When a sink drains it forms a vortex. Does that vortex speeds drainage. If yes, how does the vortex speed drainage?
Jim is basically correct in what he has said. Consider a vertical cylindrical pipe which has an air-tight plunger at some distance from the top. Now you fill a section of that pipe with a small volume of incompressible fluid (water will do for this analogy as it is [for all intents and purposes] incompressible) resting on the plunger. If you now seal the top of that pipe, you will not be able to move the plunger upwards (actually in either direction), as to do so you will have to compress the air pocket above it.
Now, take a small volume of the fluid in this same pipe which is initially at rest. Below the fluid volume is air. If the bottom of the pipe was not sealed the fluid, when released, would fall right out of the bottom; however, if it was sealed the same effect that was outlined above would come into play. Although, with the more dense fluid above the less dense fluid, we would quickly witness the air rising/bubbling through the more dense liquid [due to gravitationally induced instabilities].
Now for vortex flow. The fluid motion in a vortex creates a dynamic pressure (in addition to any hydrostatic pressure) that is lowest in the core region, closest to the axis, and increases as one moves away from it. Combined this with the hydrostatic pressure and this forces the air below through the core and above the more dense fluid. Essentially the core provides a great path of least resistance for the air to use which helps both fluids travel far more efficently.
This is quite hard to explain with out diagrams, so I hope its helps.
It may be hard to explain without diagrams, but it's easy to experiment with. Fill 2 similar bottles half with water and turn them upside down. Give one of the bottles a swing to start a vortex. You'll see that the bottle with vortex will empty fast and smooth. The other bottle will glug and shake as the water resists air streaming in through the literal bottleneck.
The bottles are equivalent to your drain as you have a bottleneck (not literal for drains, though) a reservoir of water and 2 reservoirs of air seperated by bottle and water or drain, pipes and water respectively. Most drains have an overflow leading to the same pipe and you can often hear it glug because it's another "bottleneck".