Finally, the solution is an orhoradial velocity depending only of the distance to the axis r :
From this laminar result, we can consider several cases:
-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.
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
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
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.