Discussion of the Question 04/99
CUP-A-TEAThe question was:
When you finish stirring sugar into your cup of tea the water comes to rest in a few seconds. Is the decay of the water's rotation caused primarily by the walls of the cup or by its bottom?
(9/99) Andrew Wiggin (e-mail awiggin@hotmail.com) sent us the following e-mail (slightly edited):
There's no order of magnitude difference between the two effects (bottom vs. walls), but I'm still fairly confident which way that factor of 2 or 3 goes.
We have definite evidence for convection (piling up of sugar at the center of the cup), and we know that takes up energy, so the bottom definitely has a thing going for it. On the other hand the walls do have a bigger area, but if we get into the dependence on the dimensions of the cup, things will get nasty...
Also, I tried another experiment; if you shake the cup, the oscillations do seem to go on for longer than the rotation when you stir it. Admittedly, it's hard to do this experiment in an unbiased way, and the geometry complicates it a bit. But think about shaking a rectangular aquarium, or just disturbing water in a baby's bath tub, the way I remember such situations, there is definitely less damping there
(10/00)John Moore from Bio-Imaging Research (e-mail jmoore@birinc.com) sent us the follwoing e-mail (slightly edited):
The local frictional force (to a first approximation) depends on the velocity, which is probably constant at its peripheral value V along the sidewall of height H, but decreases linearly from the outer radius R towards the center on the bottom. Integrating the local force times the area, we have V x 2R x H on the sidewalls, and V x 2(R2)/3 on the bottom. Thus, if R > 3H, the bottom will have more influence. But few cups are six time wider than they are high, so the snswer favors the walls. Note : the flow pattern won't be as simple as assumed above, but it's unlikely to change the conclusion for reasonable cups.
Note: An exact solution will depend on how the rotational flow was set up: was the cup rotated as a whole (and for how long?), or was it stirred (and from the center or the outside, and with a spoon or something else?)? However it is started, the three-dimensional flow field will be continuously altering, which means that the sidewall-to-bottom force ratio will also be changing. If one insists on an exact answer, the initial conditions must also be specified exactly.
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