A NUMERICAL-SIMULATION OF STEADY FLOW IN A 2-D COLLAPSIBLE CHANNEL

We have studied steady flow in a 2-D channel with one plane rigid wall and with a segment of the other wall replaced by an elastic membrane. Numerical solutions of the full governing equations have been obtained for Reynolds number Re = 1-600. The numerical method was to solve the Navier-Stokes equations for given membrane shape by using the finite element scheme FIDAP, and use the membrane equation to iterate for the membrane shape. The control parameters are the downstream transmural pressure, P-d, the longitudinal tension, T, and the Reynolds Number, Re. For a given Re and P-d, where P-d = p(external) - P-internal > 0, there exists a limit of T, say T-c, below which no converged solution was found. There is a somewhat higher value, T-b, such that for T-c < T < T-b, the membrane bulges out at its upstream end while the downstream part still remains collapsed. It is extremely difficult, however, to obtain converged solutions with our numerical scheme as we decrease the tension to T-b and below. To investigate whether the breakdown of the solution could be of physical origin, we analysed a simple 1-D model of the same flow, similar to that of Jensen and Pedley (1989). The results confirm that, for given Re and P-d, there is a value of T (T-b), below which the upstream part of the membrane bulges out, with collapse only in the downstream part. Similarly, for fixed T, there is a value of Re (Re-b) above which no fully collapsed solutions are attainable. The values of T-b at given Re and P-d agree very well with the numerical results, especially for higher Re. Further, a qualitative comparison of our analytical predictions with the experimental measurements in a collapsible tube by Bonis and Ribreau (1978), show that it is near the bulging points that steady flow gave way to self-excited oscillations.