Flows with velocity profiles very different from the parabolic velocity profile can occur
in the entrance region of a tube as well as in tubes with converging/diverging cross-sections. In this paper, asymptotic and numerical studies are undertaken to analyse the
temporal stability of such ‘non-parabolic’ flows in a
flexible tube in the limit of high
Reynolds numbers. Two specific cases are considered: (i) developing flow in a flexible
tube; (ii) flow in a slightly converging flexible tube. Though the mean velocity profile
contains both axial and radial components, the flow is assumed to be locally parallel
in the stability analysis. The fluid is Newtonian and incompressible, while the flexible
wall is modelled as a viscoelastic solid. A high Reynolds number asymptotic analysis
shows that the non-parabolic velocity profiles can become unstable in the inviscid
limit. This inviscid instability is qualitatively different from that observed in previous
studies on the stability of parabolic flow in a flexible tube, and from the instability of
developing flow in a rigid tube. The results of the asymptotic analysis are extended
numerically to the moderate Reynolds number regime. The numerical results reveal
that the developing flow could be unstable at much lower Reynolds numbers than
the parabolic flow, and hence this instability can be important in destabilizing the
fluid flow through flexible tubes at moderate and high Reynolds number. For flow in
a slightly converging tube, even small deviations from the parabolic profile are found
to be sufficient for the present instability mechanism to be operative. The dominant
non-parallel effects are incorporated using an asymptotic analysis, and this indicates
that non-parallel effects do not significantly affect the neutral stability curves. The
viscosity of the wall medium is found to have a stabilizing effect on this instability.
Asymptotic analysis of internal relaxation oscillations in a conceptual climate model
