I agree that it is obvious, but how does one develop a perturbation expression in which the zeroth order approximation is $x_0=\delta/(\varepsilon+\delta)$ and the first order approximation gives the error term $\mathcal{O}(\varepsilon)$ to order $\mathcal{O}(\varepsilon^2)$. I don't know how to choose an expansion which gives this particular approximation -__-

Thinking further, the 2D problem (equivalent to a fluid in a rotating cylindrical shell) is very easy to solve, and the time for homogenisation is on the order of $r^2/\nu$, where $r$ is the cylinder's radius and $\nu$ is the kinematic viscosity. It seems reasonable that we'd expect a similar time scale for a rotating sphere also. For liquid water, this gives us roughly a 1 second homogenisation time for a droplet of 1mm radius. It seems a lot can happen inside rain-drops, but not inside droplets of diameter less $500 \mu m$ or so.

@WillieWong That is so simple I dismissed it off hand :) Given that this is the long-term solution, I guess the better question would be how quickly is it dynamically approached from the time the shell begins rotating. I imagine that would be a much more difficult problem to solve analytically though...

I found this characterization very helpful, thanks. The only thing is, as you mention, some elements of $U\otimes V$, like $x_1\otimes y_1 + x_2\otimes y_2$, don't correspond directly to things you can plug into a bilinear map. So perhaps the elements of $U\otimes V$ are the things you can build from the things you can plug into a bilinear map, using definitions of addition and scalar multiplication on these objects consistent with bilinearity? :D