# Equilibrium of of water level in punctured bucket?

Would a bucket with a hole in the bottom being filled at different rates with water come to to different equilibrium levels with in the bucket? Or would at any fill rate above initial flow rate of the hole cause the bucket to overflow?

An answer with more math is preferred.

My thoughts:

If flow rates of water into bucket less than initial flow rate of hole then water level stays at or below height of hole in bucket.

If water flow rates begin to increase, so does the depth of the water above the hole then since the product of the hole's size and pressure is positively proportional to flow rate, the water leaving and coming reach a equilibrium point with in bucket x height above hole.

If water entering bucket becomes to fast then a equilibrium point in the size of the current bucket can not be reached, so the bucket over flows.

• Yes, equilibrium can be reached. The water level height above the drain hole affects the drain rate so given a constant fill rate, it is possible to reach equilibrium. To figure out what the equilibrium is requires a differential equation. – Brandon Enright Feb 26 '14 at 20:37

Bernoulli's principle and conservation of mass can give you a simple approximation to an equilibrium solution to the problem: $$\frac{1}{2}v^2+gh+\frac{p}{\rho} = constant$$ and $$\dot{m}_{in} = \rho A_{hole} v_{hole}$$ Where $\dot{m}_{in}$ is the mass flux of fluid into the bucket. Assuming zero velocity at the fill point of the bucket and pressure at the top surface equal to that at the exit hole gives $$gh_{water}=\frac{1}{2}v_{hole}^2$$ Substituting for $v_{hole}$ from conservation of mass then yields the approximate equilibrium height $$h_{water}= \frac{1}{2g}(\frac{\dot{m}_{in}}{\rho A_{hole}})^2$$

For most plumbing applications, each obstruction has a $C_V$ value and the flow is $C_V \Delta p^2$, where $\Delta p$ is the pressure drop across the obstruction. As you need to flow more water through the hole, you need more pressure behind it, which means more depth of water in the bucket. In this approximation, as long as the bucket is tall enough, you will be able to come to equilibrium.