  Amplification of deterministic disturbances in inertialess shear-driven
channel flows of viscoelastic fluids is examined by analyzing the frequency
responses from spatio-temporal body forces to the velocity and polymer stress
fluctuations. In strongly elastic flows, we show that disturbances with large
streamwise length scales may be significantly amplified even in the absence of
inertia. For fluctuations without streamwise variations, we derive explicit
analytical expressions for the dependence of the worst-case amplification (from
different forcing to different velocity and polymer stress components) on the
Weissenberg number ($We$), the maximum extensibility of the polymer chains
($L$), the viscosity ratio, and the spanwise wavenumber. For the Oldroyd-B
model, the amplification of the most energetic components of velocity and
polymer stress fields scales as $We^2$ and $We^4$. On the other hand, finite
extensibility of polymer molecules limits the largest achievable amplification
even in flows with infinitely large Weissenberg numbers: in the presence of
wall-normal and spanwise forces the amplification of the streamwise velocity
and polymer stress fluctuations is bounded by quadratic and quartic functions
of $L$. This high amplification signals low robustness to modeling
imperfections of inertialess channel flows of viscoelastic fluids. The
underlying physical mechanism involves interactions of polymer stress
fluctuations with a base shear, and it represents a close analog of the lift-up
mechanism that initiates a bypass transition in inertial flows of Newtonian
fluids.
