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

TL;DR

This paper investigates the structure, bifurcations, and morphological variations of exact traveling-wave solutions in viscoelastic flow, revealing complex behaviors and new phenomena as parameters like Weissenberg number and domain length change.

Contribution

It provides a detailed analysis of arrowhead solutions in viscoelastic flow, uncovering their bifurcation structure, morphological features, and the conditions for their existence and proliferation.

Findings

Branch topology varies with domain length and Weissenberg number.

Multiple arrowhead solutions can coexist at certain parameters.

A train of arrowheads can form at high Weissenberg numbers.

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.