THE TAYLOR NUMBER

Some comparisons can be done between several hydrodynamic instabilities since they are called instabilities with threshold. They will be sumed up in the table below. As for the Rayleigh-Benard instabilities, an adimensionnal number so-called the Taylor number can be defined.

To define it, a little sphere of rayon r0 is considered. In a Couette flow it has an orthoradial velocity vc. This sphere will lose a momentum due to the viscosity in a time . The dynamic law applied to the sphere in its motion in a fluid gives: where the mass of the sphere is and the  second member is given by the Stockes law for a sphere moving in a fluid: .

After a time the movement of the particule is such as . The driving force for the instabilities of Taylor-Couette is the centrifugal force. So in the present case, a characteristic quantity of the driving force is the variation of centrifugal force on the distance .
So Fm can be defined such as: and: A instability condition can be written as: For the case of Taylor-Couette flow, that yields: And so: The Taylor number is the adimensionnal group where Omega is the rotation velocity of the interior cylinder, R the radius of the interior cylinder, a the distance between both Couette cylinders and Nu the viscosity of the fluid.

It first can be remarked that instabilities appear for a critical value Tac. That is why Taylor-Couette instabilities are called instabilities with threshold : they happen for  certain value. It is not the case for all hydrodynamic instabilities.

Comparisons between different instabilities with threshold
Another kind of instabilities called Benard-Marangoni exists. This is very similar to Rayleigh-Benard's one, but instead of putting a top wall the geometry is a free surface. A gradient of temperature gives some instabilities in the free surface. Hexagonal cells appear with a motion of the fluid inside.
This example is another case of instability with threshold. Comparisons are shown in the table below:
the 