  A comprehensive study of the effect of wall heating or cooling on the linear,
transient and secondary growth of instability in channel flow is conducted. The
effect of viscosity stratification, heat diffusivity and of buoyancy are
estimated separately, with some unexpected results. From linear stability
results, it has been accepted that heat diffusivity does not affect stability.
However, we show that realistic Prandtl numbers cause a transient growth of
disturbances that is an order of magnitude higher than at zero Prandtl number.
Buoyancy, even at fairly low levels, gives rise to high levels of subcritical
energy growth. Unusually for transient growth, both of these are
spanwise-independent and not in the form of streamwise vortices. At moderate
Grashof numbers, exponential growth dominates, with distinct Rayleigh-Benard
and Poiseuille modes for Grashof numbers upto $\sim 25000$, which merge
thereafter. Wall heating has a converse effect on the secondary instability
compared to the primary, destabilising significantly when viscosity decreases
towards the wall. It is hoped that the work will motivate experimental and
numerical efforts to understand the role of wall heating in the control of
channel and pipe flows.
