How can you increase the water flow in a slow sink drain?

Remove whatever is obstructing the drain. -- First, get a bottle of one of the liquid drain openers from any drugstore chain. Follow the directions on the label and see what happens. If the first application doesn't open the drain, do it a second time. If that doesn't do it, then ... -- Assume the obstruction is a solid object stuck in the drain pipe. Look down under the sink for a section of pipe that goes down--up--down. That's the 'trap'; things that fall down the drain are supposed to stay in the trap. There's a big pipe-nut above and below the trap. Loosen both, and drop the trap section out of the pipe. Water and gunk dribble out on you and on the floor. Clean out the trap, re-assemble everything, and see if anything changed. If the drain is still not running, then the clog is farther down, past the trap. There's one more thing you can do if you still want to work on it yourself: -- Borrow or buy a so-called "snake". It's a long, flexible metal thing that can be run down into the drain pipe and rotated, to dig out whatever is stuck way down in the pipe. If you start it at the sink drain, it probably won't be able to make the bends past the trap, so you should open the trap again and feed the snake into the straight pipe below the trap. -- If none of this does the job, it's time to call a plumber.