The stability of the flow of an incompressible, viscous fluid through
a pipe of circular
cross-section, curved about a central axis is investigated in a weakly
nonlinear regime.
A sinusoidal pressure gradient with zero mean is imposed, acting along
the pipe.
A WKBJ perturbation solution is constructed, taking into account the need
for an
inner solution in the vicinity of the outer bend, which is obtained by
identifying the
saddle point of the Taylor number in the complex plane of the cross-sectional
angle
coordinate. The equation governing the nonlinear evolution of the leading-order
vortex
amplitude is thus determined. The stability analysis of this flow to axially
periodic
disturbances leads to a partial differential system dependent on three
variables, and
since the differential operators in this system are periodic in time, Floquet
theory may
be applied to reduce it to a coupled infinite system of ordinary differential
equations,
together with homogeneous uncoupled boundary conditions. The eigenvalues
of this
system are calculated numerically to predict a critical Taylor number consistent
with
the analysis of Papageorgiou (1987). A discussion of how nonlinear effects
alter the
linear stability analysis is also given. It is found that solutions to
the leading-order
vortex amplitude equation bifurcate subcritically from the eigenvalues
of the linear
problem.