On a solid substrate, superfluid He-4 may show stratified liquid films
(above a solid 4He layer). Assuming the existence of these films, we
analyse the dynamics of wetting for a two-layer droplet which tends to
spread and transform into a one-layer droplet (at low temperature: no
vapour phase is present). The permeation process between layers, whic
h exists in classical liquids, is replaced here by a Josephson couplin
g. Spreading should occur via Josephson vortices that drift towards th
e centre of the droplet.