Stability analysis of a bilayer contained within a cylindrical tube

Abstract

Airways in the lung are coated with a liquid bilayer consisting of a serous layer adjacent to a more viscous mucus layer which is contiguous with the air core. An instability due to surface tension at the interfaces may lead to the formation of a liquid plug that blocks the passage of air. This is known as airway closure. A stability analysis is carried out for the case when a Newtonian and immiscible liquid bilayer coats a compliant tube in the presence of an insoluble surfactant monolayer at the mucus-gas interface. A surface active material such as surfactant lowers the surface tension and also generates a surface stress at the interface, both of which are stabilizing, while the wall compliance may accelerate the formation of the liquid bridge. A system of nonlinear coupled equations for the deflections of the interfaces and the surfactant concentration is derived by using an extended lubrication theory analysis. A linear stability study using normal modes is conducted by linearizing the nonlinear evolution equations. A linear eigenvalue problem for the perturbation amplitudes is obtained. Non-trivial solutions are obtained provided the determinant of a linear system is singular. A fourth order polynomial for the growth rate of the disturbances is derived, whose coefficients depend on the wavenumber of the perturbation, the wall characteristics, the Marangoni number, the thickness of the bilayer, the aspect thickness ratio, the viscosity ratio of two liquid layers, and the surface tension ratio. Both stabilizing and destabilizing effects of various system parameters are investigated. A classical lubrication theory model is also derived for cases where a bilayer coats a rigid tube with insoluble surfactant along the liquid-gas interface, and a bilayer coating in a compliant tube with a clean liquid-gas interface. Results serve as a validation of the extended lubrication theory model. The accuracy of the extended lubrication theory model as the bilayer thickness increases is tested by considering a more general approach that is valid for arbitrary bilayer thickness. A system of two Orr-Sommerfeld equations is obtained using this more general approach, and together with the boundary conditions yields an eigenvalue problem for the growth rate. Validations and comparisons with lubrication theory models (both extended and classical ones) are provided. The nonlinear evolution equations are also solved numerically beyond the linear regime for the case of a bilayer coating a compliant tube together with surfactant along the mucus-gas interface in the last part of this thesis. These equations are solved numerically using the method of lines. Numerical results show that the closure time, that is the time requires for a liquid plug to form, goes up with Marangoni number. It is well known that for a single layer, the closure time can increase by a factor of four or five due to surfactant which immobilizes the gas-liquid interface. However, for a bilayer, surfactant may delay closure by a factor of twenty or more.