Stability of high-Reynolds-number flow in a collapsible channel

We study high-Reynolds-number flow in a two-dimensional collapsible channel in the asymptotic limit of wall deformations confined to the viscous boundary layer. The system is modelled using interactive boundary-layer equations for a Newtonian incompressible fluid coupled to the freely moving elastic wall under constant tension and external pressure. The deformation of the membrane is assumed to have small amplitude and long wavelength, whereas the flow comprises the inviscid core and the viscous boundary layers on both walls coupled to each other and to the membrane deformation. Firstly, by linking the interactive boundary-layer model to the small-amplitude, long-wavelength inviscid analysis, we conclude that the model is valid only when the pressure perturbations are fixed downstream from the wall indentation, contrary to the common assumption of classical boundary-layer theory. Next we explore possible steady states of the system, showing that a unique steady solution exists when the pressure is fixed precisely at the downstream end of the membrane, but there are multiple states possible if the pressure is specified further downstream. We examine the stability of these states by solving the generalized eigenvalue problem for perturbations to the nonlinear steady solutions and also by performing time integration of the full boundary-layer equations. Surprisingly, we find that no self-excited oscillations develop in the collapsible channel systems with finite-amplitude deformations. Instead, for each point in the parameter space, with the exception of points subject to numerical instabilities associated with the boundary-layer equations, exactly one of the steady states is predicted to be stable. We discuss these findings in relation to the results reported previously in the literature.