How does the size of the hole affect the exit velocity? I took water bottles and drilled several different sizes of holes on the bottom. 
During the experiment, I found that the water bottle with a smaller hole takes a lot longer time to leak than those with larger holes.
Does this mean that the exit velocity of the liquid (water) decrease as the size of hole decreases due to the viscosity of water? And should the exit velocity be directly proportional to the area of the hole? 
 A: 
I found that the water bottle with a smaller hole takes a lot longer
  time to leak than those with larger holes. Does this mean that the
  exit velocity of the liquid (water) decrease as the size of hole
  decreases due to the viscosity of water?

The biggest effect is due to the size of the hole. 
The water that leaves in one second is $A \times v$. There can be small differences in $v$ as you change $A$, but the biggest effect of a larger hole is that you have a larger hole: $A$ is bigger!
A: You must also include surface tension into the mix. If you make a really small hole, the water may not flow at all because it's energy would be sufficiently lower than when it exits the bottle immediately after. The speed of water alone tells you nothing about how much water is escaping the bottle. You must also consider the width of the jet. Also the main factor is the height of the water inside the bottle. The more water is pushing down on the water near the hole, the higher the jet velocity.
A: By Torricelli's Law, the velocity of incompressible, ideal fluids escaping through a sharp edged hole at a depth of $h,$ is given by the equation:
$$
v~=~\sqrt{2gh}
$$
As you can see, for ideal fluids, the rate should remain the same; however, when the fluids are compressible, have surface tension, and have viscosity, the calculations don't work out.
Also, as long as the size of the hole is negligible compared to the surface area of the water, the equation should be accurate.
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