There is a common myth that water flowing out from a sink should rotate in direction governed by on which hemisphere we are; this is shown false in many household experiments, but how to show it theoretically?
Fluid Dynamics – How to Show That the Coriolis Effect is Irrelevant for the Vortex in the Sink/Bathtub?
coriolis-effectfluid dynamicsvortexwater
Related Solutions
The whirl is due to the net angular momentum the water has before it starts draining, which is pretty much random.
If the circulation were due to Coriolis forces, the water would always drain in the same direction, but I did the experiment with my sink just now and observed the water to spin different directions on different trials.
The Coriolis force is proportional to the velocity of the water and the angular velocity of Earth. Earth's angular velocity is $2\pi/24\ {\rm hours}$, or about $10^{-4}\ s^{-1}$. If water's velocity as it drains is $v$ the Coriolis acceleration is about $10^{-4} v\ s^{-1}$.
The water moves about a meter while draining, which takes a time $1\ m/v$, so the total velocity imparted by Coriolis forces could be at most $10^{-4} v\ s^{-1} * 1\ m/v = 10^{-4} \ m/s$.
So the Coriolis effect is quite a small effect. But this first-order Coriolis effect does not cause the water to rotate.
The direction of Coriolis force depends on your direction of motion. All the water in your tub is moving the same direction, so the Coriolis force pushes it all the same direction. The effect is that if the bathtub starts out perfectly flat and begins draining (and it points north), all the water will get pushed east. The two edges of the tub will have very slightly different depths of water, because the Coriolis force is pushing sideways.
The Coriolis force could create "spinning" on uniformly-moving water, but only as a second-order effect. As you move away from the equator, the Coriolis force changes. This change in the Coriolis force is because the angle between "north" and the angular velocity vector of Earth changes as you move around; as you go further north (in the Northern Hemisphere) the "north" direction gets closer and closer to making a right angle with the angular velocity vector, so the Coriolis force increases in strength. The size of this effect would be proportional to the ratio of the size of your tub to the radius of Earth. That ratio is $10^{-7}$, so this effect is completely negligible.
The Coriolis force could also create some "spinning" if different parts of the water are moving different speeds. If the tub is draining to the north in the northern hemisphere, and water near the drain is moving faster than water far away, then the water near the drain would be pushed east more than water far away is. If you subtracted out the average effect of the Coriolis force, what remained would be an easterly push near the drain and a westerly push far away. This gives a clockwise spin as viewed from above.
We've already estimated the typical velocities as $\omega L$, so the angular momentum per unit mass induced this way would be on the order of $\omega L^2$ (but maybe smaller by a factor of 10). That's only $10^{-4}\ m^2/s$. To get an equivalent effect, in a tub of $100\ L$, you could give just one liter of water on the edge of the pool a velocity of a few cm/s, something you surely do many time over when removing your body from the tub.
This effect is too small to affect your bathtub, but it's still observable under the right conditions. According to Wikipedia, Otto Tumlirz conducted several experiments in the early 20th century that demonstrated the effects of the Coriolis forces on a draining tub of water. The tub was allowed to settle for 24 hours in a controlled environment before the experiment began. This was enough to damp out the residual angular momentum left over from filling the tub up to the point where Coriolis effects were dominant.
You are observing a hydraulic jump.
The Wikipedia article is very good, so I won't try to out-do it.
In brief summary, when the water starts running out from the place where it hits the sink, the same flux is spread out over a larger and larger circumference as you move out. This means the flow gets shallower and moves more slowly as you move further out.
If a wave propagates in this flow, its wave speed depends on the height of the water. Its speed relative to the flow also depends on the flow speed. So the propagation of wave changes as we move further out as the flow underneath the wave changes.
The wave itself changes the height of the water - the water is deep at the peak of the wave and shallow at a trough. Different parts of the wave moves at different speeds. This is clearly a non-linear phenomenon. Similar to waves crashing at the beach, eventually the propagating wave crashes over on itself. This causes a "hydraulic jump". The main effects are
- The speed of the flow goes down.
- The height of the water increases, converting some kinetic energy to potential.
- Some energy is lost to heat through turbulence.
The physics of water in your sink is not very easy - since the flow is so shallow, surface tension has considerable importance. You can learn more details in the Wikipedia article linked above.
Best Answer
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