The pulsatile motion of a semi-infinite bubble in a channel: flow fields, and transport of an inactive surface-associated contaminant

Abstract

We investigate a theoretical model of the pulsatile motion of a contaminant-doped semi-infinite bubble in a rectangular channel. We examine the fluid mechanical behaviour of the pulsatile bubble, and its influence on the transport of a surface-inactive contaminant (termed surfinactant). This investigation is used to develop a preliminary understanding of surfactant responses during unsteady pulmonary airway reopening. Reopening is modelled as the pulsatile motion of a semi-infinite gas bubble in a horizontal channel of width 2$a$ filled with a Newtonian liquid of viscosity $\mu$ and constant surface tension $\gamma$. A modified Langmuir sorption model is assumed, which allows for the creation and respreading of a surface multilayer. The bubble is forced via a time-dependent volume flux $Q(t)$ with mean and oscillatory components ($Q_{M}$ and $Q_{\omega }$, respectively) at frequency $\omega $. The flow behaviour is governed by the dimensionless parameters: Ca$_{M} \,{=}\,\mu Q_{M}/(2a\gamma $), a steady-state capillary number, which represents the ratio of viscous to surface tension forces; Ca$_{\Omega } \,{=}\,\mu Q_{\omega }/(2a\gamma $), an oscillatory forcing magnitude; $\Omega \,{=}\,\omega \mu a/\gamma $, a dimensionless frequency that represents the ratio of viscous relaxation to oscillatory-forcing timescales; and $A\,{=}\,2\hbox{\it Ca}_{\Omega }/\Omega $, a dimensionless oscillation amplitude. Our simulations indicate that contaminant deposition and retention in the bubble cap region occurs at moderate frequencies if retrograde bubble motion develops during the oscillation cycle. However, if oscillations are too rapid the ensuing large forward tip velocities cause a net loss of contaminant from the bubble tip. Determination of an optimal oscillation range may be important in reducing ventilator-induced lung injury associated with infant and adult respiratory distress syndromes by increasing surfactant transport to regions of collapsed airways.