Hydrodynamic instabilities

Taylor-Couette instability

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.
Exact Navier-Stockes solutions
Couette 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(theta)= a*r+b/r with the constants a=(Omega2*R22 - Omega1*R12 )/(R22 -R12) and b=R12 R22(Omega1-Omega2)/(R22 -R12)
The velocity field in a laminar case is purely orthoradial.