Stability Problem in the Theory of Flow through Collapsible Tubes

摘要

An investigation of the interaction of wave propagation in the flow through a nearly cylindrical collapsible tube filled with an incompressible, inviscid fluid with the flow in the upstream and downstream rigid segments of the hydraulic circuit in which the collapsible tube is mounted is reported. This leads to the study of eigenmodes of the classical wave equation satisfying somewhat unusual boundary conditions dictated by the matching of rigid and flexible parts of the circuit. The characteric equation, determining the frequencies of the eigenmodes, is of a transcendental quasi-polynominal type and the location of its roots is analyzed. It appears that for subcritical flow (S is less than 1) all eigenmodes are damped, whereas for supercritical flow (S is greater than 1) also growing eigenmodes exist. Here S is the speed index, being the ratio V/C, where V is the fluid speed in the collapsible tube and C the speed of wave propagation. The analysis leads to the conclusion that for subcritical flow the system is stable. (Copyright (c) 1988 by Faculty of Technical Mathematics and Informatics, Delft, The Netherlands.)