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In certain parameter regimes, steady flow through flexible tubes is unstable to self-excited oscillations. Whittaker et al. (2010, Proc. Roy. Soc. A 466) solved an asymptotic model for the onset of self-excited oscillations in a long, thin-walled,flexible tube clamped between two rigid tubes, with a large axial tension. This work neglected effects such as wall inertia, axial bending, and in-plane shear forces. Whittaker (2015, IMA J. Appl. Math.) reintroduced in-plane shearing and found a shear-relaxation boundary layer at the tube ends. In this thesis,wall inertia and axial bending are reintroduced into these models. In Chapter 2, wall inertia terms are added to the governing equations for the wall mechanics, and a new ‘tube law’ describing the wall motion is derived. Combining this with a description of the fluid mechanics, the effect of wall inertia on the oscillations is quantified. Wall inertia is found to be a destabilising effect. In Chapters 3–7, axial bending is reintroduced allowing ‘clamped’ boundaryconditions to be satisfied at the tube ends. Three different regimes dependent on the dimensionless length and wall thickness of the tube are found. Chapters 4–5 concentrate on the two regimes where the shear layer found by Whittaker (2015) must be considered. An axial bending boundary layer that induces higher-order corrections to the shear layer and bulk solution is found in these regimes. In Chapters 6–7, a final regime is considered where the shear layer no longer needs consideration, but a new model for the wall mechanics is needed. Deriving and solving a linearised 2D model for bending a semi-infinite block under tension, corresponding to a 2D cross-section of the tube wall, a new transverse shear-relaxation layer is found. This boundary layer allows clamped boundary conditions to be satisfied and induces higher-order corrections to the bulk solution. |
