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Modelling flow and oscillations in collapsible tubes. (English) Zbl 0931.74024
The object of modelling is the fluid flow through collapsible tubes which has several physiological applications. This article describes the standard situation of the flow along a finite elastic tube attached at its ends to rigid tubes and contained in a chamber whose pressure can be varied independently. At first, the authors develop one-dimensional models that describe steady flow and its instability and predict a variety of oscillation types. More complete description gives a related two-dimensional configuration of flow in a channel with parallel walls, a segment of a wall is replaced by a membrane under longitudinal tension \(T\). The authors calculate the membrane displacements by using the lubrication theory, Stokes equations, and steady and unsteady Navier-Stokes equations. It is shown that for a given Reynolds number \(Re\) steady flow becomes unstable below a critical value of \(T\), and period-doubling bifurcation arises. Analogous results take place for fixed \(T\) when \(Re\) exceeds a critical value. The authors investigate the effect of wall inertia and show that this effect is negligible if the flowing fluid is water, but, when the fluid is air, the high frequency flutter arises.

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76D05 Navier-Stokes equations for incompressible viscous fluids
76D08 Lubrication theory
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