Velocity of Water in a Glass: Exploring the Myth of 0 Velocity at the Top

[Gyroscope]

Homework Statement

What is the velocity of the water at the top of a glass of water? Is it really 0?
For example, there is a lot of problems which asks with what velocity does the water would leave that glass if I make a hole on the bottom. For these, we consider the velocity at the top to be zero. Why is that? Is it approximately zero or really zero? I am thinking now, if it is an incompressible fluid, it must be zero, because if the water is confined to that volume, cannot have speed. Nevertheless, it is clear that the water molecules are randomly moving. Can you clarify my doubts? Thanks in advance.

 

[berkeman]

When there is no hole in the bottom of the glass, the velocity of all of the water is zero. When there is a hole suddenly made in the bottom of the glass that goes all the way across (hole diameter = glass diameter), all the water accelerates together down out of the glass, and the velocity of the top surface is the same as the velocity of the bottom surface (ignoring the wetting effects on the walls).

When there is a hole in the bottom that is smaller than the diameter, then the flow rate out the hole will determine how fast the top surface goes down (through volume change calculations).

 

[Gyroscope]

I had an exercise on my book where I should show that the velocity at the bottom of the glass where there is a small hole is $$\sqrt{2gh}$$, h is the height of the water level. This is only true, if I consider the velocity at the surface to be zero, when applying Bernoulli's equation. Right?

 

[denverdoc]

which it will nearly be if the hole is relatively small.