Linear stability of stratified, rotating, viscous plane Couette–Poiseuille flow

The linear stability of plane Couette–Poiseuille flow (CPF) is studied with the physical effects of stratification, rotation and viscosity all included for the first time together. With no stratification, two instability mechanisms are present due to the shear and rotation which, for the most part, do not interact as they favour different forms of two-dimensionality. However, there are some small parts of parameter space where new three-dimensional instability appears indicating that Rayleigh's criterion is also violated in parameter space beyond where shear instability is expected. No fully localised centrifugal instabilities can be found for CPF, but they are shown to exist if the base flow shear has a maximum in the domain (the base flow needs to be at least cubic in the cross-stream variable rather than just quadratic as in CPF). With stable stratification present, new instabilities are found due to the combined effects of stratification and rotation, but only some appear to be of the resonance-type associated with the strato-rotational instability. The other unstable branches are more accurately interpreted as a stratification-modified centrifugal instability. Three-dimensional versions of this violate Rayleigh's criterion even when this is extended to include stratification. When stratification is stronger than rotation, the resonance-type instabilities are only dominant for cyclonic flows.