Published paperAbstract A range of problems is investigated, involving the gravity-driven inertial flow of a thin viscous liquid film over a planar surface containing topographical features, modelled via a depth-averaged form of the governing unsteady Navier-Stokes equations. The discrete analogue of the resulting coupled equation set, employing a staggered mesh arrangement for the dependent variables, is solved accurately using an efficient Full Approximation Storage (FAS) algorithm and Full Multigrid (FMG) technique together with adaptive timestepping and proper treatment of the nonlinear convective terms. A unique, comprehensive set of results is presented for both one-and two-dimensional topographical features, and errors quantified via detailed comparisons drawn with complementary experimental data and predictions from finite element analyses where they exist. It is found in the case of one-dimensional (spanwise) topography that for small Reynolds number and shallow/short features the depth-averaged form produces results that are in close agreement with corresponding finite element solutions of the full free-surface problem. For the case of flow over two-dimensional (localised) topography the free-surface disturbance observed is influenced significantly by the presence of inertia. It leads, as in the case of spanwise topography, to an increase in the magnitude and severity of the capillary ridge/trough patterns which form.

. (2015) 'Free-surface lm ow over topography : full three-dimensional nite element solutions. ', Computers and uids., Further information on publisher's website: Liquid lm ow, Finite elements, Topography, Long-wave approximation, Navier-Stokes. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. for the free-surface disturbance experienced, when the underpinning formal restrictions on geometry and capillary number are not exceeded.

. (2012) 'Electried thin lm ow at nite Reynolds number on planar substrates featuring topography.', International journal of multiphase ow., 44 . pp. 48-69. Further information on publisher's website:http://dx.doi.org/10.1016/j.ijmultiphaseow.2012.03.010Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in International Journal of Multiphase Flow. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in International Journal of Multiphase Flow, 44, September 2012, 10.1016/j.ijmultiphaseow.2012 Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractThe flow of a gravity-driven thin liquid film over a substrate containing topography, in the presence of a normal electric field, is investigated. The liquid is assumed to be a perfect conductor and the air above it a perfect dielectric. Of particular interest is the interplay between inertia, for finite values of the Reynolds number, Re, and electric field strength, expressed in terms of the Weber number, We, on the resultant free-surface disturbance away from planarity. The hydrodynamics of the film are modelled via a depth-averaged form of the Navier-Stokes equations which is coupled to a Fourier series separable solution of Laplace's equation for the electric potential: detailed steady-state solutions of the coupled equation set are obtained numerically.The case of two-dimensional flow over different forms of discrete and periodically varying spanwise topography is explored. In the case of the familiar free-surface capillary peaks and depressions that arise for steep topography, and become more pronounced with increasing Re, greater electric field strength affects them differently. In particular, it is found that for topography heights commensurate with the long-wave approximation: (i) the capillary ridge associated with a step-down topography at first increases before decreasing, both monotonically, with increasing We; (ii) the free-surface hump which arises at a step-up topography continues to increase monotonically with increasing We, the increase achieved being smaller the larger the value of Re.A series of results for the more practically relevant problem of three-dimensional film flow over substrate containing a localised squa...

. (2011) 'Inertial two-and three-dimensional thin lm ow over topography.', Chemical engineering and processing : process intensication., 50 (5-6). pp. 537-542. Further information on publisher's website:http://dx.doi.org/10.1016/j.cep.2010.08.008Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Chemical Engineering and Processing: Process Intensication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. A denitive version was subsequently published in Chemical Engineering and Processing: Process Intensication, 50, 5-6, May 2011, 10.1016/j.cep.2010.08.008. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractThe effect of inertia on gravity-driven thin film free-surface flow over substrates containing topography is considered. Flow is modelled using a depth-averaged form of the governing Navier-Stokes equations and the discrete analogue of the coupled equations solved accurately using an efficient full approximation storage (FAS) algorithm and a full multigrid (FMG) technique. The effect of inertia on free-surface disturbances induced by topographic features is illustrated by considering examples of gravity-driven flow over and around peak, trench and occlusion topography.Results are given which demonstrate how increasing Reynolds number can significantly enhance the magnitude of free-surface disturbances induced, a feature which may have important consequences for the wide range of coating processes which aim to maximise free-surface planarity.

A computational investigation is conducted concerning the stability of free-surface gravity-driven liquid film flow over periodic corrugated substrate. The underpinning mathematical formulation constitutes an extension of the weighted residual integral boundarylayer (WIBL) method proposed by C. Ruyer-Quil and P. Manneville [Eur. Phys. J. B 15, 357 (2000)] and S.J.D. D'Alessio et al. [Phys. Fluids 21, 062105 (2009)] to include thirdand fourth-order terms in the long-wavelength expansion. Steady-state solutions for the free-surface and corresponding curves of neutral disturbances are obtained using Floquet theory and validated against corresponding experimental data and full Navier-Stokes (N-S) solutions. Sinusoidal and smoothed rectangular corrugations with variable steepness are considered. It is shown that the model is capable of predicting characteristic patterns of stability, including shortwave nose and isles of stability/instability as reported experimentally for viscous film flow over inclined topography, providing an attractive trade-off between the accuracy of a full N-S computation and the efficiency of an integral method. The range of parameter values for which the WIBL model remains valid is established, in particular it is shown that its accuracy decreases with Reynolds number and corrugation amplitude, but increases with the steepness parameter and ratio of wavelength to capillary length.