Single-walled carbon nanotubes (SWNTs) are of significant interest in the electronic materials research community due to their excellent electrical properties and many promising applications. However, SWNTs grow as mixture of both metallic and semiconducting tubes and this heterogeneity frustrates their practical use in high performance electronics. Recently developed purification techniques based on nanoscale thermocapillary flow of thin film overcoats enables complete elimination of metallic SWNTs from as-grown arrays. We studied the thermocapillary flow to purify SWNTs analytically and established a simple scaling law for the film thickness profile in terms of the geometry (e.g., film thickness), material (e.g., thermal conductivity and viscosity), and loading (e.g., power density) parameters. The results show that the normalized thickness profile only depends on one nondimensional parameter: the normalized power density. These findings may serve as useful design guidelines for process optimization.