Chapter 7

# Poiseuille flow

## Complément : A video (in french) about Poiseuille's flow (with some fun science experiments)

Video about Poiseuille's flow

## Fondamental : Velocity Fields and pressures

In 1835 a French physicist Poiseuille made a series of experiments to determine how a viscous fluid flows in a straight pipe.

His goal was to understand the dynamics of blood flow in humans knowing that the blood plasma behaves like a Newtonian fluid.

A density of incompressible viscous fluid flows through a cylindrical tube of length and radius .

The pressure at the input of the tube ( ) is .

The pressure at the output of the tube ( ) is .

Poiseuille flow

We will calculate the velocity and pressure fields within the tube being placed in the steady state.

Laminar flow is assumed :

It is assumed that (invariance by rotation about the axis Oz) and the gravity is neglected.

Applying the second law of Newton to a system consisting of the fluid contained in the cylinder of radius at time and the mass that enters the cylinder between and .

At the moment , this system is made up of fluid in the cylinder of radius at and the same mass  that exits between and (incompressible flow).

Poiseuille flow

In the case of steady state, the change in momentum of the system is then simply zero :

Where :

By integration, taking account of  :

This gives the velocity profile in a Poiseuille flow.

Velocity profile in a Poiseuille flow

The graph above shows the parabolic profile of the velocity field and the variation of the pressure.

Called "charge loss" the amount .

## Fondamental : Volumetric flow rate

The volumetric flow rate is here :

Either :

We define a "hydraulic resistance", by analogy with an ohmic resistor ( ) :

Either :

The hydraulic resistance is as much higher as the viscosity of the fluid is large and the radius of the tube is small.

Volumetric flow rate

## Exemple : Charge loss and hydraulic resistance

Placing a succession of vertical tubes along the pipe.

The measurement of water levels used to determine the pressure drop :

Pressure drop and hydraulic resistance

If the flow rate is , the hydraulic resistance can be deduced as :

## Simulation : JAVA animation by JJ.Rousseau

• Flow in a tube : click HERE

Previous
Reynolds number
Next
Flow viscometer