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 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 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. 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₌₈₍ required to sustain an arrowhead is found to vary non-monotonically with Wi ; for {Wi} 20, detached-strand states control L₌₈₍ with a relation L₌₈₍ 0. 125\, {Wi}+1. 5. And fourth, 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.
Zhu et al. (Wed,) studied this question.