Transient two-dimensional coating flow of a viscoelastic fluid film on a substrate of arbitrary shape

Abstract

The interplay between inertia and viscoelasticity, as well as the influence of gravity and substrate topography are examined in this study for the transient two-dimensional flow of a thin film. The fluid is assumed to emerge from a channel, as it is driven by a pressure gradient, which is maintained inside the channel. The substrate is assumed to be stationary. The lubrication equations are generalized for a viscoelastic fluid obeying the Oldroyd-B constitutive model. These equations are solved by expanding the flow field in Fourier modes in the vertical direction and using the Galerkin projection, combined with a time-stepping implicit scheme, and integration along the flow direction. It is found that the viscosity ratio, fluid elasticity, 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 and/or highly elastic 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 viscoelastic high-viscosity fluids. In the presence of gravity, steady-state conditions are observed to be difficult to reach, even near the channel exit. 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.