Thin liquid film resulting from a distributed source on a vertical wall

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, further 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.