# Fluid Mechanics

# Oscillations in a U-tube

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

The oscillations of an incompressible fluid is studied in a small cross section of the U-tube.

It is assumed that the free surfaces remain in the rectilinear and vertical portions of the tube.

It is desired to determine the oscillation period of the liquid in the tube.

## Question

First method : by applying Bernoulli's theorem in a regime depending on the time.

### Hint

### Solution

The cote of the free surface of fluid in the right branch of the tube is denoted .

The fluid is incompressible, the speed at any point is :

It integrates the Euler equation of a current line :

Either, noting the liquid length :

Either :

The period is then :

## Question

Second method : by performing an energy balance.

### Hint

### Solution

The kinetic energy of all of the fluid is : ( is the U-tube cross section)

The potential energy is divided into three parts : it is assumed that at time , the drop was (the left side is lowered relative to the equilibrium position).

So (the center of mass is in the middle of the column of a considered fluid) :

By derivation :

Either :

We find the same equation.