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

We examine elastic travelling-wave (‘arrowhead’) solutions in a viscoelastic, unidirectionally body-forced flow, focusing on their existence and morphological changes as the Weissenberg number ${\textit{Wi}}$ and streamwise duct length $L$ are varied. We find that, First, branch topology varies from an isola at low $L$ through a two-sided reconnection at intermediate $L$ to a branch that exists at asymptotically large ${\textit{Wi}}$ for larger $L$ . At intermediate $L$ , more than two arrowhead solutions can coexist at a given $({\textit{Wi}},L)$ choice due to extra saddle–node bifurcations. Second, 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 to/detached from the arrowhead arch. Third, a minimal domain length $L_{min }$ required to sustain an arrowhead is found to vary non-monotonically with ${\textit{Wi}}$ ; for ${{\textit{Wi}}}\geqslant 20$ , detached-strand states control $L_{min }$ with a relation $L_{min }\approx 0.125\,{{\textit{Wi}}}+1.5$ . And fourth, in sufficiently long domains, the upper branch becomes a localised single arrowhead whose streamwise extent depends on ${\textit{Wi}}$ , whereas the lower branch can proliferate into a train of arrowheads at high ${\textit{Wi}}$ , a phenomenon not previously reported.