Stability of non-parabolic flow in a flexible tube

Flows with velocity profiles very different from the parabolic velocity pro file can occur in the entrance region of a tube as well as in tubes with co nverging/diverging cross-sections. In this paper, asymptotic and numerical studies are undertaken to analyse the temporal stability of such 'non-parab olic' flows in a flexible tube in the limit of high Reynolds numbers. Two s pecific cases are considered: (i) developing flow in a flexible tube; (ii) flow in a slightly converging flexible tube. Though the mean velocity profi le contains both axial and radial components, the flow is assumed to be loc ally parallel in the stability analysis. The fluid is Newtonian and incompr essible, while the flexible wall is modelled as a viscoelastic solid. A hig h Reynolds number asymptotic analysis shows that the non-parabolic velocity profiles can become unstable in the inviscid limit. This inviscid instabil ity is qualitatively different from that observed in previous studies on th e 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 numeri cal results reveal that the developing flow could be unstable at much lower Reynolds numbers than the parabolic flow, and hence this instability can b e important in destabilizing the fluid flow through flexible tubes at moder ate 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-paralle l effects are incorporated using an asymptotic analysis, and this indicates that non-parallel effects do not significantly affect the neutral stabilit y curves. The viscosity of the wall medium is found to have a stabilizing e ffect on this instability.