STABILITY OF THE FLOW OF A FLUID THROUGH A FLEXIBLE TUBE AT INTERMEDIATE REYNOLDS-NUMBER

The stability of the flow of a fluid in a flexible tube is analysed ov er a range of Reynolds numbers 1 < Re < 10(4) using a linear stability analysis. The system consists of a Hagen-Poiseuille flow of a Newtoni an fluid of density rho, viscosity eta and maximum velocity V through a tube of radius R which is surrounded by an incompressible viscoelast ic solid of density rho, shear modulus G and viscosity eta(s) in the r egion R < r < HR. In the intermediate Reynolds number regime, the stab ility depends on the Reynolds number Re = rho VR/eta, a dimensionless parameter Sigma = rho GR(2)/eta(2), the ratio of viscosities eta(r) = eta(s)/eta, the ratio of radii H and the wavenumber of the perturbatio ns k. The neutral stability curves are obtained by numerical continuat ion using the analytical solutions obtained in the zero Reynolds numbe r limit as the starting guess. For eta(r) = 0, the flow becomes unstab le when the Reynolds number exceeds a critical value Re-c, and the cri tical Reynolds number increases with an increase in C. In the limit of high Reynolds number, it is found that Re-c proportional to Sigma(alp ha), 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 stress es are confined to a boundary layer of thickness Re-1/3 for Re much gr eater than 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 nu mber also does not follow a simple scaling relation. The effect of var iation of eta(r) on the stability is analysed, and it is found that a variation in eta(r) could qualitatively alter the stability characteri stics. At relatively low values of Sigma (about 10(2)), the system cou ld become unstable at all values of eta(r), but at relatively high val ues of Sigma (greater than about 10(4)), an instability is observed on ly when the viscosity ratio is below a maximum value eta(rm).