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.