Bifurcation phenomena in steady flows of a viscous fluid. I. Theory

The investigation deals with questions relating to the existence of multiple solutions to hydrodynamic problems, especially questions about bifurcations of solutions and about stability. Although the aim is to resolve some of these questions generally, particular reference is made to events observed in the Taylor experiment on Couette flow between rotating cylinders, for which a new rationale is developed. In § 2 certain results for the abstract nonlinear problem of steady motion in a bounded fluid are summarized, including a set of generic properties that can be associated with real flows. In § 3 these results are applied to the interpretation of observable phenomena, and several predictions are made which are open to experimental checks. It is shown that new insights into the Taylor experiment are gained by considering its dependence on two parameters, one the Reynolds number R and the other the length l of the flow domain. Attention is paid to the initial development of cells as R is gradually increased from small values (§3.3), to a hysteresis phenomenon accompanying morphogenesis of the cellular structure at critical values of l (§3.4), and to properties of secondary modes possible above respective critical values of R (§ 3.5). In the appendix some corresponding interpretations are noted for the Bénard problem of incipient convective motion.