The influence of surfactant on two-phase flow in a flexible-walled channel under bulk equilibrium conditions

Abstract

A preliminary model of pulmonary airway reopening is developed that includes the physicochemical influence of surfactant under bulk-equilibrium conditions. The airway is modeled following Gaver et al. [J. Fluid. Mech. 319, 25–65 (1996)] as a flexible-walled channel, where walls are membranes under longitudinal tension T, and supported with elasticity E with a stress-free separation distance 2H. The lining fluid has viscosity μ and surface tension γ*. Airway reopening occurs when a semi-infinite bubble of air with pressure Pb* progresses steadily at velocity U and separates the walls. Surfactant exists in the lining fluid (C*) and at the air–liquid interface (Γ*). Bulk equilibrium is assumed (C*=C0) and the kinetic transfer of surfactant between the bulk and interface occurs with a rate k. The equilibrium relationship between Γ* and C* is based upon Henry’s isotherm (Γeq=KC0). The surface tension equation of state, a relationship between γ* and Γ*, is assumed to be linear near Γeq. Marangoni stresses develop from the transport of surfactant at the interface, leading to interfacial rigidification and changes in the airway reopening behavior. The behavior is governed by the following dimensionless parameters: the capillary number Ca=μU/γeq, the surface elasticity number El=−(dγ*/dΓ*)(Γeq/γeq), the modified Stanton number Stλ=(k/K)/(U/H), the wall elastance parameter β=EH2/γeq, the wall tension ratio η=T/γeq, and the surface Pèclet number Peint=UH/Dint. The results indicate that El can have a dual, contrasting influence on the airway reopening behavior. By increasing El through an increase in dγ*/dΓ* (method 1), larger Pb* are predicted from the resulting interfacial surfactant gradients and interfacial rigidification. In contrast, increasing Γeq (method 2) increases El but reduces Pb* due to the global reduction of γ*; however, the reduction in Pb* is augmented by the increasing importance of viscous, elastic, tension and Marangoni stresses. Furthermore, for Stλ>10 the interface remains mobile due to rapid surfactant adsorption and the elimination of Marangoni stresses, which minimizes Pb*. This behavior may be important in the development of improved exogenous surfactants for the treatment of a variety of pulmonary diseases.