Complément : A video (in french) about Poiseuille's flow (with some fun science experiments)
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 .
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).
In the case of steady state, the change in momentum of the system is then simply zero :
By integration, taking account of :
This gives the 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 .