Thermocapillary instability of a surfactant-laden shear-imposed film flow

We examine the linear thermocapillary instability of a two-dimensional surfactant-laden gravity-driven shear-imposed film flowing over a uniformly heated inclined wall. We observe that the surfactant Marangoni number stabilizes, while the thermal Marangoni number destabilizes H-mode, S-mode, P-mode, and shear mode. These opposing impacts establish an analytical relationship between surfactant and thermal Marangoni numbers, for which the critical Reynolds numbers for the H-mode instability of the non-isothermal and isothermal film flows coincide. When comparing results with and without inertia, we predict that inertia stabilizes surfactant mode.