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Re: Bernoulli's relationship (not posted)



Neat problem, Robert. I have a lot of trouble with it before I get to
any "Bernoulli" argument (which I would avoid anyway).

This problem is unphysical if viscosity is assumed to be zero. An
active propulsive force would be unnecessary in that case, so your
propeller would become trivial.

First, consider the laminar flow of incompressible liquid in a
horizontal channel in a region where there is no active propulsion.
Let's take a look at *average* velocity (which I'll call "velocity"
from now onward) as a function of distance along the channel. I know
this is unreasonable, and maybe that will surface. If we look at the
steady-state situation we see that there must exist a velocity
gradient in the direction of motion. The dissipative nature of the
viscous force means that the velocity gradient must be negative in
that direction. Continuity (depth times velocity is a independent of
position along the channel) demands, then, that there be a positive
surface level gradient; the water gets deeper as it progresses along
the channel. This is a counterintuitive result, perhaps, but I've
been hosing out my rain gutters lately (more to do Saturday) and I
can testify that it certainly works over fairly long finite troughs.

I'm going to resort to a theoretician's trick here; I'm going to
impose periodic boundary conditions. I need a distributed source of
propulsion along the channel to make up for dissipation, so I'll put
in many of your propellers at equal intervals along the channel.
Moreover, I'm going to idealize the propeller planes by simply
characterizing it as imposing a discontinuous increase in velocity at
the plane. This implies that there is a discontinuous decrease in
depth just downstream from the propeller. (This propeller plane will
also establish roughly constant velocity over the cross section of
the channel, making the average velocity model more appropriate.) The
surface profile on both sides of the propeller plane has roughly the
same positive gradient. If the trough is horizontal on the bottom
then the propellers make the water flow uphill. The water pressure
along the trough increases with distance, of course.

[I think this conceptual model serves well to construct a qualitative
picture. It has no counterintuitive features, and if you think it
has, please come over to my house Saturday and help me clean out my
rain gutters; I'll show you.]

The direction of flow through your hypothetical hole depends entirely
upon how far downstream from a propeller plane you drill it. Just
beyond the propeller water will flow into the faster moving stream
because it is shallower than the stationary trough water. Just before
the propeller plane water will flow into the stationary trough
because the depth is greater and the pressure is higher in the moving
water. I guess that for "high" velocities this might be incorrect
(Bernoulli's effect) but I would have to perform a tougher
calculation to see that. I would consider that result unlikely, and
it doesn't come easily out of my simple model.

As I said before, it is a neat problem in that it makes me construct
models with which I am unfamiliar. This is merely a first iteration
in that process, and I'm sure that the many creative souls here will
have much to add. I just gave a midterm, so I'm gonna go away.

It's your turns, JD & JM.

Leigh