M. D. Korzec and P. L. Evans. From bell shapes to pyramids: a reduced continuum model for self-assembled quantum dot growth. Phys. D, 239(8):465-474, 15 April 2010. [ bib | http | .pdf ]
A continuum model for the growth of self-assembled quantum dots that incorporates surface diffusion, an elastically deformable substrate, wetting interactions and anisotropic surface energy is presented. Using a small slope approximation a thin-film equation for the surface profile that describes faceted growth is derived. A linear stability analysis shows that anisotropy acts to destabilize the surface. It lowers the critical height of flat films and there exists an anisotropy strength above which all thicknesses are unstable. A numerical algorithm based on spectral differentiation is presented and simulations are carried out. These clearly show faceting of the growing islands and a power law coarsening behavior.
P. L. Evans, M. D. Korzec, A. Münch, and B. Wagner. Thin film models for quantum dot structures in solid films. In EUROMECH Colloqium 497 - Recent Developments and New Directions in Thin-Film Flow, Edinburgh, U.K., July 2009. ICMS. [ bib ]
M. D. Korzec, P. L. Evans, A. Münch, and B. Wagner. Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations. SIAM J. Appl. Math., 69(2):348-374, November 2008. Preprint arXiv:0712.2482v1 available at http://arxiv.org/abs/0712.2482. [ bib | http ]
P. L. Evans, M. D. Korzec, A. Münch, and B. Wagner. Higher-order equations for quantum dot structures in thin solid films. In XXII International Congress of Theoretical and Applied Mechanics (ICTAM 2008), Adelaide, Australia, August 2008. [ bib ]
P. L. Evans, A. Münch, and J. R. King. Asymptotic structure of a dewetting, slipping, thin liquid film rim. In EUROMECH 490 Workshop Dynamics and Stability of Thin Liquid Films and Slender Jets, London, U.K., September 2007. Imperial College London. [ bib ]
P. L. Evans and A. Münch. Meniscus solutions and shock dynamics for Marangoni-driven flows. In 7th European Coating Symposium (ECS 2007), Paris, France, September 2007. Will be published in a special issue of the European Physical Journal (EPJ Special Topics). [ bib | .html ]
P. L. Evans, A. Münch, and J. R. King. Asymptotic structure of a dewetting thin liquid film. In 13th International Coating Science and Technology Symposium (ISCST 2006), Denver, Colorado, U.S.A., September 2006. [ bib ]
P. L. Evans and A. Münch. Asymptotic structure of a dewetting, slipping, thin liquid film rim. In Proceedings of the GAMM 2006 Annual Meeting, Berlin, Germany, March 2006. 77th Annual Meeting of the Gesellschaft für Angewandte Mathematik und Mechanik (GAMM 2006). [ bib ]
P. L. Evans, J. R. King, and A. Münch. Intermediate-asymptotic structure of a dewetting rim with strong slip. AMRX, 2006(Article ID 25262), 2006. [ bib | http | .pdf ]
When a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behavior of a film, when there is strong slip at a liquid-substrate interface. The film can be modeled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.
P. L. Evans and Andreas Münch. Interaction of advancing fronts and meniscus profiles formed by surface-tension-gradient-driven liquid films. SIAM J. Appl. Math., 66(5):1610-1631, 2006. [ bib | http | .pdf ]
On a tilted heated substrate, surface tension gradients can draw liquid up out of a reservoir. The resulting film thickness profile is controlled by two parameters which depend on the tilt of the substrate, the imposed temperature gradient, and the thickness of a postulated thin precursor layer. The evolution of this film in time is studied using a lubrication model. A number of distinct behaviours are possible as the substrate tilt angle and other parameters are varied. Recent results for the multiple stationary profiles possible near the meniscus are used, and the interaction of these profiles with the advancing front is examined. It is shown how to systematically determine the evolution of the entire film profile from the meniscus to the apparent contact line. This allows a categorisation of the range of behaviours for a transversely-uniform profile, in a two-dimensional parameter space. In addition to capillary fronts, and double shock structures, new combinations that arise for certain ranges of large substrate tilt and precursor thickness are described. These include profiles involving rarefaction fans, connecting to either an undercompressive or a classical wave at the advancing front.
