A simplified model for the pulmonary alveolus that imitates the rhythmical expansion of the alveolus and the periodic shear flow in the adjacent airway is explored. The model consists of two eccentric cylinders and incompressible fluid that occupies the gap between them. The two cylinders undergo a simultaneous rhythmical expansion and contraction (mimicking the alveolus expansion) while the inner cylinder performs a periodic rotation about its axis (inducing shear flow mimicking airway ductal flow). An analytical solution is obtained for the creeping flow induced by the simultaneously expanding cylinders. It is shown that above a certain critical value of rotation to expansion velocity ratio, the flow exhibits characteristic features such as a saddle point and closed streamlines about a centre, similar to those existing inside a single alveolus during inhalation and exhalation. Poincaré maps of the trajectories of fluid particles demonstrate that, under various flow conditions, chaotic trajectories may exist, provided that expansion and rotation are slightly out of phase. This is similar to normal breathing conditions where the periodic expansion of the alveolus and the tidal flow (i.e. shear flow above the mouth of the alveolus) may be slightly out of phase. A novel definition of overall convective mixing efficiency is also suggested that inherently discounts reversible processes that do not contribute to mixing. It is demonstrated that two different convective mechanisms, related to the irreversibility of exhalation and inhalation and the onset of chaos, govern mixing efficiency in lung alveoli.