# Continuum Mechanics - Bernouilli's Equation in a Cylindrical Tank

1. Apr 23, 2012

### lemonkey

1. The problem statement, all variables and given/known data

A large cylindrical tank of radius R is full of water to a height h(t). This drains under gravity
out of the bottom of the tank through a small hole of radius r. The acceleration due to gravity
is g. The pressure of the air can be assumed to be the same at the top and at the bottom of the tank.

Show that if the uniform velocity of the ﬂuid at the top of the tank is U and the uniform
velocity of the ﬂuid leaving through the small hole is u then:
$$u = U\;\frac{R^2}{r^2}$$

By applying Bernoulli’s Theorem on a streamline from the top of the ﬂuid to the bottom,
show that:
$$\frac{dh}{dt} = -\sqrt{2gh}\;(\frac{R^4}{r^4} - 1)^{-\frac{1}{2}}$$

2. Relevant equations
We've been given Bernoulli's Theorem as:
$$\frac{1}{2}u\bullet u + \frac{P}{\rho} + V = constant$$, where P is pressure, rho is density and V is potential.

3. The attempt at a solution
I tried using Bernoulli's Theorem, but wasn't sure what V was. I'm really not sure what to do. :(