Thermocapillary stability of a Cattaneo–Christov film with deformable free surface coating a thick substrate

We investigate the linear thermocapillary stability of a Newtonian liquid film governed by the Cattaneo–Christov heat-flux model, coating a thick wall of finite thermal conductivity and bounded by a deformable free surface. Finite thermal relaxation effects, quantified by the Cattaneo number C f , are found to play a central role in mode selection at onset: increasing C f stabilizes stationary disturbances, while at the same time destabilizing the oscillatory modes. Surface deformability, measured by the crispation number Cr , predominantly favors stationary longwave instabilities and has a comparatively weak effect on shortwave and oscillatory branches. The fluid-to-wall thermal conductivity ratio χ , together with the wall-thickness-to-fluid-depth ratio d , controls the transition between longwave and shortwave dominated stationary onset, providing a continuous link between the classical limits of a perfectly conducting wall and an adiabatic wall. Notably, very large but finite values of χ are shown to qualitatively alter the dominant instability mechanism relative to the commonly assumed infinite χ approximation. For sufficiently large χ and C f , codimension-two points emerge at which stationary and oscillatory modes compete at threshold. In the oscillatory regime, increasing C f results in larger convective cells and lower oscillation frequencies. Taken together, these results reveal a rich interplay between stationary longwave, stationary shortwave and oscillatory mechanisms, highlighting the essential role of finite wall conductivity and non-Fourier heat transport in thermocapillary instability.