Keywords: multigrid, adaptive time-stepping, parallelisation, thin film flow, lubrication approximation.

The deposition and flow of continuous thin liquid films over man-made or naturally occurring functional surfaces containing regions of micro-scale topography (which may be fully submerged or extend through the surface of the film itself) performs an important role in numerous engineering and biologically related fields. For example, in the context of engineering processes thin film flows play a key part in photo-lithography [1] and precision coating processes [2], while in biological systems they occur in areas as diverse as the liquid linings of lungs [3] and plant disease control [4].

Current analytic and experimental methods are incapable, and likely to remain so for the foreseeable future, of either meeting the considerable challenges posed in unravelling the underlying physics of so wide a range of real problems or of providing the necessary insight for improving man-made functional surfaces and the design and creation of novel ones. Equally, it would be naive to pretend that there currently exists an off-the-shelf computational answer involving a combined multi-scale modelling approach comprised of a strategic mix of molecular dynamics, meso-scale and continuum methods. The reality is that at present, the modelling of three-dimensional thin film flows over surfaces containing complex topography is still at an early stage of development, for which exploitation of the long wave lubrication approximation is the focus [5]. Its simplifying feature is that it enables the reduction of the governing Navier-Stokes equations to a coupled system of two-dimensional partial differential equations for film thickness and pressure.

The research reported describes the development and application of an efficient and scalable parallel multigrid solver, utilising a lubrication approximation, for the accurate numerical simulation of thin film flows over engineered and natural occurring surfaces containing complex small-scale topographical features, implemented on the following HPC architectures: HECToR, HPCx and BlueGene/P. Examples considered include: (i) flow past multiple sparsely distributed occlusions, in which the height of the solid topography far exceeds that of the undisturbed asymptotic film thickness; (ii) flow over a complex man-made multiple-connected surface pattern, where free-surface planarisation is important in processes such as photo-lithography with the optimum orientation of the flow direction explored in order to minimised free-surface disturbances; (iii) the analysis of flow over a natural complex topography in the form of a skin sample, which is important in the context of the efficacy of protective coatings.

1

W.K. Ho, A. Tay, L.L. Lee, C.D. Schaper, "On control of resist film uniformity in the microlithography process", Control. Eng. Prac., 12(7), 881-892, 2004. doi:10.1016/j.conengprac.2003.12.001

2

A. Clarke, "Coating on a rough surface", A.I.Ch.E. Journal, 48, 2149-2156, 2002. doi:10.1002/aic.690481006

3

D.P. Gaver, J.B. Grotberg, "The Dynamics of a Localized Surfactant on a Thin-Film", J. Fluid Mech., 213, 127-148, 1990. doi:10.1017/S0022112090002257

4

D.R. Walters, "Disguising the leaf surface: the use of leaf coatings for plant disease control", Euro. J. Plant Pathology, 114, 255-260, 2006. doi:10.1007/s10658-005-5463-7

5

Y.C. Lee, H.M. Thompson, P.H. Gaskell, "An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features", Comp. & Fluids, 36, 838-855, 2007. doi:10.1016/j.compfluid.2006.08.006

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