Dynamics and transport of a localized soluble surfactant on a thin film

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Abstract
The flow induced by a localized droplet of soluble surfactant on the surface of a thin film is analysed, motivated by an interest in the interaction between inhaled droplets and the lung's thin liquid lining after an aerosol lands on its surface. The spreading is driven by gradients of surface tension and results in flow of the droplet and underlying liquid film. This induced flow field plays an important role in the transport of dissolved species from the droplet, through the film, and to the tissue for absorption. Evolution equations for the film thickness, surface and bulk liquid concentrations are derived using lubrication theory, since the depth of the film is much smaller than the characteristic radius of the droplet. Solutions are obtained numerically using the method of lines for a variety of surface Péclet numbers. We find that the effect of solubility is to decrease both film disturbances and surface concentrations, and to induce an absorption-driven backflow. In addition, there is a gravity-driven backflow from hydrostatics. At large surface Péclet numbers, large film disturbances are obtained and more surfactant is able to diffuse across the rigid permeable wall, while surface diffusion causes more rapid spreading and decreases film disturbances. Gravity acts as a restoring force by creating a bidirectional flow, and hence disenhances the vertical flux of surfactant across the air-liquid interface. This model may have implications for the delivery of drugs by aerosol inhalation.

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