Symmetry and stability in TaylorCouette flow
Abstract
The flow of a fluid between concentric rotating cylinders (the Taylor problem) is studied by exploiting the symmetries of the system. The NavierStokes 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 sixdimensonal space which can be shown to commute with an action of the symmetry group 0(2) x S0(2). The group structure is used 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, transitions which seem to agree with recent experiments of Andereck, Liu, and Swinney (1984) are obtained. 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.
 Publication:

SIAM Journal of Mathematical Analysis
 Pub Date:
 March 1986
 Bibcode:
 1986SJMA...17..249G
 Keywords:

 Couette Flow;
 Flow Stability;
 Rotating Fluids;
 Taylor Instability;
 Branching (Mathematics);
 Counter Rotation;
 Eigenvalues;
 Group Theory;
 Kernel Functions;
 Rotating Cylinders;
 Vortices;
 Fluid Mechanics and Heat Transfer