The nonlinear growth of surface-tension-driven instabilities of a thin annular film

The stability and initial growth rate of disturbances on an annular film lining a cylindrical tube have been the focus of several previous works. The further development of these disturbances as they grow to form stable unduloids or liquid bridges is investigated by means of a thin-film integral model. The model is compared both with perturbation theories for early times, and a numerical solution of the exact equations (NEKTON) for later times. The thin-film model gave results that were in good agreement with solutions of the exact equations. The results show that linear perturbation theory can be used to give good estimates of the times for unduloid and liquid bridge formation. The success of the model derives from the dominant influence of narrow draining regions that feed into the growing unduloid, and these regions remain essentially one-dimensional throughout the growth of the instability. The model is used to analyse the evolution of the liquid layer lining the small airways of the lung during a single breath. The timescales for formation of unduloids and liquid bridges are found to be short enough for the liquid layer to be in a virtually quasi-equilibrium state throughout the breathing cycle. This conclusion is only tentative, however, because the model assumes that the surface tension of the airway liquid lining does not change with changes in interfacial area despite the known presence of pulmonary surfactant.