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

Abstract

The stability of the flow of a fluid in a flexible tube is analysed over a range of Reynolds numbers 1<Re<104 using a linear stability analysis. The system consists of a Hagen–Poiseuille flow of a Newtonian fluid of density ρ, viscosity η and maximum velocity V through a tube of radius R which is surrounded by an incompressible viscoelastic solid of density ρ, shear modulus G and viscosity ηs in the region R<r<HR. In the intermediate Reynolds number regime, the stability depends on the Reynolds number Re=ρVR/η, a dimensionless parameter [sum ]=ρGR2/η2, the ratio of viscosities ηr= ηs/η, 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 ηr=0, the flow becomes unstable when the Reynolds number exceeds a critical value Rec, and the critical Reynolds number increases with an increase in [sum ]. In the limit of high Reynolds number, it is found that Rec∝[sum ]α, where α 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[Gt ]1, and the shear stress, scaled by ηV/R, increases as Re1/3. However, no simple scaling law is observed for the normal stress even at 103<Re<105, and consequently the critical Reynolds number also does not follow a simple scaling relation. The effect of variation of ηr on the stability is analysed, and it is found that a variation in ηr could qualitatively alter the stability characteristics. At relatively low values of [sum ] (about 102), the system could become unstable at all values of ηr, but at relatively high values of [sum ] (greater than about 104), an instability is observed only when the viscosity ratio is below a maximum value η*rm.