Local perturbation of thin film flow

Abstract

The flow of a thin viscous film over a profiled substrate under the action of an external pressure profile along its free surface, in addition to gravity and capillarity, is considered. Specifically, the linear response of the film flow to localized small pressure perturbations is analyzed. The linear evolution equation of the response analysis is derived from the nonlinear flow equation of the film, which is formulated for the present purpose by reinterpreting the individual terms in the equation of film flow over flat substrates in terms of film thickness or surface curvature. This equation displays an equivalence, within the range of its validity, the lubrication approximation, between a particular, stationary persistent pressure action and the action of a (fixed) substrate topography on the film flow; the equivalence relation is formulated in the paper. The core of the paper is the derivation and analysis of the response of the film to an instantaneous pressure perturbation. For a point source of perturbation this response is the fundamental solution of the linear evolution equation. It is evaluated analytically, exactly or asymptotically, for the two extreme situations of either gravitational or capillary dominance of the flow (i.e. for large and small Bond numbers, respectively), and numerically in the intermediate situations. From the response signal to the instantaneous perturbation the film response to a (harmonically) persistent perturbation is then constructed by superposition. For a steadily acting point source this superposition is a ‘Green’s’ function. Its analytic evaluation for gravitational dominance provides, in addition to those of the fundamental solution, useful support of validation and interpretation for numerical signal evaluations. Such evaluations are given for steady persistent excitation. Finally, the signal analysis is discussed in a process-engineering perspective.