  The stability of an evaporating thin liquid film on a solid substrate is
investigated within lubrication theory. The heat flux due to evaporation
induces thermal gradients; the generated Marangoni stresses are accounted for.
Assuming the gas phase at rest, the dynamics of the vapour reduces to
diffusion. The boundary condition at the interface couples transfer from the
liquid to its vapour and diffusion flux. A non-local lubrication equation is
obtained; this non-local nature comes from the Laplace equation associated with
quasi-static diffusion. The linear stability of a flat film is studied in this
general framework. The subsequent analysis is restricted to moderately thick
films for which it is shown that evaporation is diffusion limited and that the
gas phase is saturated in vapour in the vicinity of the interface. The
stability depends only on two control parameters, the capillary and Marangoni
numbers. The Marangoni effect is destabilising whereas capillarity and
evaporation are stabilising processes. The results of the linear stability
analysis are compared with the experiments of Poulard et al (2003) performed in
a different geometry. In order to study the resulting patterns, the amplitude
equation is obtained through a systematic multiple-scale expansion. The
evaporation rate is needed and is computed perturbatively by solving the
Laplace problem for the diffusion of vapour. The bifurcation from the flat
state is found to be a supercritical transition. Moreover, it appears that the
non-local nature of the diffusion problem unusually affects the amplitude
equation.
