https://doi.org/10.1007/s100510170306
Asymptotic analysis of wall modes in a flexible tube revisited
Department of Chemical
Engineering, Indian Institute of Science, Bangalore 560 012,
India
Corresponding author: a kumaran@chemeng.iisc.ernet.in
Received:
12
June
2000
Revised:
8
November
2000
Published online: 15 February 2001
The stability of wall modes in fluid flow through a flexible tube
of radius R surrounded by a viscoelastic material in the
region R < r < H R is
analysed using a combination of asymptotic and numerical methods.
The fluid is Newtonian, while the flexible wall is modelled as an
incompressible viscoelastic solid. In the limit of high Reynolds
number (Re), the vorticity of the wall modes is confined to a
region of thickness in the fluid near the wall of the
tube. Previous numerical studies on the stability of Hagen-Poiseuille flow in a flexible tube to axisymmetric disturbances
have shown that the flow could be unstable in the limit of high Re,
while previous high Reynolds number asymptotic analyses have
revealed only stable modes. To resolve this discrepancy, the present
work re-examines the asymptotic analysis of wall modes in a flexible
tube using a new set of scaling assumptions. It is
shown that wall modes in Hagen-Poiseuille flow in a flexible tube
are indeed unstable in the limit of high Re in the scaling regime
. Here Σ is a nondimensional parameter
characterising the elasticity of the wall, and
, where ρ and η are the density and
viscosity of the fluid, and G is the shear modulus of the wall medium.
The results from the present asymptotic analysis are in excellent
agreement with the previous numerical results. Importantly,
the present work shows that the different types of unstable modes at high
Reynolds number reported in previous numerical studies are
qualitatively the same: they all belong to the class of unstable
wall modes predicted in this paper.
PACS: 83.50.-v – Deformation; material flow / 47.15.Fe – Stability of laminar flows / 47.60.+i – Flows in ducts, channels, nozzles and conduits
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2001