  We study the effect of acceleration and deceleration on the stability of
channel flows. To do so, we derive an exact solution for laminar profiles of
channel flows with arbitrary, time-varying wall motion and pressure gradient.
This solution then allows us to investigate the stability of any unsteady
channel flow. In particular, we investigate the nonnormal growth of
perturbations in flows with exponentially decaying acceleration and
deceleration, with comparisons to growth in a constant flow (i.e., the
time-invariant simple shear or parabolic profile). We apply this acceleration
and deceleration through the velocity of the walls and through the flow rate.
For accelerating flows, disturbances never grow larger than disturbances in a
constant flow, while decelerating flows show massive amplification of
disturbances -- at a Reynolds number of 800, disturbances in the decelerating
flow grow $\mathscr{O}(10^4)$ times larger than disturbances grow in a constant
flow. This amplification increases as we raise the rate of deceleration and the
Reynolds number. We find that this amplification arises due to a transition
from spanwise perturbations leading to the largest amplification, to streamwise
perturbations leading to the largest amplification that only occurs in the
decelerating flow. By evolving the optimal perturbations through the linearized
equations of motion, we reveal that the decelerating case achieves this massive
amplification through a down-gradient Reynolds stress mechanism, which
accelerating and constant flows cannot maintain. Finally, we end the paper by
validating that these perturbations exhibit the same growth behaviors in direct
numerical simulations.
