Thermocapillary flows on heated substrates with sinusoidal topography

Two-dimensional steady thermocapillary flows in a liquid layer over a substrate, which has a uniform temperature and sinusoidal topography, are investigated by asymptotic theory. Here, the buoyancy effect is negligible and the interface is not significantly disturbed under low Marangoni number and low capillary number. A temperature gradient along the gas/liquid interface causes recirculating flows. For a small aspect ratio, which yields a sinusoidal topography with a long wavelength relative to the mean depth of the liquid layer, the second-order solutions are obtained analytically. The basic solutions show vertical diffusion of heat and vorticity from the substrate and interface, respectively. In the second corrections, the horizontal diffusion of heat weakens the overall flow and the convection of heat intensifies it.