  Beneitez et al. (2023b) have recently discovered a new linear "polymer
diffusive instability" (PDI) in inertialess viscoelastic rectilinear shear flow
of a FENE-P fluid with polymer stress diffusion. Here, we examine the impact of
inertia on the PDI, which we delineate for both plane Couette and channel
configurations under varying Weissenberg number $W$, polymer stress diffusivity
$\varepsilon$, solvent-to-total viscosity $\beta$ and Reynolds number $Re$,
considering Oldroyd-B and FENE-P 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. This
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 stabiliser, 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.

Fluid Dynamics