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

Abstract

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, $\mathrm{Wi}$, 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 $\mathrm{Wi}$ for larger $L$. At intermediate $L$ more than two arrowhead solutions can coexist at a given $(\mathrm{Wi}, L)$ 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 $L_{\min}$ required to sustain an arrowhead is found to vary non-monotonically with $\mathrm{Wi}$; for $\mathrm{Wi}\ge 20$, detached-strand states control $L_{\min}$ with a relation $L_{\min}\approx 0.125\mathrm{Wi}+1.5$. And fourthly, in sufficiently long domains, the upper branch becomes a localised single arrowhead whose streamwise extent depends on $\mathrm{Wi}$, whereas the lower branch can proliferate into a train of arrowheads at high $\mathrm{Wi}$, a phenomenon not previously reported.