  We examine the effects of confinement on the dynamics of premelted films
driven by thermomolecular pressure gradients. Our approach is to modify a
well-studied setting in which the thermomolecular pressure gradient is driven
by a temperature gradient parallel to an interfacially premelted elastic wall.
The modification treats the increase in viscosity associated with the thinning
of films, studied in a wide variety of materials, using a power law and we
examine the consequent evolution of the confining elastic wall. We treat (1) a
range of interactions that are known to underlie interfacial premelting and (2)
a constant temperature gradient wherein the thermomolecular pressure gradient
is a constant. The difference between the cases with and without the proximity
effect arises in the volume flux of premelted liquid. The proximity effect
increases the viscosity as the film thickness decreases thereby requiring the
thermomolecular pressure driven flux to be accommodated at higher temperatures
where the premelted film thickness is the largest. Implications for experiment
and observations of frost heave are discussed.
