[From Bill Powers (931220.0710 MST)]

Martin Taylor (931219) --

On further consideration, I believe your analysis of how a vortex

forms is incomplete. It is not the Bernoulli effect, but

conservation of angular momentum that gives rise to a bathtub

vortex.

In the laminar flow case, fluid approaches the center radially;

there must be a radial height gradient to sustain the radial

flow. If there is a slight deflection of any part of this flow so

it is not radial, a packet of water will follow a path tangential

to some small circle around the center. The initial angular

momentum will be (save for friction) preserved, so as the packet

approaches the center, it will have greater and greater angular

velocity as the radius decreases. Shear friction will then

accelerate packets of water with a greater radius, ultimately

giving the entire body of water a spin that decreases in angular

velocity as radius increases. Equilibrium will be reached when

the distribution of angular velocity is sufficient to make the

shear forces times speed of rotation everywhere equal to the work

done by the descending flow. The work done by the vertical

component of velocity is then equal to the energy dissipated by

shear friction over the whole volume. There must, of course, be

sufficient head to produce enough energy to overcome frictional

losses, which are a nonlinear function of shear rate.

The rotation sets up a counterpressure due to centrifugal force.

As the vortex develops, the flow through the drain hole

decreases, because of the increasing counterpressure. I

discovered as a child that the last of the bathwater will drain

much faster if you keep the vortex from forming. Equilibrium

occurs when the vortex has a shape such that the gravitational

force minus the centrifugal force, both projected into lines of

flow, is just balanced by frictional forces and acceleration due

to the converging flow, everywhere within the affected volume.

This gives the typical curved funnel shape, with the greatest

angular velocity at the surface of the funnel nearest the drain

hole, and thus the greatest counterpressure at that height.

The angular momentum of the stable vortex is, I suspect, zero if

there is no continuing tangential disturbance. Assume that the

vortex as a whole is rotating clockwise. If you follow any

orbiting packet, you find that a packet just inside that radius

is moving faster, and one just outside it is moving slower.

Drawing a line between the inside and outside packets, you would

see that relative to the center packet there is a

counterclockwise rotation superimposed on the general clockwise

rotation. Conservation of momentum suggests that the sum of all

these angular momenta is zero.

While a vortex involves a complex set of actions and reactions in

three dimensions, it is still an equilibrium system in which the

steady state involves a balance of forces acting in both

directions at once. The forward forces are everywhere identical

to the reverse forces when friction and acceleration are taken

into account. A perturbation of the vortex forces it into a

nonequilibrium state, which is then restored as the excess energy

(or deficiency of energy) is rebalanced. All of the restoring

forces come from the deviation caused by the perturbation, and

the interaction of all parts of the flow. While I don't know how

to calculate the power gain involved, I strongly suspect that it

is less than 1.

You are incorrect, by the way, in saying that a tendency to

return to equilibrium after a disturbance is removed requires a

loop gain greater than 1. It requires only a loop gain greater

than zero. A control system with a loop gain of 0.5 will produce

an output that is 0.5/1.5 or 1/3 the magnitude of a disturbance.

When the disturbing influence is removed, the output will restore

the system to equilbrium.

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I really object to calling a vortex a "self-organizing" system in

the context of any quantitative discussion. Organization is a

perception; who is to say whether a vortex impresses everyone as

more organized than a laminar flow, or less? The "organized-ness"

is based on the familiarity of the vortex shape. Equating

organization to a function of entropy is gratuitous; One might

as well just compute entropy and leave "organization" out of it

because that, in effect, is what is done.

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Best,

Bill P.