# A reminder of theory

## Cylindrical Couette Flow

We consider two infinite coaxial cylinder border a fluid. R1 and R2 their radius, W1 and W2 their rotating speed. This is an example of exact Navier-Stokes solution. We suppose this is a steady flow and there is a revolution symmetry around the axis of rotation.

Finally, the solution is an orhoradial velocity depending only of the distance to the axis r :

, withand.

From this laminar result, we can consider several cases:

-Only the outer cylinder is moving, the velocity is orthoradial and the flow stable, until this velocity becomes too high and the flow starts to be turbulent.

-Only the inner cylinder is moving, there is a threshold for R2 from which some instabilities appear.Indeed, some tore rolls appear around the inner cylinder. In fact, we are going to see now, that these instabilitie are ruled by a number called the Taylor Number.

## The Taylor Number

Basically, these tore rolls appear when the centrifugal force becomes stronger than the viscous force. It is only visible when the inner cylinder is moving and the outer cylinder is maintened fix becausec of the kinetic moment more important close to the axis.

In theory, to see these instabilities, it needs a little perturbation. For this, we consider a little sphere, R0 is radius and Vc is velocity, which is radial. Now we are going to work out the centrifugal force Fm and the viscous force Fv.

The viscous force is :

And this sphere will lose a momentum mVcdue to the viscosity in a time where A is a geometric constant.

Concerning the centrifugal force, after this time, the movement of the sphere is dr=Vc tV.

The driving force corresponds to the variation of the centrifugal force mW2 on the distance dr. Finally, we have the centrifugal force :  with B a geometric constant, R=(R1+R2)/2 and a =(R1-R2)/2 .

Geoffroy Ingram Taylor (right) with his assistant Walter Thomson .

The instability condition is Fm>Fv i.e. : .

So Fm/Fv=Ta, with  and K a constant depending of geometric parameters. In theory the instabilities start from Ta=1712.

Now, we can establish a parallel between this instability and Rayleigh-Benard instablity or Benard-Marangoni instability. In fact these instabilities are simply a question of balance between stabilizing and unstabilizing force, like viscou force and centrifugal force in Taylor-Couette flow.

## 1.Rayleigh-Benard:

For this instability, a heavy viscous fluid between two parallel plates which are at different temperatures. If the temperature of the top plate is lower than the temperauture of the bottom plate, with the influence of the gravity, we can observe some instabilities. In this case the stabilizing force is thermal diffusion and unstabilizing force is Archimede flotability. This instability is managed by the Rayleigh number:

2.Benard-Marangoni:

This instability occurs when the bottom side of a free surface liquid is heated.this instability apear when there is a temperature perturbation on the top surface of the liquid. the interface forces introduce a volume motion. If we put a soap drop at the surface of dusty watter we see that the dusty go in the radial direction to the outside of the flow. this produce , indirectely the apparition of a superficiel layerin the watter. the forces make the particals attracted by the zones of the less concetration and the most superficiel tension.