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 thresholdAnother 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: