Stability of inviscid flow in a flexible tube

Abstract

The equivalents of the classical theorems of hydrodynamic stability are derived for inviscid flow through a flexible tube. An important difference between flows in plane and cylindrical geometries is that the Squire transformation, which states that two-dimensional perturbations in plane parallel flows are always more unstable than three-dimensional perturbations, is not valid for tube flows. Therefore, it is necessary to analyse both axisymmetric and non-axisymmetric perturbations in flows in cylindrical geometries. Perturbations of the form exp[ik(χ –ct) + inϕ] are imposed on a steady axisymmetric mean flow V(r), and the stability of the mean velocity profiles and bounds for the phase velocity of the unstable modes are determined. Here r, ϕ and x are the radial, polar and axial directions, and k and c are the wavenumber and phase velocity. The flexible wall is represented by a standard constitutive equation which contains inertial, elastic and dissipative terms. Results for general velocity profiles are derived in two limiting cases: axisymmetric flows (n = 0) and highly non-axisymmetric flows (n [Gt ] k). The results indicate that axisymmetric perturbations are always stable for (V″ – r−1V′) V < 0 and could be unstable for (V″ – r−1V′) V < 0, while highly non-axisymmetric perturbations are always stable for (V″ + r−1V′) V [ges ] 0 and could be unstable for (V″ + r−1V′) V < 0. In addition, bounds on the real part (cr) and imaginary part (ci) of the phase velocity are also derived. For the practically important case of Hagen–Poiseuille flow, the present analysis indicates that axisymmetric perturbations are always stable, while highly non-axisymmetric perturbations could be unstable. This is in contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones.