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Multiple solutions and flow limitation in collapsible channel flows. (English) Zbl 0993.76018
Summary: Steady and unsteady numerical simulations of two-dimensional flow in a collapsible channel were carried out to study the flow limitation which typically occurs when the upstream transmural pressure is held constant while flow rate and pressure gradient along the collapsible channel can vary independently. Multiple steady solutions are found for a range of upstream transmural pressures and Reynolds number using an arclength control method. The stability of these steady solutions is tested in order to check the correlation between flow limitation and self-excited oscillations (the latter being a consequence of unstable steady solutions). Both stable and unstable solutions are found when flow is limited. Self-excited oscillations and divergence instabilities are observed in certain solution branches. The instability of the steady solutions seems to depend on the unsteady boundary conditions used, i.e. on which parameters are allowed to a vary. However, steady solutions associated with the solution branch before flow limitation where the membrane wall bulges are found to be stable for each of the three different boundary conditions employed. We conclude that there is no one-to-one correlation between the two phenomena in this two-dimensional channel model.

MSC:
76D99 Incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
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