A model of flow and surfactant transport in an oscillatory alveolus partially filled with liquid

The flow and transport in an alveolus are of fundamental importance to partial liquid ventilation, surfactant transport, pulmonary drug administration, cell-cell signaling pathways, and gene therapy. We model the system in which an alveolus is partially filled with liquid in the presence of surfactants. By assuming a circular interface due to sufficiently strong surface tension and small surfactant activity, we combine semianalytical and numerical techniques to solve the Stokes flow and the surfactant transport equations. In the absence of surfactants, there is no steady streaming because of reversibility of Stokes flow. The presence of surfactants, however, induces a nontrivial cycle-averaged surfactant concentration gradient along the interface that generates steady streaming. The steady streaming patterns (e.g., number of vortices) particularly depend on the ratio of inspiration to expiration periods (⁠$I:E$ ratio) and the sorption parameter $K$⁠. For an insoluble surfactant, a single vortex is formed when the $I:E$ ratio is either smaller or larger than 1:1, but the recirculations have opposite directions in the two cases. A soluble surfactant can lead to more complex flow patterns such as three vortices or saddle-point flow structures. The estimated unsteady velocity is $10−3cm∕s$⁠, and the corresponding Péclet number for transporting respiratory gas is $O(1)$⁠. For a cell-cell signaling molecule such as surfactant-associated protein-A for regulating surfactant secretion, the Péclet number could be $O(10)$ or higher. Convection is either comparable to or more dominant than diffusion in these processes. The estimated steady velocity ranges from $10−6to10−4cm∕s$⁠, depending on $I:E$ and $K$⁠, and the corresponding steady Péclet number is between $10−8∕Dm$ and $10−6∕Dm$ (⁠$Dm$ is the molecular diffusivity with units of $cm2∕s$⁠). Therefore, for $Dm⩽10−8cm2∕s$⁠, the convective transport dominates.