What force causes the sand in the bottom of a spinning bucket of water
to move to the center of the bucket as the bucket is stopped from
spinning?
For example, when the bucket starts spinning, the centrifugal force
sends the sand to the outside edge of the bucket, where the edge
applies the centripetal force to contain the sand.  The magnitude of
these forces being mv^2/r (m times v squared over r) where m is the
mass, v is the speed and r is the radius.  The question above is with
water in the bucket, since with only air, the sand remains at the edge
as the bucket stops spinning.

I have no evidence or formula, only my personal thoughts:

The water in the bucket looks like a  V   when it is spinning in the
bucket due to the centrifigal force.  Similar to this (but in 3D):
|        | 
|\      /|
| \    / |
|  \  /  |
|___\/___|

When the bucket stops spinning the force holding the water in the V
formation is gone and the water rushes to the middle of the bucket to
fill the void (as any liquid would do).  While rushing to the middle,
the sand is swept along with it until it all crashes in the center of
the bucket where the sand then remains.

I also have no evidence or formula, but personal thoughts.

If the bucket's been spinning, when it's stopped, wouldn't
the water continue spinning in a whirlpool, sucking the 
sand to the center?

First the difference between the air and water case can be attributed
to the Ekman number, E = kin.viscosity/(omega*l^2)  l being the
characteristic length. omega being the rate of rotation (rad/s).

Second during spindown the turbulent boundary layer at the wall and at
the bottom of the bucket will grow.  This should attribute to pushing
particles initially away.

A further reason particles may move toward the center of your bucket
is Rossby waves.  They move from one side of the bucket to the other
during 'spin-down'. The thoery of their formation is due to a
imperfect cylinder base, even a small deviation from a 90 degree
wall/floor may produce them.

Note that when you stop spinning the bucket the flow will also have
inherit instabilities and may have a turbulent decay.

Resources: 
-Personal Knowledge
-Rossby, CG 1939 Relatoins between variations in intensity of the
zonal circulation of the atmosphere and the displacements of
semi-permanent centers of actions, J Mar. Res 2 38-55.  (pages 86)
-Schlichting, H Boundary Layer Theory, 8th edition, springer.
-Batchelor, GK The THeory of Rotating Fluids, 1968, cambridge at the
university press.

I highly recommend the Theory of Rotating Fluids book - I think you
would enjoy it.  I know I did! :)

The comments by Jack_of_few_trades-ga and sublime1-ga were similar to
my thoughts until I filled the bucket to the rim and placed a lid on
it.  (I used a plastic bucket on a phonograph turntable... if you are
old enough to know what that is...) Turned on the turntable and the
sand went to the edge.  Turned off the turntable and the sand still
went to the center.  I first discovered this phenomenon with a wading
pool 10' in diameter and 2' deep.  I would walk around with the leaf
strainer next to the edge to get the grass particles out.  I noticed
that when I stopped all the dirt went to the center of the pool.  So I
started a siphon with a garden hose and sucked all the dirt out of the
center.  I kept the pool clean in this manner from then on.

The comments by Saem_aero-ga may be in the right direction.  I'm going
to read the book he suggested to see what I can find out.

rwgoodwin,
  I'm glad your going to take a look at this book.  However I have to
warn you that it is an advanced text. Just wanted to drop you a line
so that you weren't surprised.

Best,
Steve

Steve,
Thanks for your input, probably good enough for the $20.  I have a
degree in math and minored in physics for teaching many years ago. 
This problem has always puzzled me and I hoped for a solution here. 
You have at least sent me in the right direction, I think.  Thanks
again, R. Wilson Goodwin

What if you added some particles that were close to water density so
you could see the water movement?

xcarlx-ga,
I did try some fishing floats that I could alter the density by adding
water to them.  It didn't seem to make any difference.  The force is
greater than the movement of the water.  When the turntable was turned
on the sand on the bottom went to the edge.  However, once the system
had stabilized, (left on for 5 minutes) when I added more sand it went
to the bottom and stayed.  Meaning it did NOT move to the edge.  When
turned off, the force looks just as great to the center as it did to
the edge when turned on.  Wilson