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.