Taylor-Couette instability
Introduction
The geometry that consists in 2 concentric cylinders which border a fluid, is well known in fluid mechanics. Indeed it is a famous exemple of exact solution of the Navier-Stockes equations.These results are reminded here. The shape of the velocity field is orthoradial. But for certain cases some instabilities appear with a shape of toroidal rolls.
The first flow was studied by M. Couette in1901 while
G.I. Taylor discovered the instabilities for the first time in 1923. This
problem looks like the Rayleigh-Benard problem. In this problem rolls appear
because of the buoyancy forces due to the instable stratification of density.
Here the velocity gradients stand for the temperature
gradients. Then the variation of kinetic momentum create a radial gradient
of centrifuge force.
An adimensionnal number can be defined such as the Rayleigh number. Here it is the Taylor number defined as:
Exact Navier-Stockes solutionsCouette flow is a basic example of exact Navier-Stockes solution in a laminar case. Some results can be reminded.
With the boundary conditions v(r=R1)=Omega1*R1 and v(r=R2)=Omega2*R2, the velocity field is:V(r)=0The velocity field in a laminar case is purely orthoradial.
V(x)=0
V(theta)= a*r+b/r with the constants a=(Omega2*R2^{2} - Omega1*R1^{2} )/(R2^{2} -R1^{2}) and b=R1^{2} R2^{2}(Omega1-Omega2)/(R2^{2} -R1^{2})