https://doi.org/10.1007/s100510050409
Asymptotic analysis of wall modes in a flexible tube
Department of Chemical Engineering,
Indian Institute of Science, Bangalore 560 012, India
Corresponding author: a kumaran@chemeng.iisc.ernet.in
Received:
5
November
1997
Revised:
10
March
1998
Accepted:
29
April
1998
Published online: 15 August 1998
The stability of wall modes in a flexible tube of radius R
surrounded by a viscoelastic material in the region R < r < H R in
the high Reynolds number limit is studied using asymptotic
techniques. The fluid is a Newtonian fluid,
while the wall material is modeled as an incompressible
visco-elastic solid. In the limit of high Reynolds number, the vorticity of
the wall modes is confined to a region of thickness in
the fluid near the wall of
the tube, where the small parameter
, and
the Reynolds number is
,
ρ and η are the fluid density and viscosity,
and V is the maximum fluid velocity. The regime
is considered
in the asymptotic analysis, where G is
the shear modulus of the wall material. In this limit, the
ratio of the normal stress and normal displacement in the wall,
, is only a function of H and scaled wave number
. There are multiple solutions for the growth rate which
depend on the parameter
.
In the limit
, which is equivalent to using a
zero normal stress boundary condition for the fluid, all the
roots have negative real parts, indicating that the wall modes
are stable. In the limit
, which corresponds to
the flow in a rigid tube, the stable roots of previous studies
on the flow in a rigid tube are recovered. In addition, there is one
root in the limit
which does not reduce to any of the rigid
tube solutions determined previously.
The decay rate of this solution decreases
proportional to
in the limit
, and the
frequency increases proportional to
.
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, 1998