Transient Thin-Film Flow on a Moving Boundary of Arbitrary Topography

The transient two-dimensional flow of a thin Newtonian fluid film over a moving substrate of arbitrary shape is examined in this theoretical study. The interplay among inertia, initial conditions, substrate speed, and shape is examined for a fluid emerging from a channel, wherein Couette–Poiseuille conditions are assumed to prevail. The flow is dictated by the thin-film equations of the “boundary layer” type, which are solved by expanding the flow field in terms of orthonormal modes depthwise and using the Galerkin projection method. Both transient and steady-state flows are investigated. Substrate movement is found to have a significant effect on the flow behavior. Initial conditions, decreasing with distance downstream, give rise to the formation of a wave that propagates with time and results in a shocklike structure (formation of a gradient catastrophe) in the flow. In this study, the substrate movement is found to delay shock formation. It is also found that there exists a critical substrate velocity at which the shock is permanently obliterated. Two substrate geometries are considered. For a continuous sinusoidal substrate, the disturbances induced by its movement prohibit the steady-state conditions from being achieved. However, for the case of a flat substrate with a bump, a steady state exists.