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

Abstract

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-dimensional amplitude Δ and a Reynolds number R. At small values of the Reynolds number the flow is synchronous with the wall motion and is stable. If the amplitude of oscillation is held fixed and the Reynolds number is increased there is a symmetry-breaking bifurcation at a finite value of R. When R is further increased, additional bifurcations take place, but the structure which develops, essentially chaotic flow resulting from a Feigenbaum cascade or a quasi-periodic flow, depends on the amplitude of oscillation. The flow in the different regimes is investigated by a combination of asymptotic and numerical methods. In the small-amplitude high-Reynolds-number limit we show that the flow structure develops on two time scales with chaos occurring on the longer time scale. The chaos in that case is shown to be associated with the unsteady breakdown of a steady streaming flow. The chaotic flows which we describe are of particular interest because they correspond to Navier–Stokes solutions of stagnation-point form. These flows are relevant to a wide variety of flows of practical importance.