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 [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=[rho]VR/[eta], a dimensionless parameter [sum L: summation operator]=[rho]GR²/[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 Rec, and the critical Reynolds number increases with an increase in [sum L: summation operator]. In the limit of high Reynolds number, it is found that Rec[is proportional to][sum L: summation operator][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[minus sign]^1/3 for Re[dbl greater-than sign]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 10³<Re<10⁵, 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 [sum L: summation operator] (about 10²), the system could become unstable at all values of [eta]r, but at relatively high values of [sum L: summation operator] (greater than about 10⁴), an instability is observed only when the viscosity ratio is below a maximum value [eta]^∗_rm.