  The convective instability in a plane liquid layer with time-dependent
temperature profile is investigated by means of a general method suitable for
linear stability analysis of an unsteady basic flow. The method is based on a
non-normal approach, and predicts the onset of instability, critical wave
number and time. The method is applied to transient Rayleigh-Benard-Marangoni
convection due to cooling by evaporation. Numerical results as well as
theoretical scalings for the critical parameters as function of the Biot number
are presented for the limiting cases of purely buoyancy-driven and purely
surface-tension-driven convection. Critical parameters from calculations are in
good agreement with those from experiments on drying polymer solutions, where
the surface cooling is induced by solvent evaporation.
