Absolute and "upstream" convective instabilities in plane Couette-Poiseuille flow

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.