Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography

Abstract

Article:Gaskell, P.H., Jimack, P.K., Sellier, M. et A range of two-and three-dimensional problems is explored featuring the gravitydriven flow of a continuous thin liquid film over a non-porous inclined flat surface containing well-defined topography. These are analysed principally within the framework of the lubrication approximation, where accurate numerical solution of the governing nonlinear equations is achieved using an efficient multigrid solver.Results for flow over one-dimensional steep-sided topographies are shown to be in very good agreement with previously reported data. The accuracy of the lubrication approximation in the context of such topographies is assessed and quantified by comparison with finite element solutions of the full Navier-Stokes equations, and results support the consensus that lubrication theory provides an accurate description of these flows even when its inherent assumptions are not strictly satisfied. The NavierStokes solutions also illustrate the effect of inertia on the capillary ridge/trough and the two-dimensional flow structures caused by steep topography.Solutions obtained for flow over localized topography are shown to be in excellent agreement with the recent experimental results of Decré & Baret (2003) for the motion of thin water films over finite trenches. The spread of the 'bow wave', as measured by the positions of spanwise local extrema in free-surface height, is shown to be well-represented both upstream and downstream of the topography by an inverse hyperbolic cosine function.An explanation, in terms of local flow rate, is given for the presence of the 'downstream surge' following square trenches, and its evolution as trench aspect ratio is increased is discussed. Unlike the upstream capillary ridge, this feature cannot be completely suppressed by increasing the normal component of gravity. The linearity of free-surface response to topographies is explored by superposition of the free surfaces corresponding to two 'equal-but-opposite' topographies. Results confirm the findings of Decré & Baret (2003) that, under the conditions considered, the responses behave in a near-linear fashion.

Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography

“…While the generality of these results may be restricted somewhat by the limitations of the lubrication analysis, it has previously been shown that the results of lubrication theory are in good agreement with boundary integral computations of the unsimplified Stokes equations for the flow of an isothermal film over topography for small capillary numbers ͑CaϽ 0.01͒, even for very steep topographical features. 42 Finite element solutions 43 of the full Navier-Stokes equations further confirm the accuracy of the lubrication solutions in this regime even though some of the restrictions inherent to the reduction in dimensionality of lubrication theory are not entirely satisfied. The results presented here are therefore expected to be valid for such small capillary numbers that correspond to thin films.…”

“…Lubrication theory was found to be accurate within this regime even for very steep features, as the slope of the free surface, and not that of the topographical feature, is the relevant constraint. 42 Comparisons to finite element solutions 43 of the full Navier-Stokes equations and to experiments 44 further confirm that lubrication theory provides surprisingly accurate results even when it is applied to situations in which its inherent restrictions are seemingly violated.…”

“…III. 39 It has been shown, however, through comparisons to Stokes flow 42 and finite element 43 solutions and experiment 43,44 that the evolution equation derived using lubrication theory ͓Eq. ͑2͔͒ is a valid approximation for small Ca ͑approximately smaller than 10 −2 ͒ even for steep features.…”

Adaptive Multigrid Solutions of Thin Film Flows over Topography

“…Topographies are defined via arctangent functions which enable the steepness of their sides to be controlled easily and the boundary conditions are that the flow is fully developed upstream and downstream. Further details are given in [4].…”

“…Recently, however, a Multigrid approach with adaptive time-stepping has been developed as a more efficient alternative to ADI schemes, having already been applied successfully to droplet spreading flows [3] and continuous film flow with [4] or without evaporation [5]. This paper highlights the additional benefits of adopting an automatic local grid refinement procedure within the Multigrid algorithm which allows fine grids to be used only where they are needed, i.e.…”