Shear flow over a self-similar expanding pulmonary alveolus during rhythmical breathing

Alternating shear flow over a self-similar, rhythmically expanding hemisphe rical depression is investigated. It provides a fluid-mechanical model for an alveolated respiratory unit, by means of which the effect of lung rhythm ical expansion on gas mixing as well as aerosol dispersion and deposition c an be studied. The flow is assumed to be very slow and governed by the quas i-steady linear Stokes equations. Consequently, superposition of the follow ing two cases provides an easy route toward characterizing the aforemention ed flow field. The first case treats the flow field that is generated by a rhythmically expanding spherical cap (the alveolus). The cap is attached at its rim to a circular opening in an expanding unbounded plane bounding a s emi-infinite fluid region. The rate of expansion of the cap and the plane a re chosen such as to maintain the system's configurational self-similarity. The second case addresses the flow disturbance that is generated by an alt ernating shear flow encountering a rigid hemispherical cavity in a plane bo unding a semi-infinite fluid domain. For the first case, a stream-function representation employing toroidal coo rdinates furnishes an analytical solution, whereas the second case was solv ed numerically by Pozrikidis (1994). Linear superposition of the two flow c ases results in particularly rich streamline maps. In the symmetry plane (b isecting the cap and parallel to the mean shear flow), for a certain range of shear to expansion-rate ratios, the streamline maps are self-similar and display closed orbits and two internal stagnation points. One of the stagn ation points is a 'centre' surrounded by closed streamlines whereas the oth er constitutes a 'saddle point'. For other planes, no stagnation points exi st in the field, but the streamlines associated with the saddle point displ ay complex looping patterns. These unique flow structures, when subjected t o a small perturbation (e.g. a small asynchrony between ductal and alveolar entering flows) cause highly complex stochastic particle trajectories even in the quasi-static Stokes alveolar flow. The observed irreversible flow p henomena in a rhythmically expanding alveolus may be partially responsible for the 'stretch-and-fold' flow mixing patterns found in our recent flow vi sualization studies performed in excised animal lung acini.