The centre-mode instability and arrowheads in channel flow of a Giesekus fluid

Introducing a small concentration of elastic polymers into a Newtonian fluid introduces a new `centre mode’ instability (Garg et al, Phys. Rev. Lett. 121, 2018) which persists at zero Reynolds number (Re), albeit at wildly unphysical parameter values. This talk explores the characteristics of the centre mode in pressure-driven channel flow of a Giesekus fluid, where a mobility parameter α is used to control the amount of second normal stress difference. We first study the linear stability, which is seen to differ significantly from simpler models (Oldroyd-B, FENE-P) and connects to Re=0 for Weissenberg numbers as low as Wi~40 at realistic values of α. Weakly non-linear analysis shows that the instability is subcritical at low Re but can become supercritical in the elasto-inertial regime for Re>50 and α>10^(-3). We then employ full branch continuation to trace the unstable solutions from the bifurcation point down to the saddle node at Wi~10 (at Re=0). Our findings demonstrate that the subcritical nature of the instability is maintained in the more complex model, with the nonlinear unstable region extending well below linear stability predictions. Following the branches, we find that the structure of the upper branch arrowhead is qualitatively unchanged from the FENE-P model, as is the location of the saddle node at least at low Re.