  Modal stability analysis provides information about the long-time growth or
decay of small-amplitude perturbations around a steady-state solution of a
dynamical system. In fluid flows, exponentially growing perturbations can
initiate departure from laminar flow and trigger transition to turbulence.
Although flow of a Newtonian fluid through a pipe is linearly stable for very
large values of the Reynolds number ($Re \sim 10^7$), a transition to
turbulence often occurs for $Re$ as low as $1500$. When a dilute polymer
solution is used in the place of a Newtonian fluid, the transitional value of
the Reynolds number decreases even further. Using the spectral collocation
method and Oldroyd-B constitutive equation, Garg et al. (Phys. Rev. Lett.
121:024502, 2018) claimed that such a transition in viscoelastic fluids is
related to linear instability. Since differential matrices in the collocation
method become ill-conditioned when a large number of basis functions is used,
we revisit this problem using the well-conditioned spectral integration method.
We show modal stability of viscoelastic pipe flow for a broad range of fluid
elasticities and polymer concentrations, including cases considered by Garg et
al. Similarly, we find that plane Poiseuille flow is linearly stable for cases
where Garg et al. report instability. In both channel and pipe flows, we
establish the existence of spurious modes that diverge slowly with finer
discretization and demonstrate that these can be mistaken for grid-independent
modes if the discretization is not fine enough.
