  Accurate and computationally accessible models of liquid film flows allow for
optimizing coating processes such as hot-dip galvanization and vertical
slot-die coating. This paper extends the classic three-dimensional integral
boundary layer (IBL) model for falling liquid films (FF) to account for a
moving substrate (MS). We analyze the stability of the liquid films on
vertically moving substrates in a linear and a nonlinear setting. In the linear
analysis, we derive the dispersion relation and the temporal growth rates of an
infinitesimal disturbance using normal modes and linearized governing
equations. In the nonlinear analysis, we consider disturbances of finite size
and numerically compute their evolution using the set of nonlinear equations in
which surface tension has been removed. We present the region of (linear)
stability of both FF and MS configurations, and we place the operating
conditions of an industrial galvanizing line in these maps. A wide range of
flow conditions was analyzed and shown to be stable according to linear and
nonlinear stability analyses. Moreover, the nonlinear analysis, carried out in
the absence of surface tension, reveals a nonlinear stabilizing mechanism for
the interface dynamics of a liquid film dragged by an upward-moving substrate.
