Weakly nonlinear analysis of the viscoelastic instability in channel flow for finite and vanishing Reynolds numbers

Abstract

The recently discovered centre-mode instability of rectilinear viscoelastic shear flow (Garget al.,Phys. Rev. Lett., vol. 121, 2018, 024502) has offered an explanation for the origin of elasto-inertial turbulence that occurs at lower Weissenberg numbers ($Wi$). In support of this, we show using weakly nonlinear analysis that the subcriticality found in Pageet al.(Phys. Rev. Lett., vol. 125, 2020, 154501) is generic across the neutral curve with the instability becoming supercritical only at low Reynolds numbers ($Re$) and high$Wi$. We demonstrate that the instability can be viewed as purely elastic in origin, even for$Re=O(10^3)$, rather than ‘elasto-inertial’, as the underlying shear doesnotfeed the kinetic energy of the instability. It is also found that the introduction of a realistic maximum polymer extension length,$L_{max}$, in the FENE-P model moves the neutral curve closer to the inertialess$Re=0$limit at a fixed ratio of solvent-to-solution viscosities,$\beta$. At$Re=0$and in the dilute limit ($\beta \rightarrow 1$) with$L_{max} =O(100)$, the linear instability can be brought down to more physically relevant$Wi\gtrsim 110$at$\beta =0.98$, compared with the threshold$Wi=O(10^3)$at$\beta =0.994$reported recently by Khalidet al.(Phys. Rev. Lett., vol. 127, 2021, 134502) for an Oldroyd-B fluid. Again, the instability is subcritical, implying that inertialess rectilinear viscoelastic shear flow is nonlinearly unstable – i.e. unstable to finite-amplitude disturbances – for even lower$Wi$.