Abstract. We investigate wave solutions of a lubrication model for surfactant-driven flow of a thin liquid film down an inclined plane. We model the flow in one space dimension with a system of nonlinear PDEs of mixed hyperbolic-parabolic type in which the effects of capillarity and surface diffusion are neglected. Numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves, some with jumps in height and surfactant concentration gradient. The various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane. Jump conditions admit new shock waves sustained by a linear surfactant wave traveling upstream. The stability of these waves is investigated analytically and numerically. For initial value problems, a critical ratio of upstream to downstream height separates two distinct long-time wave patterns. Below the critical ratio, there is also an exact solution in which the height is piecewise constant and the surfactant concentration is piecewise linear and has compact support. [12,13,17,18].In much of the research into surfactant spreading, the effect of gravity in driving the flow has been assumed to be negligible [10]. In this paper, we consider flow on an inclined plane, building on the work of Edmonstone, Craster, and Matar [6,7] that explores the effect of adding gravity to the driving force. Our results demonstrate that gravity has a profound effect and probably should not be neglected in simulations of surfactant spreading.For constant Marangoni force, as in studies of thermally driven thin films [2,4,8], the equations of motion are reasonably represented by a scalar fourth order PDE, known as the thin film equation. In the presence of surfactant, however, the Marangoni force is not constant; the density and motion of the surfactant molecules are modeled by an additional equation. The full model, derived using lubrication theory in [3], consists of a system of two nonlinear coupled PDEs for the film height and the surfactant concentration.The PDE system exhibits a complicated combination of wave-like structures in the solution of initial value and boundary value problems [6,7]. The complexity comes

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerateparabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

Modeling the flow of fluid in the lungs, even under baseline healthy conditions, presents many challenges. The complex rheology of the fluids, interaction between fluids and structures, and complicated multi-scale geometry all add to the complexity of the problem. We provide a brief overview of approaches used to model three aspects of pulmonary fluid and flow: the surfactant layer in the deep branches of the lung, the mucus layer in the upper airway branches, and closure/reopening of the airway. We discuss models of each aspect, the potential to capture biological and therapeutic information, and open questions worthy of further investigation. We hope to promote multi-disciplinary collaboration by providing insights into mathematical descriptions of fluid-mechanics in the lung and the kinds of predictions these models can make.