Inertial enhancement of the polymer diffusive instability

Abstract

Beneitez et al. (Phys. Rev. Fluids, vol. 8, 2023, L101901) have recently discovered a new linear ‘polymer diffusive instability’ (PDI) in inertialess rectilinear viscoelastic shear flow using the finitely extensible nonlinear elastic constitutive model of Peterlin (FENE-P) when polymer stress diffusion is present. Here, we examine the impact of inertia on the PDI for both plane Couette and plane Poiseuille flows under varying Weissenberg number ${W}$ , polymer stress diffusivity $\varepsilon$ , solvent-to-total viscosity ratio $\beta$ and Reynolds number ${Re}$ , considering the FENE-P and simpler Oldroyd-B constitutive relations. Both the prevalence of the instability in parameter space and the associated growth rates are found to significantly increase with ${Re}$ . For instance, as $Re$ increases with $\beta$ fixed, the instability emerges at progressively lower values of $W$ and $\varepsilon$ than in the inertialess limit, and the associated growth rates increase linearly with $Re$ when all other parameters are fixed. For finite $Re$ , it is also demonstrated that the Schmidt number $Sc=1/(\varepsilon Re)$ collapses curves of neutral stability obtained across various $Re$ and $\varepsilon$ . The observed strengthening of PDI with inertia and the fact that stress diffusion is always present in time-stepping algorithms, either implicitly as part of the scheme or explicitly as a stabilizer, implies that the instability is likely operative in computational work using the popular Oldroyd-B and FENE-P constitutive models. The fundamental question now is whether PDI is physical and observable in experiments, or is instead an artifact of the constitutive models that must be suppressed.