  In this work we consider a new class of oscillatory instabilities that
pertain to thermocapillary destabilization of a liquid film heated by a solid
substrate. We assume the substrate thickness and substrate-film thermal
conductivity ratio are large so that the effect of substrate thermal diffusion
is retained at leading order in the long-wave approximation. As a result,
system dynamics are described by a nonlinear partial differential equation for
the film thickness that is nonlocally coupled to the full substrate heat
equation. Perturbing about a steady quiescent state, we find that its stability
is described by a non-self adjoint eigenvalue problem. We show that, under
appropriate model parameters, the linearized eigenvalue problem admits complex
eigenvalues that physically correspond to oscillatory (in time) instabilities
of the thin film height. As the principal results of our work, we provide a
complete picture of the susceptibility to oscillatory instabilities for
different model parameters. Using this description, we conclude that
oscillatory instabilities are more relevant experimentally for films heated by
insulating substrates. Furthermore, we show that oscillatory instability where
the fastest-growing (most unstable) wavenumber is complex, arises only for
systems with sufficiently large substrate thicknesses.
