We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose radius is changing in a prescribed manner. We construct a class of exact solutions of the Navier-Stokes equations in the case when the vessel radius is a function of time alone so that the cross-section is circular and uniform along the pipe axis. The Navier-Stokes equations admit solutions which are governed by nonlinear partial differential equations depending on the radial coordinate and time

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... ne, and forced by the wall motion. These solutions correspond to a wide class of bounded radial stagnation-point flows and are of practical importance. In dimensionless terms, the flow is characterized by two parameters: ∆, the amplitude of the oscillation, and R, the Reynolds number for the flow. We study flows driven by a time-periodic wall motion, and find that at small R the flow is synchronous with the forcing and as R increases a Hopf bifurcation takes place. Subsequent dynamics, as R increases, depend on the value of ∆. For small ∆ the Hopf bifurcation leads to quasi-periodic solutions in time, with no further bifurcations occurring -this is supported by an asymptotic high-Reynolds-number boundary layer theory. At intermediate ∆, the Hopf bifurcation is either quasi-periodic (for the smaller ∆) or subharmonic (for larger ∆), and the solutions tend to a chaotic attractor at sufficiently large R; the route to chaos is found not to follow a Feigenbaum scenario. At larger values of ∆, we find that the solution remains time periodic as R increases, with solution branches supporting periods of successive integer multiples of the driving period emerging. On a given branch the flow exhibits a self-similarity in both time and space and these features are elucidated by careful numerics and an asymptotic analysis. In contrast to the two-dimensional case (see Hall & Papageorgiou 1999) chaos is not found at either small or comparatively large ∆.