Keywords: lubrication theory, Marangoni shear stress, capillarity, traveling waves
Andreas Münch and P. L. Evans. Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate. Phys. D, 209(1-4):164-177, 15 September 2005. [ bib | http | .pdf ]
We revisit the situation of a thin liquid film driven up an inclined substrate by a thermally induced Marangoni shear stress against the opposing parallel component of gravity. In contrast to previous studies, we focus here on the meniscus region, in a case where the substrate is nearly horizontal. Our numerical simulations show that the time-dependent lubrication model for the film profile can reach a steady state in the meniscus region that is unlike the monotonic solutions found in [Münch, SIAM J. Appl. Math., 62(6):2045-2063, 2002]. A systematic investigation of the steady states of the lubrication model is carried out by studying the phase space of the corresponding third-order ODE system. We find a rich structure of the phase space including multiple non-monotonic solutions with the same far-field film thickness.
P. L. Evans and A. Münch. Meniscus solutions and shock dynamics for Marangoni-driven flows. In H. Benkreira, editor, 6th European Coating Symposium (ECS 2005), Bradford, United Kingdom, September 2005. [ bib ]
P. L. Evans, L. W. Schwartz, and R. V. Roy. Three-dimensional solutions for coating flow on a rotating horizontal cylinder: theory and experiment. Phys. Fluids, 17(7):072102, July 2005. [ bib | http | .pdf ]
We present three-dimensional numerical simulations of the flow of a thin liquid coating on a rotating horizontal right circular cylinder. The liquid motion is described using a lubrication model. The model evolution equation is discretized and solved numerically using an alternating-direction implicit algorithm. The cylinder rotates about its axis, carrying liquid around its circumference, resulting in the formation of a relatively thick coating where the cylinder surface moves upward. For coatings which are initially nearly uniform along the cylinder axis, this results in a ridge of liquid aligned with the cylinder axis. Over time, this ridge may break up into one of several possible configurations, including drops near the underside, and rings enveloping the cylinder. Simulations show that on larger cylinders, under certain circumstances this ridge may develop undulations which grow to form long fingers. These fingers drain down the cylinder. The simulation results are compared with a simple laboratory experiment, which exhibits similar fingering.
P. L. Evans and Andreas Münch. Thin liquid coatings on a tilted heated plate: Lubrication modelling and analysis. In Proceedings of the GAMM 2005, volume 5, pages 617-618, Luxembourg, 29 March-1 April 2005. GAMM, Wiley-VCH. [ bib | http ]
T. Witelski, D. Schwendeman, and P. L. Evans. Analysis of pressurized porous air bearings. Study Group Report, US Workshop on Mathematical Problems in Industry, 20th MPI, 2005. [ bib | http ]
P. L. Evans, L. W. Schwartz, and R. V. Roy. Steady and unsteady solutions for coating flow on a rotating horizontal cylinder: Two-dimensional theoretical and numerical modeling. Phys. Fluids, 16(8):2742-2756, August 2004. [ bib | http | .pdf ]
A model for the evolution of a thin liquid coating on a horizontal cylinder is presented. The cylinder rotates about its axis, carrying liquid around its circumference. For a viscous coating, this leads to formation of a relatively thick coating where the cylinder surface moves upward. The model is based on lubrication theory, as the coating is thin compared to the cylinder radius, and includes the effects of cylinder rotation, gravity, surface tension, and flow along the cylinder axis. A two-dimensional numerical scheme based on finite differences is produced, for investigation of the case when axial flow is neglected. This numerical scheme is validated in appropriate limiting cases. Coating cross sections are obtained over a range of cylinder rotation rates, for realistic parameter values. These show a transition from pendant drops hanging beneath the cylinder to a nearly-uniform coating wrapped around it as rotation rate is increased.
