The onset of chaos in a class of Navier-Stokes solutions

The flow between parallel walls driven by the time-periodic oscillation of one of the walls is investigated. The flow is characterized by a non-dimens ional amplitude Delta and a Reynolds number R. At small values of the Reyno lds number the flow is synchronous with the wall motion and is stable. If t he amplitude of oscillation is held fixed and the Reynolds number is increa sed there is a symmetry-breaking bifurcation at a finite value of R. When R is further increased, additional bifurcations take place, but the structur e which develops, essentially chaotic flow resulting from a Feigenbaum casc ade or a quasi-periodic flow, depends on the amplitude of oscillation. The flow in the different regimes is investigated by a combination of asymptoti c and numerical methods. In the small-amplitude high-Reynolds-number limit we show that the flow structure develops on two time scales with chaos occu rring on the longer time scale. The chaos in that case is shown to be assoc iated with the unsteady breakdown of a steady streaming flow. The chaotic f lows which we describe are of particular interest because they correspond t o Navier-Stokes solutions of stagnation-point form. These flows are relevan t to a wide variety of flows of practical importance.