  Here we report some interesting new features of the spatio-temporal
instability of the incompressible plane Couette-Poiseuille flow (CPF). First of
all, this flow represents the first instance of a "non-inflectional" absolute
instability, within constant-viscosity formulation, which is triggered when one
of the plates moves opposite to the bulk motion. More strikingly, with further
increase in the negative plate motion, the absolute instability ($\textrm{AI}$)
transitions to an "upstream" convective instability ($\textrm{CI}^-$), wherein
an unstable wave packet moves opposite to the direction of the bulk flow. Thus,
the CPF exhibits a unique $\textrm{CI}^+ \to \textrm{AI} \to \textrm{CI}^-$
transition, for a given Reynolds number ($Re$), where $\textrm{CI}^+$ denotes
the commonly-observed case of a "downstream" convective instability. This type
of transition has not been reported for other known examples of absolutely
unstable flows. We compute the leading and trailing edge velocities for an
amplifying wave packet and find that, for the plane Poiseuille flow, both these
velocities approach zero as $Re \to \infty$. As a result, at high $Re$, even
the slightest of negative plate motions is sufficient to trigger $\textrm{AI}$
and subsequently $\textrm{CI}^-$, as observed for the CPF. The wave-packet
dispersion first increases with $Re$, followed by a decrease, which points to a
peculiar "dual" role of viscosity in sustaining $\textrm{AI}$ in the CPF,
namely, viscosity promotes sustenance of $\textrm{AI}$ at moderate Reynolds
numbers but suppresses it at low and high Reynolds numbers. These results can
be well understood within the Ginzburg-Landau framework, and therefore can be
expected to have a wider applicability.
