In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an
infinitely wide liquid film draining under gravity down an inclined plane are obtained. The
first solution, which predicts that the dry patch has a parabolic shape and that the transverse
profile of the free surface always has a monotonically increasing shape, is appropriate for
weak surface-tension effects and far from the apex of the dry patch. The second solution,
which predicts that the dry patch has a quartic shape and that the transverse profile of the
free surface has a capillary ridge near the contact line and decays in an oscillatory manner far
from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly
vertical) and near (but not too close) to the apex of the dry patch. With the average volume
flux per unit width (or equivalently with the uniform height of the layer far from the dry
patch) prescribed, both solutions contain a free parameter. For each value of this parameter
there is a unique solution in the first case and either no solution or a one-parameter family of
solutions in the second case. The solutions capture some of the qualitative features observed
in experiments.
INFINITE INTERVAL PROBLEMS ARISING IN THE MODEL OF A SLENDER DRY PATCH IN A LIQUID FILM DRAINING UNDER GRAVITY DOWN AN INCLINED PLANE
