An analysis of the three-dimensional instability of two-dimensional
viscoelastic elliptical flows is presented, extending the inviscid analysis
of Bayly (1986) to include
both viscous and elastic effects. The problem is governed by three parameters:
E, a geometric parameter related to the ellipticity; Re,
a wavenumber-based Reynolds
number; and De, the Deborah number based on the period of the
base
flow. New modes and mechanisms of instability are discovered. The flow
is generally susceptible
to instabilities in the form of propagating plane waves with a rotating
wavevector,
the tip of which traces an ellipse of the same eccentricity as the flow,
but with the
major and minor axes interchanged. Whereas a necessary condition for purely
inertial
instability is that the wavevector has a non-vanishing component along
the vortex
axis, the viscoelastic modes of instability are most prominent when their
wavevectors
do vanish along this axis. Our analytical and numerical results delineate
the region
of parameter space of (E, ReDe) for which
the new instability exists. A simple model
oscillator equation of the Mathieu type is developed and shown to embody
the
essential qualitative and quantitative features of the secular
viscoelastic instability.
The cause of the instability is a buckling of the ‘compressed’
polymers as they are
perturbed transversely during a particular phase of the passage of the
rotating plane
wave.