Surface deformations in dynamic thermocapillary convection under partial slip.

A prescribed, horizontal temperature gradient is imposed upon a horizontal liquid layer bounded from above by a deformable, liquid-gas interface and bounded from below by a partial-slip, rigid surface. A steady shear flow driven by thermocapillary motion emerges. This dynamic liquid layer is susceptible to the onset of oblique three-dimensional hydrothermal waves, purely two-dimensional hydrothermal waves, longitudinal traveling waves, and longitudinal rolls depending on the capillary number. A low capillary number analysis finds that surface deformations are destabilizing for all modes of instability. There is a preference for two-dimensional hydrothermal waves when there are surface deformations. Though longitudinal traveling waves are never selected as the preferred mode of instability, these waves offer a convenient way to understand the behavior of oblique hydrothermal waves, which are near-longitudinal. This is especially the case for low capillary numbers, but oblique hydrothermal waves instead tend to align themselves with the direction of flow as the capillary number increases. Surface deformations affect longitudinal waves most significantly out of all the modes of instability, especially for low Prandtl numbers. The typical length scales shorten and the critical Marangoni numbers increase with the capillary number for all types of modes. Notably, the system selects long waves near a critical Prandtl number when the interface is nondeformable and when the layer is subject to partial slip, but this is no longer the case when the upper surface is deformable.