Often in sinks, toilets, bathtubs, etc., the water is already moving, though
maybe very slowly. As the water drains, that slow rotation becomes more
visible. In the case of a sink or bathtub, where the width of the pool
decreases as it drains, the rotation increases speed as it drains. This is
because angular momentum is conserved (think of a figure skater speeding up
as she pulls her arms into her body -- it is the same effect). In a draining
tank, the water by the drain has the least momentum. As it leaves, the
remaining water with more momentum takes its place. If an object with
angular momentum moves closer to the center of its rotation, it speeds up.
So, the water gradually picks up speed, and you see rotation.
Even if there is *no* angular momentum to start with, fluid can start to
rotate. This is called a "secondary flow". To understand why secondary flows
develop, you have to understand the nature of viscosity. When molecules of a
liquid are attracted to each other, they resist being pulled apart. For
example, when you move your hand through water, you are dragging molecules
of water with your hand. Some of the resistance is simply the inertia of the
water, but much of it is viscosity. The molecules that you accelerate pull
on the ones next to them, and those next to them, etc. Moving water
molecules across each other, a 'sideways' force, is called 'shear'.
Resistance to shear is called viscosity.
Imagine a large cylindrical vessel full of water with a small drain at the
bottom. When you open the drain, water starts to flow down the hole. Of
course you have taken great care to ensure the water is completely
stationary first, and that opening the drain does not perturb it. As the
water flows downward, it drags the molecules around it due to viscosity. At
the point of the drain, some of the molecules go down the drain, but other
ones cannot fit. Yet, they have still gained some energy by being dragged by
the ones that did go down the drain. They have to go somewhere, and since
they cannot go down, and gravity makes it hard to go up, they go sideways.
Over time, they start a rotational flow, called 'secondary flow'. Over a
short period of time, viscosity, caused by the molecules' mutual attraction
to each other, ensures that they move together in the same direction. In
time, a vessel with a drain full of stationary water will develop a quite
noticeable rotation due to secondary flow.
Another place where secondary flows commonly occur is in tea. Here, the
opposite occurs: a rotational flow causes vertical motion. If you drink
green tea, watch the leaves as you stir it. Even though you are stirring the
tea in a rotational direction, you can see the leaves are pushed upward. The
upward motion, caused by viscosity in response to the rotation, is another
example of secondary flow. Of course, depending on how you stir, it could be
your spoon moving them up, not secondary flow. So be careful. :)
A lot of research has been performed to understand how and when secondary
flows occur. It turns out that any viscosity gradient can cause a secondary
flow. Sometimes secondary flows are hard to see (they can be very
small/slow), but they are there! This is the underlying reason for the
'swirl' you see.
If you travel from the northern hemisphere to the southern, you will notice that water 'swirls' counterclockwise. Simply enough, it is because of the rotation of the earth acting on the water in conjunction with gravity.