  In this article, we present a model of heat transfer occurring through a
li\-quid film flowing down a vertical wall. This new model is formally derived
using the method of asymptotic expansions by introducing appropriately chosen
dimensionless variables. In our study the small parameter, known as the film
parameter, is chosen as the ratio of the flow depth to the characteristic
wavelength. A new Nusselt solution should be explained, taking into account the
hydrodynamic free surface variations and the contributions of the higher order
terms coming from temperature variation effects. Comparisons are made with
numerical solutions of the full Fourier equations in a steady state frame. The
flow and heat transfer are coupled through Marangoni and temperature dependent
viscosity effects. Even if these effects have been considered separately
before, here a fully coupled model is proposed. Another novelty consists in the
asymptotic approach in contrast to the weighted residual approach which have
been formerly applied to these problems.
