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 alone, 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: $\D$, 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 $\Delta$. For small $\Delta$ 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 $\Delta$, the Hopf bifurcation is either quasi-periodic (for the smaller $\Delta$) or subharmonic (for larger $\Delta$), 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 $\Delta$, 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 $\Delta$.