https://doi.org/10.1007/s100510170045
Effect of tangential interface motion on the viscous instability in fluid flow past flexible surfaces
Department of Chemical Engineering, Indian Institute of
Science, Bangalore 560 012, India
Corresponding author: a kumaran@chemeng.iisc.ernet.in
Received:
8
November
2000
Revised:
20
March
2001
Published online: 15 October 2001
The stability of linear shear flow of a Newtonian fluid past a
flexible membrane is analysed in the limit of low Reynolds number as
well as in the intermediate Reynolds number regime for two different
membrane models. The objective of this paper is to demonstrate the
importance of tangential motion in the membrane on the stability
characteristics of the shear flow. The first model assumes the wall
to be a "spring-backed" plate membrane, and the displacement of the
wall is phenomenologically related in a linear manner to the change
in the fluid stresses at the wall. In the second model, the
membrane is assumed to be a two-dimensional compressible
viscoelastic sheet of infinitesimal thickness, in which the
constitutive relation for the shear stress contains an elastic part
that depends on the local displacement field and a viscous component
that depends on the local velocity in the membrane. The stability
characteristics of the laminar flow in the limit of low {Re} are
crucially dependent on the tangential motion in the membrane wall.
In both cases, the flow is stable in the low Reynolds number
limit in the absence of tangential motion in the membrane. However,
the presence of tangential motion in the membrane destabilises the
shear flow even in the absence of fluid inertia. In this
case, the non-dimensional velocity () required for
unstable fluctuations is proportional to the wavenumber k
(
) in the plate membrane type of wall while it
scales as k2 in the viscoelastic membrane type of wall
(
) in the limit
. The results
of the low Reynolds number analysis are extended numerically to the
intermediate Reynolds number regime for the case of a viscoelastic
membrane. The numerical results show that for a given set of wall
parameters, the flow is unstable only in a finite range of Reynolds
number, and it is stable in the limit of large Reynolds number.
PACS: 83.50.-v – Deformation and 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