a pipe, what survives a little way down it is one shape. The disturbance's higher modes decay as the square of their mode number, so the sixth is gone in thirteen centimetres where the first survives four and a half metres — and the entrance length is that one mode's decay rate and an arbitrary threshold.

Compressible flow

The gradient the heat never hears

On a flat plate at a Prandtl number of one, a boundary layer's total enthalpy is a straight-line function of its velocity, whatever the wall's temperature. Put the same layer in a pressure gradient and the straight line fails everywhere except on an insulated wall, because the gradient enters the velocity's equation and not the enthalpy's — and a favourable gradient can leave a band of gas colder than the free stream above a wall three times hotter than it.

A blade passing the vortex it shed a passage ago. The lift the blade section feels as it passes a tip vortex at five per cent of the radius. The pulse is a doublet rather than a bump — upwash on one side and downwash on the other — so the blade is pushed one way and then the other in the width of a few chords.

Circulation and lift

A blade that flies through what it shed

A rotor blade meets the tip vortex the blade in front of it left, a fifth of a second earlier, at a few per cent of the radius. What it feels is a doublet — five and a half degrees of upwash and then five and a half of downwash within a few chords — and the peak goes as one over the miss distance.

A force that depends on the molecular scale only through its logarithm. The force per unit length needed to move a 30° contact line of water at 1 mm/s, out to 1 mm, against the slip length on a logarithmic axis, from a picometre to a tenth of a millimetre. Each tenfold change in the slip length moves the force by the same fixed amount, so eight decades of the most uncertain length in the problem change the answer by a factor of about twenty — and at zero slip length the line keeps rising without end. The dashed curve takes the exact wedge's angle factor with a sharp cutoff; the solid one the thin-film wedge with Navier slip.

What is taught wrongly

The drop a no-slip wall would never let spread

Liquid touching a solid moves with it, and nearly everywhere that is as close to exact as anything in fluid mechanics. At the edge of a spreading drop it cannot be: the stress in the corner rises as one over the distance from the edge, and the force needed to move the edge is infinite. Something slips over a nanometre, and because the answer depends on that length only through its logarithm, a drop spreads at almost the same rate whatever the something is.

Three profiles that do not depend on the radius. The radial, azimuthal and axial velocities of the flow above a rotating disc, as functions of one similarity variable. The radial one is a jet: fluid thrown outward by the swirl it has picked up, peaking at 0.181 of the local disc speed a fifth of the way through the layer. The azimuthal one falls from the disc's own speed to nothing. And the axial one is the surprise — it does not vanish far from the disc but tends to a constant, so the disc draws fluid down onto itself at 0.8845 times the square root of the viscosity times the rotation rate, at every radius and for ever.

Viscosity

The solution that keeps its nonlinear term

Every exact solution before this one has been exact because the nonlinear term vanished. A rotating disc's does not vanish — at the wall it is the whole of the balance — and the reduction is exact anyway, because the radius divides out of all three momentum equations at once.

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