Marangoni instability of a thin liquid film resting on a locally heated horizontal wall

Long-wave Marangoni instabilities can be induced thermally on a thin liquid layer overlying a horizontal solid substrate with either a uniform or a nonuniform base temperature. For a nonuniform base temperature, the film height thickens near the region where temperature gradients are negligible and severely thins upstream; "fingering" patterns are observed in this region. These states are related to the patterns observed in the isothermal case, which are reasonably well understood. The stability of these spatiotemporally evolving states to transverse disturbances is investigated using a transient growth-type analysis. It is found that the band of unstable wave numbers exhibiting growth is strongly dependent on the lateral extent of the heating source. Inspection of surface reconstructions of the film thickness profiles reveals the existence of three-dimensional patterns in the thinning region behind the thickened front