# Stability of the flow of a fluid through a flexible tube at intermediate Reynolds number

Kumaran, V (1998) Stability of the flow of a fluid through a flexible tube at intermediate Reynolds number. In: Journal of Fluid Mechanics, 357 . pp. 123-140.

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## Abstract

The stability of the flow of a fluid in a flexible tube is analysed over a range of Reynolds numbers $1 < Re < 10^4$ using a linear stability analysis. The system consists of a Hagen-Poiseuille flow of a Newtonian fluid of density $\rho$, viscosity $\eta$ and maximum velocity V through a tube of radius R which is surrounded by an incompressible viscoelastic solid of density $\rho$, shear modulus G and viscosity $\eta_s$ in the region R < r < HR. In the intermediate Reynolds number regime, the stability depends on the Reynolds number $Re = \rhoVR/\eta$, a dimensionless parameter $\Sigma = \rhoGR^2=\eta^2$, the ratio of viscosities $\eta_r = \eta_s/\eta$, the ratio of radii H and the wavenumber of the perturbations k. The neutral stability curves are obtained by numerical continuation using the analytical solutions obtained in the zero Reynolds number limit as the starting guess. For $\eta_r = 0$, the flow becomes unstable when the Reynolds number exceeds a critical value $Re_c$, and the critical Reynolds number increases with an increase in $\Sigma$. In the limit of high Reynolds number, it is found that $Re_c \propto\Sigma^{\alpha}$, where $\alpha$ varies between 0:7 and 0:75 for H between 1:1 and 10:0. An analysis of the flow structure indicates that the viscous stresses are confined to a boundary layer of thickness $Re^{-1/3}$ for $Re \gg 1$, and the shear stress, scaled by $\eta V / R$, increases as $Re^{1/3}$. However, no simple scaling law is observed for the normal stress even at $10^3 < Re < 10^5$, and consequently the critical Reynolds number also does not follow a simple scaling relation. The effect of variation of $\eta_r$ on the stability is analysed, and it is found that a variation in $\eta_r$ could qualitatively alter the stability characteristics. At relatively low values of $\Sigma$ (about $10^2$), the system could become unstable at all values of $\eta_r$ , but at relatively high values of $\Sigma$ (greater than about $10^4$), an instability is observed only when the viscosity ratio is below a maximum value $\eta^{\bullet}_{rm}$.

Item Type: Journal Article Copyright of this article belongs to Cambridge University Press. Division of Mechanical Sciences > Chemical Engineering 04 Dec 2006 19 Sep 2010 04:32 http://eprints.iisc.ernet.in/id/eprint/9030