Starting from stationary bifurcations in Couette–Dean flow, we compute stationary
nontrivial solutions in the circular Couette geometry for an inertialess finitely extensible
nonlinear elastic (FENE-P) dumbbell fluid. These solutions are isolated from
the Couette flow branch arising at finite amplitude in saddle–node bifurcations as
the Weissenberg number increases. Spatially, they are strongly localized axisymmetric
vortex pairs embedded in an arbitrarily long ‘far field’ of pure Couette flow, and are
thus qualitatively, and to some extent quantitatively, similar to the ‘diwhirl’ (Groisman
& Steinberg 1997) and ‘flame’ patterns (Baumert & Muller 1999) observed experimentally.
For computationally accessible parameter values, these solutions appear
only above the linear instability limit of the Couette base flow, in contrast to the
experimental observations. Correspondingly, they are themselves linearly unstable.
Nevertheless, extrapolation of the trend in the bifurcation points with increasing
polymer extensibility suggests that for sufficiently high extensibility the diwhirls will
come into existence before the linear instability, as seen experimentally.
Based on the computed stress and velocity fields, we propose a fully nonlinear self-sustaining
mechanism for these flows. The mechanism is related to that for viscoelastic
Dean flow vortices and arises from a finite-amplitude perturbation giving rise to a
locally unstable profile of the azimuthal normal stress near the outer cylinder at
the symmetry plane of the vortex pair. The unstable stress profile, in combination
with a ‘tubeless siphon’ effect, nonlinearly sustains the patterns. We propose that
these solitary, strongly nonlinear structures comprise fundamental building blocks for
complex spatiotemporal dynamics in the flow of elastic liquids.