Stability of wall modes in a flexible tube

Abstract

The asymptotic results (Kumaran 1998b) obtained for Λ∼1 for the flow in a flexible tube are extended to the limit Λ[Lt ]1 using a numerical scheme, where Λ is the dimensionless parameter Re^1/3 (G/ρV²), Re=(ρVR/η) is the Reynolds number, ρ and η are the density and viscosity of the fluid, R is the tube radius and G is the shear modulus of the wall material. The results of this calculation indicate that the least-damped mode becomes unstable when Λ decreases below a transition value at a fixed Reynolds number, or when the Reynolds number increases beyond a transition value at a fixed Λ. The Reynolds number at which there is a transition from stable to unstable perturbations for this mode is determined as a function of the parameter Σ=(ρGR²/η²), the scaled wavenumber of the perturbations kR, the ratio of radii of the wall and fluid H and the ratio of viscosities of the wall material and the fluid η_r. For η_r=0, the Reynolds number at which there is a transition from stable to unstable perturbations decreases proportional to Σ^1/2 in the limit Σ[Lt ]1, and the neutral stability curves have a rather complex behaviour in the intermediate regime with the possibility of turning points and isolated domains of instability. In the limit Σ[Gt ]1, the Reynolds number at which there is a transition from stable to unstable perturbations increases proportional to Σ^α, where α is between 0.7 and 0.75. An increase in the ratio of viscosities η_r has a complex effect on the Reynolds number for neutrally stable modes, and it is observed that there is a maximum ratio of viscosities at specified values of H at which neutrally stable modes exist; when the ratio of viscosities is greater than this maximum value, perturbations are always stable.