#
__A reminder of theory__

##
__Cylindrical Couette Flow__

We consider two infinite coaxial
cylinder border a fluid. *R*_{1} and *R*_{2}
their radius, *W*_{1} and *W*_{2}
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 *R*_{2} 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,
*R*_{0}
is radius and *V*_{c} is velocity, which is radial.
Now we are going to work out the centrifugal force *F*_{m}
and the viscous force
*F*_{v}.

The viscous force is :

And this sphere will lose a
momentum *mV*_{c}due 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=V*_{c
}t_{V}.

The driving force corresponds to the variation of the
centrifugal force
*mW*^{2} on the distance *dr*.
Finally, we have the centrifugal force :
with *B* a geometric constant, *R=(R*_{1}+R_{2})/2
and *a =(R1-R2)/2* .

__Geoffroy Ingram Taylor (right) with his assistant
Walter Thomson __.

The instability condition is *F*_{m}>F_{v}
i.e. : .

So *F*_{m}/F_{v}=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.

##
__Comparison with other Hydraulic Instabilities__

##
**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.