Bifurcations and symmetries in viscous flow
The results of an experimental study of phenomena which occur in the flow of a viscous fluid in closed domains with discrete symmetries are presented. The purpose is to investigate the role which ideas from low-dimensional dynamical systems have to play in describing qualitative changes that take place with variation of the governing parameters. Such a descriptive framework already exists for the case of the Taylor-Couette system, where the domain possesses a continuous azimuthal symmetry group. The present investigation is aimed at establishing the typicality of previously reported behaviour under progressive reductions of azimuthal symmetry. In the first investigations, the fixed outer circular cylinder of the standard system is replaced with one of square cross-section. Thus there is now discrete Ζ4 symmetry in the azimuthal direction. Knowledge of the two-dimensional flow field is used to establish the nature of the steady three-dimensional motion equivalent to Taylor vortex flow. It is shown that similar bifurcation sequences exist in both standard and square systems for the case of very small aspect ratio where a single Taylor cell is formed. This flow develops as the result of a bifurcation which breaks the Ζ2 symmetry that is imposed on the annulus by two solid stationary ends. The study is then extended to consider time-dependent effects in the square system. Two different oscillatory single-cell flows are identified, and it is shown that each is the result of a Hopf bifurcation. Selection of a particular dynamic mode is found to depend on the aspect ratio of the system. A low-dimensional bifurcation structure is uncovered which connects the two modes in parameter space, and involves a novel type of steady single-cell flow. Finally, observations are reported of a nontrivial type of dynamical behaviour which bears strong resemblance to motion found in a circularly symmetric Taylor-Couette system that is related to the Šilnikov mechanism for finite-dimensional chaos. A second variant on the Taylor-Couette system is considered where the outer cylinder is shaped like a stadium. The effect is to reduce further the overall symmetry of the domain to a Ζ2 × Ζ2 group. The two-dimensional flow field is investigated using both numerical and experimental techniques. Time-dependent phenomena are then investigated in the three-dimensional flow over a relatively wide range of aspect ratio. It is found that a sequence of a Hopf bifurcation followed by period-doubling bifurcations exists up to a certain aspect ratio, beyond which there is an apparently sudden and reversible transition between regular and irregular dynamical behaviour. Although this transition is not of a low-dimensional nature, the experimental results suggest that it exists as the result of a coalescence of the bifurcations which are found at lower values of aspect ratio.