Burr Zimmerman, a scientist at the US Dept. of Energy explains how the angular momentum in a vortex gets started when water goes down a drain.
“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.”
—From DOE Ask a Scientist:
(source no longer available: http://www.stab-iitb.org/newton-mirror/askasci/phy05/phy05136.htm)