R. J. Valkenburg and P. L. Evans. Lens distortion calibration by straightening lines. In Proceedings of Image and Vision Computing New Zealand 2002, pages 65-69, Auckland, New Zealand, November 2002. IVCNZ2002. [ bib | .html | .pdf ]
A new method is presented for lens distortion calibration of a CCD camera system. The method relies on the principle that straight lines remain straight under perspective projection, in the spirit of the analytic plumb line technique. A two-stage process is used. The first stage models the distorted images of straight lines as smooth parametric curves. In the second stage, these curves are transformed through a lens distortion model to produce new curves. An error measure is defined based on the deviation of these curves from straightness. Optimal values of the lens distortion coefficients and image centre location are found by minimising this error. The method is flexible and allows the use of simple targets, such as taut wires. The method is demonstrated on several images of a carefully prepared wire target, and then in a typical laboratory environment.
P. L. Evans, L. W. Schwartz, and R. V. Roy. A mathematical model for crater defect formation in a drying liquid film coating. In 10th International Coating Science and Technology Symposium (ISCST 2000), Scottsdale, Arizona, U.S.A., September 2000. [ bib ]
P. L. Evans, L. W. Schwartz, and R. V. Roy. A mathematical model for crater defect formation in a drying paint layer. Journal of Colloid and Interface Science, 227(1):191-205, July 2000. [ bib | http | .pdf ]
Certain deep indentations observed in dry coatings are referred to as 'craters.' They are believed to arise from gradients in the coating surface tension. A mathematical model of surface-tension-gradient-driven flow, using the lubrication approximation for thin layers, is developed to study the formation of craters. The paint is modeled as consisting of an evaporating 'solvent' part and a nonvolatile 'resin' part. Surface tension gradients on the coating surface arise due to a nonuniform distribution of surfactant. Axisymmetric numerical simulations using the model are performed to explore two candidate crater production mechanisms: an initial release of concentrated surfactant and a steady surfactant source. The effects of changes in various properties, such as the paint drying rate, the surfactant diffusivity, and the viscosity increase during drying, are examined. The model produces craters with large diameters, pronounced rims, and central peaks, similar to those seen in practice. Drying rate has a large influence on crater diameter and depth, by limiting flow due to surface tension gradients within a given time. Reduction of the paint viscosity increase during drying causes increased flow rates, leading to larger craters. A preexisting layer of surfactant on the paint surface sharply reduces the extent of cratering. Surfactant diffusion also tends to reduce the severity of cratering by alleviating surface tension gradients. In some cases, a simplified form of the drying model may be used to quickly approximate the results of the full model. The model provides useful insights into the craters seen in industrial coating applications.
P. L. Evans. Mathematical and Numerical Investigations of Coating Flows. PhD thesis, Univ. of Delaware, Newark, Del., 2000. [ bib | .html ]
P. L. Evans, L. W. Schwartz, and R. V. Roy. Numerical simulation of coating flow on a horizontal rotating cylinder. In 9th International Coating Science and Technology Symposium (ISCST 1998), Newark, Delaware, U.S.A., May 1998. [ bib ]
G. D. Lewis, B. J. Noye, and P. L. Evans. A comparison of finite difference and Lagrangian-Stochastic methods for oil slick tracking. In R. L. May and A. K. Easton, editors, Computational Techniques and Applications: CTAC-95, pages 471-477, Singapore, 1996. World Scientific. July 1995. [ bib ]
P. L. Evans and B. J. Noye. A model for fast oil spill prediction in shallow gulfs. In O. Bellwood, H. Choat, and N. Saxena, editors, PACON '94 Proceedings: Recent Advances in Marine Science and Technology, pages 119-130, Townsville, Australia, 1995. James Cook University of North Queensland, PACON International. [ bib | http ]
P. L. Evans. A model for fast oil spill trajectory prediction, applied to Gulf St. Vincent, South Australia. Honours thesis, The University of Adelaide, South Australia. Department of Applied Mathematics., 1993. [ bib ]
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