How to show that the Coriolis effect is irrelevant for the whirl/vortex in the sink/bathtub?

The Coriolis acceleration goes like $-2\omega \times v$, which for the sake of an order of magnitude estimate we can take to be $a\sim \omega v$. But in order to get an observable effect, we don't just need an acceleration, we need a difference in acceleration between the two ends of the tub, which are separated by some distance $L\sim 1$ m. The accelerations differ because $v=\omega r$, and $r$ differs by $\Delta r\sim L$. The result is that the difference in acceleration is $\omega^2 L$, which is on the order of $10^{-8}$ m/s2. This is much too small to have any observable effect in an ordinary household experiment.

This explains why the Coriolis effect is important for hurricanes (large L) but not for bathtub drains (small L).

Detecting the Coriolis effect in a draining tub requires very carefully controlled experiments (Trefethen 1965; also see this web page by Baez). Lautrup 2005 gives numerical estimates showing that in order to see the Coriolis effect, the the water must be very still ($v\lesssim 0.1$ mm/s), the water must also be allowed to settle for several days, and precautions have to be taken in order to prevent convection.

Lautrup, Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World, p. 289

Trefethen, Letters to Nature 207 (1965) 1984, http://www.nature.com/nature/journal/v207/n5001/abs/2071084a0.html