# Fluid Mechanics

# Flow viscometer

Take 20 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.

A viscous, incompressible fluid, flows slowly from a cylindrical container of diameter into a horizontal capillary tube of diameter and length .

We neglect the effects due to the tube ends.

## Question

Can we consider the flow as quasi-permanent ? Justify.

Deduce the expression of the volume flow rate as a function of .

### Hint

### Solution

We place ourselves in the ARQS.

One has a cylindrical Poiseuille flow :

The liquid is practically at rest in the container of diameter .

We can therefore write the fluid static law :

and

Consequently :

With .

## Question

Establish a differential equation satisfied by .

Solve for the initial condition .

### Hint

### Solution

The flow is incompressible, there is conservation of volume flow rate :

Either :

This leads to the differential equation :

Whose solution is :

With :

## Question

It took a duration so that the liquid level moves from the height to the height .

Determining the kinematic viscosity of the liquid.

It gives : , , and

### Hint

### Solution

Numerical application gives :