  We present a detailed study of the linear stability of plane
Couette-Poiseuille flow in the presence of a cross-flow. The base flow is
characterised by the cross flow Reynolds number, $R_{inj}$ and the
dimensionless wall velocity, $k$. Squire's transformation may be applied to the
linear stability equations and we therefore consider 2D (spanwise-independent)
perturbations. Corresponding to each dimensionless wall velocity, $k\in[0,1]$,
two ranges of $R_{inj}$ exist where unconditional stability is observed. In the
lower range of $R_{inj}$, for modest $k$ we have a stabilisation of long
wavelengths leading to a cut-off $R_{inj}$. This lower cut-off results from
skewing of the velocity profile away from a Poiseuille profile, shifting of the
critical layers and the gradual decrease of energy production. Cross-flow
stabilisation and Couette stabilisation appear to act via very similar
mechanisms in this range, leading to the potential for robust compensatory
design of flow stabilisation using either mechanism. As $R_{inj}$ is increased,
we see first destabilisation and then stabilisation at very large $R_{inj}$.
The instability is again a long wavelength mechanism. Analysis of the
eigenspectrum suggests the cause of instability is due to resonant interactions
of Tollmien-Schlichting waves. A linear energy analysis reveals that in this
range the Reynolds stress becomes amplified, the critical layer is irrelevant
and viscous dissipation is completely dominated by the energy
production/negation, which approximately balances at criticality. The
stabilisation at very large $R_{inj}$ appears to be due to decay in energy
production, which diminishes like $R_{inj}^{-1}$. Our study is limited to two
dimensional, spanwise independent perturbations.
