  Beneitez et al. (Phys. Rev. Fluids, 8, L101901, 2023) have recently
discovered a new linear "polymer diffusive instability" (PDI) in inertialess
rectilinear viscoelastic shear flow using the FENE-P model when polymer stress
diffusion is present. Here, we examine the impact of inertia on the PDI for
both plane Couette (PCF) and plane Poiseuille (PPF) 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 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.
