Revisiting creeping viscoelastic cross-slot flow: Global linear stability and structural sensitivity analyses

Abstract

The viscoelastic instability of cross-slot flow was first observed experimentally almost half a century ago and reproduced numerically two decades ago, yet its physical origin remains unresolved. We revisit this problem for two-dimensional creeping flow of Oldroyd-B fluid by combining direct numerical simulations, global stability analysis, structural sensitivity analysis, and energy-budget analysis. Our simulations reproduce the canonical pitchfork bifurcation, and the stability analysis consistently predicts the threshold and perturbation growth rates. The leading eigenmode consists of a chiral velocity--stress perturbation that tilts and rotates the birefringent strand generated by the extensional flow. Structural sensitivity and energy-budget analyses identify narrow high-extension-rate ridges within the extensional flow as both the spatial core and energetic source of the instability. In these ridges, the stress-based wavemaker co-localizes with large positive disturbance polymeric stress power density, indicating localized transfer of stored elastic energy to the disturbance flow. Analyses of cross-slot variants with rounded corners and with a centered cylinder further reveal that neither sharp corners nor a free central stagnation point is the essential destabilizing ingredient; rather, the instability originates from elastic-energy release in extension-dominated regions characteristic of cross-slot flow.