The propagation of a liquid bolus along a liquid-lined flexible tube

We use lubrication theory and matched asymptotic expansions to model the quasi-steady propagation of a liquid plug or bolus through an elastic tube. In the limit of small capillary number, asymptotic expressions are found for the pressure drop across the bolus and the thickness of the liquid film left behind, as functions of the capillary number, the thickness of the liquid lining ahead of the bolus and the elastic characteristics of the tube wall. These results generalise the well-known theory for the low-capillary-number motion of a bubble through a rigid tube (Bretherton 1961). As in that theory, both the pressure drop across the bolus and the thickness of the film it leaves behind vary like the two-thirds power of the capillary number. In our generalised theory, the coefficients in the power laws depend on the elastic properties of the tube. For a given thickness of the liquid lining ahead of the bolus, we identify a critical imposed pressure drop above which the bolus will eventually rupture, and hence the tube will reopen. We find that generically a tube with smaller hoop tension or smaller longitudinal tension is easier to reopen. This flow regime is fundamental to reopening of pulmonary airways, which may become plugged through disease or by instilled/aspirated fluids