Influence of inertia, gravity, and substrate topography on the two-dimensional transient coating flow of a thin Newtonian fluid film

Abstract

The interplay between inertia, gravity, and substrate topography is examined in this study for the transient two-dimensional flow of a thin Newtonian film. Surface tension effect is assumed to be negligible. The fluid emerges from a channel and is driven by a pressure gradient maintained inside the channel. The substrate is assumed to be stationary and of arbitrary shape. The lubrication equations are solved by expanding the flow field in terms of orthonormal modes in the vertical direction and using the Galerkin projection, combined with a time-stepping implicit scheme, and integration along the flow direction. The leading-order mode is found to be clearly dominant. Gravity and substrate topography can have a significant effect on transient behavior, but this effect varies significantly, depending on the level of fluid inertia. The wave and flow structures are examined for high- and low-inertia fluids. It is found that low-inertia fluids tend to accumulate near the channel exit, exhibiting a standing wave that grows with time. This behavior clearly illustrates the difficulty faced with coating high-viscosity fluids. The topography of the substrate has a drastic effect on the flow. A secondary wave emerges in the presence of a bump or a depression in the substrate. The wave structure is again highly dependent on the level of inertia.