Abstract
The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed
in the limit of zero Reynolds number. The system consists of a Couette flow of a
Newtonian fluid past a viscoelastic medium of non-dimensional thickness H (the ratio
of wall thickness to the fluid thickness), and viscosity ratio μr (ratio of the viscosities
of wall and fluid media). The wall medium is bounded by the fluid at one surface
and two different types of boundary conditions are considered at the other surface of
the wall medium – for ‘grafted’ gels zero displacement conditions are applied while
for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is
permitted to move in the lateral direction. The linear stability analysis reveals that for
grafted gels the most unstable modes have α ∼ O(1), while for adsorbed gels the most
unstable modes have α → 0, where α is the wavenumber of the perturbations. The
results from the weakly nonlinear analysis indicate that the nature of the bifurcation
at the linear instability is qualitatively very different for grafted and absorbed gels.
The bifurcation is always subcritical for the case of flow past grafted gels. It is found,
however, that relatively weak but finite-amplitude disturbances do not significantly
reduce the critical velocity required to destabilize the flow from the critical velocity
predicted by the linear stability theory. For the case of adsorbed gels, it is found
that a supercritical equilibrium state could exist in the limit of small wavenumber
for a wide range of parameters μr and H, while the bifurcation becomes subcritical
at larger values of the wavenumber and there is a transition from supercritical to
subcritical bifurcation as the wavenumber is increased.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
29 articles.
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