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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?

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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.

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    $\begingroup$ ... and if you really get down to the details, the size and shape of the hole also make a slight difference. For liquid flow through an orifice you create what's called a vena contracta . Since it takes some time for the water to accelerate as it shoots out of the orifice the fluid doesn't reach max velocity until it is slightly away from the orifice. And according to Bernoulli that's where the fluid pressure will be minimum. So that leads to a 'necking down' of the fluid stream which becomes the true effective exit area; the area one should use in $Q=VA$. $\endgroup$ – docscience Feb 2 '17 at 14:54
  • $\begingroup$ Thanks. I didn't think about the surface tension. How should I determine the surface tension? Is there any equation? $\endgroup$ – user144160 Feb 2 '17 at 14:59
  • $\begingroup$ well, keeping it really simple, you have to calculate the adhesive forces of water on the bottle and on the air surrounding the water hole and adhesion of water molecules against other water molecules. Then you could understand why the water is flowing the way it does (for example spilling on the outside of bottle instead of falling to the ground). If yo utry to also keep the shape of the hole in mind this becomes a very very complex problem. But could be a lot more easily understood keeping parameters ideal and simple. $\endgroup$ – MaDrung Feb 2 '17 at 15:04
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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.

Another post on the same topic

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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!

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