Re: Light Mill / electrostatic motor

[Leigh Palmer <palmer@SFU.CA>]

I said:

> > The light mill is well understood (I even understand it) and the
> > "conventional" explanation is very wrong,

to which Chuck Britton replied:

> please share some of your understanding, preferably without
> resorting to thermotranspiration and such jargon.
> (i.e. blindly invoking the authority of J.C. Maxwell just
> don't cut it!)

Agreed!

> And why does heated air push on the dark vane of the light mill?
> This is EXACTLY my point with the light mill. What is the
> difference???? A gas molecule rebounds from the dark side with
> MORE KE and momentum and hence pushes the dark side.

The temperature of the air immediately adjacent to the dark
side is higher than that on the light side, so the number
density of molecules is lower on the dark side. Consequently
the number of collisions per unit area is reduced on the
dark side relative to the number on the light side. The easy
argument must be examined in more detail, and on doing so
Maxwell concluded it was wanting. I invoke Maxwell only to
validate my claim that the conventional argument is too
simple; more must be done.

That said, I must now reconstruct a more satisfactory
explanation, something I haven't done for several years. I
will start by suggesting a slight change in the problem.
Let's consider the motion at constant velocity through a
tenuous gas of a thin plate oriented perpendicular to its
direction of motion. A temperature difference exists
between the front and rear surfaces of the plate, the rear
surface being the warmer. We maintain this temperature
difference by making the front shiny and the back black, and
illuminating the plate. Rotation of the mill plays no
significant role in this problem, and we expect that the
motion of our hypothetical plate, if it is not intrinsically
stable (I think it is), can be made so by a gyroscopic
device. The Crookes' radiometer mechanism is the same as the
mechanism that drives this hypothetical plate.

In examining this system I will claim that only the edges of
the plate play a role in its propulsion. I believe that this
has been tested by constructing a Crookes radiometer with
vanes each having a large hole in it, but I can't cite the
work. In any event There will be a stagnation pressure
difference between the front (shiny) and the rear (black) of
the moving plate, the pressure on the shiny side being
higher. of course. We will neglect radiation pressure in
this discussion. It is the retarding force due to the
stagnation pressure difference that must be balanced by the
propulsive force due to the mechanism operating at the
edges of the plate.

Since we know that the Crookes radiometer is self starting,
we may further simplify the problem by analyzing the plate
(or rather the gas at the edge of the plate) in a static
surrounding atmosphere, avoiding problems due to a net,
albeit constant, velocity flow past the edge perpendicular
to the plate. (I don't want to hear about ramjets again.)
In the static case we have a small volume of air next to
the edge, within which there is a temperature gradient
maintained by exchange with the edge. We must now
demonstrate that maintenance of such a temperature gradient
in still air can produce thrust. By hypothesis, then, the
number of molecules leaving the cold side of the volume is
the same as the number of molecules entering the cold side.
The same is true for the warm side of the volume. Thus the
rate of expulsion of molecules from the two sides is the
same, since the surroundings are isotropic and homogeneous.
Since the molecules from the warm side carry with them on
average a greater momentum, the volume will act as a jet
engine, expelling momentum, but no net material.

This mechanism will work only if the mean free path of the
molecules is comparable to or less than the thickness of
the plate, and the highest velocity will be attained when
the viscosity of the atmosphere is a minimum consistent
with this requirement. Thus there will be an optimum
pressure for the Crookes radiometer. Once the plate begins
to move there will be air flow, but the velocity of the
plate will be much less than the molecular velocities
involved. The same analysis above follows, but there will
be net flows into and out of the region of interest. The
global conservation of gas will still hold, however.

Now I have a picture in my mind. All of this happens in
the absence of gravity, or the velocities are directed
horizontally, so convection plays no propulsive role.

OK, that's the best I can do just now (it's been on my
desktop since last week). Perhaps the Collective can dot
some i's and cross some t's to make it more palatable. I
hope that I've conveyed the main idea.

Thanks for asking, Chuck.

Leigh