On the essential structure of exact traveling-wave solutions in viscoelastic flow

We examine elastic travelling-wave (‘arrowhead’) solutions in a viscoelastic, unidirec- tionally body-forced flow, focusing on their existence and morphological changes as the Weissenberg number, Wi, and streamwise duct length, 𝐿, are varied. We find that branch topology varies from an isola at low 𝐿 through a two-sided reconnection at intermediate 𝐿 to a branch which exists at asymptotically large Wi for larger 𝐿. At intermediate 𝐿 more than two arrowhead solutions can coexist at a given (Wi, 𝐿) choice due to extra saddle node bifurcations. Secondly, the canonical arrowhead consists of two legs joined by an arched head that blocks throughflow and traps a counter-rotating vortex pair, while a polymer strand can emerge as a by-product of a strong extensional region attached/detached to the arrowhead arch. Thirdly, a minimal domain length 𝐿min required to sustain an arrowhead is found to vary non-monotonically with Wi; for Wi ⩾ 20, detached-strand states control 𝐿min with a relation 𝐿min ≈0.125Wi +1.5. And fourthly, in sufficiently long domains, the upper branch becomes a localised single arrowhead whose streamwise extent depends on Wi, whereas the lower branch can proliferate into a train of arrowheads at high Wi, a phenomenon not previously reported.