Three-dimensional instability of a liquid layer flowing down a heated vertical cylinder

The three-dimensional thermocapillary instability of a fluid film coating the outside or the inside of a cylinder is investigated in the absence and in the presence of gravity. In the absence of gravity (pure thermocapillarity), it is found that it is possible to excite high azimuthal modes as the most unstable. In the presence of gravity the thermocapillary instability of a thin fluid film flowing down a vertical cylinder is investigated. It is shown that thermocapillarity also promotes the instability of high azimuthal modes, even higher than those found in the pure thermocapillary case. This is in contrast with the isothermal case where only the zeroth mode (the longitudinal one) is the most unstable and also with previous work on flow down a rotating cylinder where, in some conditions, rotation may promote the first azimuthal mode as the most unstable. Curves of criticality and those corresponding to the maximum growth rate are given for different values of the parameters.