  The stability of a thermocapillary flow in an extended cylindrical geometry
is analyzed. This flow occurs in a thin liquid layer with a disk shape when a
radial temperature gradient is applied along the horizontal free surface.
Besides the aspect ratio, a second parameter related to the local curvature is
introduced to describe completely the geometrical effects. We recover classical
hydrothermal waves as predicted by Smith and Davis but the properties of these
waves are shown to evolve with the curvature parameter, thus leading to a non
uniform pattern over the cell. Moreover, it is shown that the problem is not
invariant with respect to the exchange of the hot and cold sides.
