The propagation of a surfactant laden liquid plug in a capillary tube

Abstract

This paper considers the propagation of a liquid plug, forced by a driving pressure ΔP, within a rigid tube. The tube is already lined with a liquid precursor film of thickness h̄ 2 . Both the plug and the precursor film, as well as the interface, contain small amounts of surfactant whose concentrations are assumed to be near equilibrium. Since the motions are slow, we seek asymptotic solutions for small capillary number, Ca≪1, and also assume that sorption kinetics control the surfactant flux to the interface compared to bulk diffusion. An additional asymptotic assumption is that the Stanton number, St, is sufficiently large such that β∝Ca 1/3 /St≪1, which relates the importance of sorption kinetics to convection. The surfactant strength is measured by the surface elasticity, E=M/β where M is the Marangoni number. The results of the analysis are that, for a given plug Ca, ΔP increases with increasing E but decreases with increasing h̄ 2 . The trailing film thickness, h̄ 1 , increases with ΔP, but at a slower rate when E is larger. For h̄ 1 <h̄ 2 , criteria for plug rupture are established. This model is relevant to delivery of surfactants into the lung by direct instillation into the bronchial network as is done in surfactant replacement therapy and the use of surfactantsolutions to carry other substances (e.g., genetic material) into the airways.