STABILITY OF THE VISCOUS-FLOW OF A FLUID THROUGH A FLEXIBLE TUBE

Authors
Citation
V. Kumaran, STABILITY OF THE VISCOUS-FLOW OF A FLUID THROUGH A FLEXIBLE TUBE, Journal of Fluid Mechanics, 294, 1995, pp. 259-281
Citations number
40
Categorie Soggetti
Mechanics,"Phsycs, Fluid & Plasmas
Journal title
ISSN journal
00221120
Volume
294
Year of publication
1995
Pages
259 - 281
Database
ISI
SICI code
0022-1120(1995)294:<259:SOTVOA>2.0.ZU;2-Y
Abstract
The stability of Hagen-Poiseuille flow of a Newtonian fluid of viscosi ty eta in a tube of radius R surrounded by a viscoelastic medium of el asticity G and viscosity eta(s) occupying the annulus R < r < HR is de termined using a linear stability analysis. The inertia of the fluid a nd the medium are neglected, and the mass and momentum conservation eq uations for the fluid and wall are linear. The only coupling between t he mean flow and fluctuations enters via an additional term in the bou ndary condition for the tangential velocity at the interface, due to t he discontinuity in the strain rate in the mean flow at the surface. T his additional term is responsible for destabilizing the surface when the mean velocity increases beyond a transition value, and the physica l mechanism driving the instability is the transfer of energy from the mean flow to the fluctuations due to the work done by the mean flow a t the interface. The transition velocity Gamma(t) for the presence of surface instabilities depends on the wavenumber k and three dimensionl ess parameters: the ratio of the solid and fluid viscosities eta(r) = (eta(s)/eta), the capillary number Lambda = (T/GR) and the ratio of ra dii H, where T is the surface tension of the interface. For eta(r) = 0 and Lambda = 0, the transition velocity Gamma(t) diverges in the limi ts k much less than 1 and k much greater than 1, and has a minimum for finite k. The qualitative behaviour of the transition velocity is the same for Lambda > 0 and eta(r) = 0, though there is an increase in Ga mma(t) in the limit k much greater than 1. When the viscosity of the s urface is non-zero (eta(r) > 0), however, there is a qualitative chang e in the Gamma(t) vs. k curves. For eta(r) < 1, the transition velocit y Gamma(t) is finite only when k is greater than a minimum value k(min ), while perturbations with wavenumber k < k(min) are stable even for Gamma--> infinity. For eta(r) > 1, Gamma(t) is finite only for k(min) < k < k(max), while perturbations with wavenumber k < k(min) or k > k( max) are stable in the limit Gamma--> infinity. As H decreases or eta( r) increases, the difference k(max)- k(min) decreases. At minimum valu e H = H-min, which is a function of eta(r), the difference k(max)-k(mi n) = 0, and for H < H-min, perturbations of all wavenumbers are stable even in the limit Gamma--> infinity. The calculations indicate that H -min shows a strong divergence proportional to exp (0.0832 eta(r)(2)) for eta(r) much greater than 1.