The effect of gravity on liquid plug propagation in a two-dimensional channel

Abstract

The effect of plug propagation speed and gravity on the quasisteady motion of a liquid plug in a two-dimensional liquid-lined channel oriented at an angle α with respect to gravity is studied. The problem is motivated by the transport of liquid plugs instilled into pulmonary airways in medical treatments such as surfactant replacement therapy, drug delivery, and liquid ventilation. The capillary number Ca is assumed to be small, while the Bond number Bo is arbitrary. Using matched asymptotic expansions and lubrication theory, expressions are obtained for the thickness of the trailing films left behind by the plug and the pressure drop across it as functions of Ca, Bo, α and the thickness of the precursor films. When the Bond number is small it is found that the trailing film thickness and the flow contribution to the pressure drop scale as Ca2∕3 at leading order with coefficients that depend on Bo and α. The first correction to the film thickness is found to occur at O(Ca) compared to O(Ca4∕3) in the Bo=0 case. Asymmetry in the liquid distribution is quantified by calculating the ratio of liquid volumes above and below the centerline of the channel, VṘ. VR=1 at Bo=0, indicating a symmetric distribution, and decreases with Bo and Ca, but increases with the plug length Lp. The decrease of VR with Ca suggests that higher propagation speeds in small airways may result in less homogenous liquid distribution, which is in contrast to the expected effect in large airways. For given values of the other parameters, a maximum capillary number Cac is identified above which the plug will eventually rupture. When the Bond number becomes equal to an orientation-dependent critical value Boc, it is found that the scaling of the film thickness and pressure drop change to Ca1∕2 and Ca1∕6, respectively. It is shown that this scaling is valid for small increments of the Bond number over its critical value, Bo=Boc+BCa1∕6, but for higher Bond numbers the asymptotic approach breaks down.