M. Golubitsky and I.N. Stewart
Symmetry and stability in Taylor-Couette flow
SIAM J. Math. Anal.
17 (2) (1986) 249-288.
We study the flow of a fluid between concentric rotating cylinders
(the Taylor problem) by exploiting the symmetries of the system.
The Navier-Stokes equations, linearized about Couette flow, possess
two zero and four purely imaginary eigenvalues at a suitable value
of the speed of rotation of the outer cylinder. There is thus a
reduced bifurcation equation on a six-dimensional space which can
be shown to commute with an action of the symmetry group 0(2) X S0(2).
We use the group structure to analyze this bifurcation equation in the
simplest (nondegenerate) case and to compute the stabilities of solutions.
In particular, when the outer cylinder is counterrotated we can obtain
transitions which seem to agree with recent experiments of Andereck,
Liu, and Swinney [1984]. It is also possible to obtain the "main sequence"
in this model. This sequence is normally observed in experiments when
the outer cylinder is held fixed.