Principle of Taylor-Couette instability

    The geometry consists in two concentric cylinders. A fluid is confined between the two cylinders.

    If the interior cylinder is mentained fix and the exterior one moves with velocity , the flow is stable and becomes clearly turbulent for one certain value of  but there is no transition and no instability.
    On the other hand, if the exterior cylinder is maintained fix and the interior one moves with velocity , for a critical value of rotation speed , instabilities can appear : there is a stacking of contrarotative cells and we speak also about tore rolls around the interior cylinder inside the fluid. The shape of the velocity field inside fluid is then orthoradial.
 
 

    If the rotative speed of the exterior cylinder increases tore rolls begin to deform to give turbulence.
 

    This firt flow was studied by Couette in 1901 and Taylor discovered instabilities for the first time in 1923 with the second flow.

    This problem looks like the Rayleigh-Benard one ; it is also an instability with threshold.

    This instability is due to the destabilizing effect of the centrifugal force, and there is a competition between this effect and the stabilizing effect of viscous drag force.

    The gradient of centrifugal force due to variation of kinetic momentum gives velocity gradients so rolls appear inside the fluid if centrifugal force is superior to viscous drag force.

    An adimensionnal number can be found such as the Rayleigh number. Here it is the Taylor number which represents the rate between the centrifugal force and the viscous drag force :

where is the rotation velocity of the interior cylinder, R the average radius of cylinders ( ), d the distance between both cylinders ( ),  the kinematic viscosity of the fluid.

    The critical value of Taylor number to develop Taylor-Couette instability is :

    Like for Benard-Marangoni instability, manifestations of Taylor-Couette instability look like manisfestations of Rayleigh-Benard instability.(See Manifestations of Rayleigh-Benard instability).

    Moreover, it is not easy to see with our eyes such instability because it is an internal instability and our eyes don't detect movements of fluid inside a big mass of this fluid if there is no colouring or tracer.