RESUMO
To assess how the presence of surfactant in lung airways alters the flow of mucus that leads to plug formation and airway closure, we investigate the effect of insoluble surfactant on the instability of a viscoplastic liquid coating the interior of a cylindrical tube. Evolution equations for the layer thickness using thin-film and long-wave approximations are derived that incorporate yield-stress effects and capillary and Marangoni forces. Using numerical simulations and asymptotic analysis of the thin-film system, we quantify how the presence of surfactant slows growth of the Rayleigh-Plateau instability, increases the size of initial perturbation required to trigger instability and decreases the final peak height of the layer. When the surfactant strength is large, the thin-film dynamics coincide with the dynamics of a surfactant-free layer but with time slowed by a factor of four and the capillary Bingham number, a parameter proportional to the yield stress, exactly doubled. By solving the long-wave equations numerically, we quantify how increasing surfactant strength can increase the critical layer thickness for plug formation to occur and delay plugging. The previously established effect of the yield stress in suppressing plug formation [Shemilt et al., J. Fluid Mech., 2022, vol. 944, A22] is shown to be amplified by introducing surfactant. We discuss the implications of these results for understanding the impact of surfactant deficiency and increased mucus yield stress in obstructive lung diseases.
RESUMO
Experimental and theoretical work reported here on the collapse of catenoidal soap films of various viscosities reveal the existence of a robust geometric feature that appears not to have been analyzed previously; prior to the ultimate pinchoff event on the central axis, which is associated with the formation of a well-studied local double-cone structure folded back on itself, the film transiently consists of two acute-angle cones connected to the supporting rings, joined by a central quasicylindrical region. As the cylindrical region becomes unstable and pinches, the opening angle of those cones is found to be universal, independent of film viscosity. Moreover, that same opening angle at pinching is found when the transition occurs in a hemicatenoid bounded by a surface. The approach to the conical structure is found to obey classical Keller-Miksis scaling of the minimum radius as a function of time, down to very small but finite radii. While there is a large body of work on the detailed structure of the singularities associated with ultimate pinchoff events, these large-scale features have not been addressed. Here we study these geometrical aspects of film collapse by several distinct approaches, including a systematic analysis of the linear and weakly nonlinear dynamics in the neighborhood of the saddle node bifurcation leading to collapse, both within mean curvature flow and the physically realistic Euler flow associated with the incompressible dynamics of the surrounding air. These analyses are used to show how much of the geometry of collapsing catenoids is accurately captured by a few active modes triggered by boundary deformation. A separate analysis based on a mathematical sequence of shapes progressing from the critical catenoid towards the Goldschmidt solution is shown to predict accurately the cone angle at pinching. We suggest that the approach to the conical structures can be viewed as passage close to an unstable fixed point of conical similarity solutions. The overall analysis provides the basis for the systematic study of more complex problems of surface instabilities triggered by deformations of the supporting boundaries.
