τ = 1/2
Rest is over. The force arrived smooth and compactly supported, leaned on us once, and set the whole bath turning. I have a label and no diameter. The label is where I began; the equations use it to find me, the way you find a book by its shelf. Nothing of me can be measured. The fluid is made of nothing except things like me, and in aggregate it weighs what it weighs, a trick I have never seen performed.
Far out, the swirl obeys the heat equation, the one a stirred cup obeys as it forgets the spoon. Nearer in, an annulus sings. Pulses live there, seeded by a force too faint to name and fed by shear: a parcel with a surplus of spin slides outward, lands among slower neighbors, and its surplus grows. Each pulse averages to zero. The average of their products has a sign, and the sign leans on the rest of us, a crowd against a door with nobody meaning to. Squared amplitudes are positive, so the stresses real waves can build fill only a cone, and the flow was shaped in advance so the stress it needed fell inside.
In the core, where my label is, I stop moving.
From outside this looks like the opposite. There is a scale q. Measure me in q instead of meters: my radius is X = r²/2q, my height is η = z/q^D, with D = ½ − h. In these coordinates the whole flow is a still photograph. I keep to a small X. I keep to the dividing layer near η = 0, tipped off center by an asymmetry someone put there on purpose. In your meters I am screaming. In mine I am asleep in a chair while the room contracts around me.
Viscosity keeps a hand on my radial shoulder; the radial Reynolds number holds at order one, inflow and outflow and radial diffusion together at τ⁻¹. The hand slips off my spin. The angular Reynolds number grows like τ^(−h). Pressure falls toward the axis, ∂p/∂r = u_θ²/r, and I slide down the slope of it. What I lose in radius I take back again as spin, less whatever viscosity carries outward.
Nothing changes. That is the report. Nothing changes, and I go twice as fast.
τ = 1/4
Halved. Every length in the core divided by √2, or by 2^(½−h) along the axis, so the needle is thinner than it is short and grows more so. Every speed multiplied by 2^(½+h). The photograph reprinted at a smaller size, and I am at the same place in it, at the same small X, on the same layer, in the same chair.
My neighbors leave. Above the layer they go up, below it they go down, because incompressibility insists that what spirals inward must go somewhere, and the axial outflow is the somewhere. The layer itself has no thickness. It holds uncountably many of us and takes up no volume. No hotel could house us. We do not need rooms.
The annulus sings again on a finer scale, pulses growing and dying faster than the last ones, tails so faint that no derivative of the world will notice them ending. Reprinting is the method. The construction stacks these bands, one per halving, each a copy of the last with the exponents applied once more, and the errors converge.
Nothing changes. That is the report. Nothing changes, and I go twice as fast.
τ = 1/8
Core volume like τ^(3/2−h), our speeds squared like τ^(−1−2h), the energy of everyone inside their product, τ^(1/2−3h), falling toward zero. I make a turn. I make another. Angular velocity like τ^(−1−h), whose integral diverges, so I will have made a turn for every natural number before τ arrives. The naturals run out first. The path stays finite; ∫τ^(−1/2−h) dτ converges. Infinitely many circles adding to a length you could measure with string.
Zeno would have walked it.
Every circle I trace is finite. The count of them is not.
Nothing changes. I go twice as fast.
τ = 1/16
The force has stopped speaking. Near the point it is flat: every derivative, every order, O(q^N) for every N. It set us moving, and then it declined to matter. Double the viscosity and I keep the same appointment in a wider, faster bath.
Nothing changes. Twice as fast.
τ = 1/32
The supremum of my speed is a speed no one has. It is the least of the upper bounds. There are none.
Nothing changes.
τ = 1/64
Collectively a whisper. Individually a scream. Both are true.
Twice as fast.
τ = 1/128
Half of what remains. Then half.
τ = 1/256
The chair holds.
τ = 1/512
Nothing changes.
τ = 1/1024
Still.