  We examine the dynamics of a thin film formed by a distributed liquid source
on a vertical solid wall. The model is derived using the lubrication
approximation and includes the effects of gravity, upward airflow and surface
tension. When surface tension is neglected, a critical source strength is found
below which the film flows entirely upward due to the airflow, and above which
some of the flow is carried downward by gravity. In both cases, a steady state
is established over the region where the finite source is located. Shock waves
that propagate in both directions away from the source region are analyzed.
Numerical simulations are included to validate the analytical results. For
models including surface tension, numerical simulations are carried out. The
presence of surface tension, even when small, causes a dramatic change in the
film profiles and the speed and structure of the shock waves. These are studied
in more detail by examining the traveling wave solutions away from the source
region.
