Stability of the viscous flow of a fluid through a flexible tube

Abstract

The stability of Hagen-Poiseuille flow of a Newtonian fluid of viscosity η in a tube of radius R surrounded by a viscoelastic medium of elasticity G and viscosity η_s occupying the annulus R < r < HR is determined using a linear stability analysis. The inertia of the fluid and the medium are neglected, and the mass and momentum conservation equations for the fluid and wall are linear. The only coupling between the mean flow and fluctuations enters via an additional term in the boundary condition for the tangential velocity at the interface, due to the discontinuity in the strain rate in the mean flow at the surface. This additional term is responsible for destabilizing the surface when the mean velocity increases beyond a transition value, and the physical 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 at the interface. The transition velocity Γ_t for the presence of surface instabilities depends on the wavenumber k and three dimensionless parameters: the ratio of the solid and fluid viscosities η_r = (η_s/η), the capillary number λ = (T/GR) and the ratio of radii H, where T is the surface tension of the interface. For η_r = 0 and λ = 0, the transition velocity Γ_t diverges in the limits k [double less-than sign] 1 and k [dbl greater-than sign] 1, and has a minimum for finite k. The qualitative behaviour of the transition velocity is the same for λ < 0 and η_r = 0, though there is an increase in λ_t in the limit k > 1. When the viscosity of the surface is non-zero (η_r < 0), however, there is a qualitative change in the λ_t vs. k curves. For η_r < 1, the transition velocity λ_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 λ [rightward arrow] [infty infinity]. For η_r > 1, λ_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 λ [rightward arrow] [infty infinity]. As H decreases or η_r increases, the difference k_max - k_min decreases. At a minimum value H = H_min which is a function of η_r, the difference k_max - k_min = 0, and for H < H_min, perturbations of all wavenumbers are stable even in the limit λ [rightward arrow] [infty infinity]. The calculations indicate that H_min shows a strong divergence proportional to exp (0.0832n_r²) for η_r [dbl greater-than sign] 1.