High-Reynolds-number steady flow in a collapsible channel

We have studied steady flow in a two-dimensional channel in which a section of one wall has been replaced by an elastic membrane under dimensionless longitudinal tension T but possessing no bending stiffness. The dimensionless upstream transmural pressure takes a value P-ext, the membrane section is assumed to be long compared with the channel width and its deformation is assumed to remain within the viscous boundary layers. Standard high-Reynolds-number asymptotic methods are applied to arrive at a coupled boundary-layer-membrane problem. A non-zero cross-stream pressure gradient, leading to flow perturbations upstream of the membrane, is included in the analysis. Linearization of the boundary-layer problem yields firstly an analytic solution at non-zero P-ext and asymptotically high T. This takes the form of an expansion in T-1 for which the membrane shape and the flow decouple at each order. Extension of this solution branch to smaller values of the tension suggests a singularity at finite tension, where the deformation of the membrane becomes very large. Secondly, when the upstream transmural pressure is zero the trivial solution is valid for all values of the tension. However, we also obtain eigensolutions where the membrane tension plays the role of eigenvalue. There are thus non-trivial Solutions of the problem at these particular values of the tension. The nonlinear coupled boundary-layer-membrane problem is then solved numerically. A finite-difference, Keller-box, marching scheme is used, together with a shooting algorithm to satisfy the boundary condition at the downstream end of the membrane. This reveals a variety of different solutions, showing the relation between the two cases captured by the linearized analysis and demonstrating the existence of parameter ranges for which no solutions exist under the specified constraints. Such parameter ranges appear not to exist if the downstream, rather than the upstream, transmural pressure is held constant. The relation to our results of solutions obtained by solving the two-dimensional Navier-Stokes equations directly is discussed. Reasonable agreement between parameters is obtained, once allowance is made for the finite Reynolds number and membrane length in those computations.