The liquid lining in small human airways can become unstable and form liquid plugs that close off the airways. Direct numerical simulations are carried out on an airway model to study this airway instability and the flow-induced stresses on the airway walls. The equations governing the fluid motion and the interfacial boundary conditions are solved using the finite-volume method coupled with the sharp interface method for the free surface. The dynamics of the closure process is simulated for a viscous Newtonian film with constant surface tension and a passive core gas phase. In addition, a special case is examined that considers the core dynamics so that comparisons can be made with the experiments of Bian et al. (J. Fluid Mech., vol. 647, 2010, p. 391). The computed flow fields and stress distributions are consistent with the experimental findings. Within the short time span of the closure process, there are large fluctuations in the wall shear stress. Furthermore, dramatic velocity changes in the film during closure indicate a steep normal stress gradient on the airway wall. The computational results show that the wall shear stress, normal stress and their gradients during closure can be high enough to injure airway epithelial cells.