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# Stability of wall modes in a flexible tube

Kumaran, V (1998) Stability of wall modes in a flexible tube. In: Journal of Fluid Mechanics, 362 . pp. 1-15.

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## Abstract

The asymptotic results (Kumaran 1998b) obtained for $\Lambda\sim1$ for the flow in a flexible tube are extended to the limit $\Lambda \ll 1$ using a numerical scheme, where $\Lambda$ is the dimensionless parameter $Re^{1/3}(G/pV^2), Re = (p^{VR}/eta)$ is the Reynolds number, $\rho$ and $\eta$ 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 $\Lambda$ decreases below a transition value at a fixed Reynolds number, or when the Reynolds number increases beyond a transition value at a fixed $\Lambda$ . 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 $\sum = (pGR^2/\eta^2)$, 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 $\eta_r$. For $\eta_r = 0$, the Reynolds number at which there is a transition from stable to unstable perturbations decreases proportional to $\sum^{1/2}$ in the limit $\sum \ll 1$, 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 $\sum \ll 1$, the Reynolds number at which there is a transition from stable to unstable perturbations increases proportional to $\sum^\alpha$, where $\alpha$ is between 0:7 and 0:75. An increase in the ratio of viscosities $\eta_r$ 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.

Item Type: Journal Article Journal of Fluid Mechanics Cambridge University Press Copyright of this article belongs to Cambridge University Press. Division of Mechanical Sciences > Chemical Engineering 16 Jan 2007 19 Sep 2010 04:34 http://eprints.iisc.ac.in/id/eprint/9499

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