Contact line motion in axial thermocapillary outward flow

We study the contact line dynamics of a viscous droplet deposited at the centre of a substrate subject to an axial thermal gradient. The temperature of the substrate decreases with distance from the centre, so the Marangoni stress induced at the liquid-air interface displaces the liquid radially outward. The flow experiences two stages. In the first stage, the droplet evolves towards an axially symmetric ring whose radius increases with time as. Using the lubrication approximation, we perform numerical simulations that confirm this law for the motion of the front and show that the maximum thickness of the profile decreases as. We explain the evolution law of the contact line by balancing Marangoni and viscous stresses. In the second stage, the contact line becomes unstable and develops smooth corrugations whose amplitude increases with time and that eventually become long fingers. The temporal evolution of the Fourier spectra of the contour shows a shift of the most unstable mode from smaller to larger azimuthal wavenumbers.