From 5ea8f829ab9ba17be7d695fb49e45420560d98a4 Mon Sep 17 00:00:00 2001 From: Zitzeronion Date: Tue, 30 Jan 2024 10:45:14 +0100 Subject: [PATCH] Printed and evaluated hopefully --- bibliography/Thesis_Swalbe.bib | 49 ++++++++------------------------- chapters/Conclusion.tex | 22 +++++++-------- chapters/Introduction.tex | 21 +++++++------- chapters/Method.tex | 16 +++++------ chapters/Theory.tex | 26 ++++++++--------- chapters/chapter_1_heading.tex | 11 ++++++++ main.sty | 1 + main.tex | 20 ++++++++++---- papers/Joss_paper.pdf | Bin 0 -> 421712 bytes papers/SuppMat.pdf | Bin 0 -> 395979 bytes papers/paper_1.pdf | Bin 0 -> 1838760 bytes papers/paper_2.pdf | Bin 0 -> 1677873 bytes papers/paper_3.pdf | Bin 0 -> 1463148 bytes 13 files changed, 81 insertions(+), 85 deletions(-) create mode 100644 chapters/chapter_1_heading.tex create mode 100644 papers/Joss_paper.pdf create mode 100644 papers/SuppMat.pdf create mode 100644 papers/paper_1.pdf create mode 100644 papers/paper_2.pdf create mode 100644 papers/paper_3.pdf diff --git a/bibliography/Thesis_Swalbe.bib b/bibliography/Thesis_Swalbe.bib index c9ccf0c..916ef74 100644 --- a/bibliography/Thesis_Swalbe.bib +++ b/bibliography/Thesis_Swalbe.bib @@ -59,7 +59,6 @@ @article{bellNumericalMethodsStochastic2007 pages = {016708}, publisher = {{American Physical Society}}, doi = {10.1103/PhysRevE.76.016708}, - urldate = {2023-07-13}, abstract = {The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This paper examines explicit Eulerian discretizations of the full LLNS equations. Several computational fluid dynamics approaches are considered (including MacCormack's two-step Lax-Wendroff scheme and the piecewise parabolic method) and are found to give good results for the variance of momentum fluctuations. However, neither of these schemes accurately reproduces the fluctuations in energy or density. We introduce a conservative centered scheme with a third-order Runge-Kutta temporal integrator that does accurately produce fluctuations in density, energy, and momentum. A variety of numerical tests, including the random walk of a standing shock wave, are considered and results from the stochastic LLNS solver are compared with theory, when available, and with molecular simulations using a direct simulation Monte Carlo algorithm.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\8RUX29JW\\Bell et al. - 2007 - Numerical methods for the stochastic Landau-Lifshi.pdf;C\:\\Users\\zitz\\Zotero\\storage\\FWD28Y4Q\\PhysRevE.76.html} } @@ -136,7 +135,6 @@ @article{doi:10.1146/annurev.fl.26.010194.002525 journal = {Annu. Rev. Fluid Mech.}, volume = {26}, number = {1}, - eprint = {https://doi.org/10.1146/annurev.fl.26.010194.002525}, pages = {529--571}, doi = {10.1146/annurev.fl.26.010194.002525} } @@ -163,7 +161,6 @@ @article{beckerThinfilmEquationRecent2005 number = {9}, pages = {S291}, doi = {10.1088/0953-8984/17/9/002}, - urldate = {2023-07-19}, abstract = {This paper is concerned with mathematical aspects of lubrication equations. In the first part, we discuss recent analytical achievements for various types of thin-film equations. Of interest are issues like (non-)uniqueness, wetting behaviour and contact line motion, in particular optimal propagation rates and waiting time or dead core phenomena. In the second part, we shall present novel numerical results for thin-film flow on heterogeneous substrates based on entropy consistent schemes. Finally, we will be concerned with new algorithmic concepts for the simulation of thin-film flow of shear-thinning liquids.} } @@ -239,7 +236,6 @@ @article{benziLatticeBoltzmannEquation1992a number = {3}, pages = {145--197}, doi = {10.1016/0370-1573(92)90090-M}, - urldate = {2024-01-05}, abstract = {The basic elements of the theory of the lattice Boltzmann equation, a special lattice gas kinetic model for hydrodynamics, are reviewed. Applications are also presented together with some generalizations which allow one to extend the range of applicability of the method to a number of fluid dynamics related problems.}, file = {C:\Users\zitz\Zotero\storage\674S8DJN\037015739290090M.html} } @@ -302,7 +298,6 @@ @article{bergstromHamakerConstantsInorganic1997 volume = {70}, pages = {125--169}, doi = {10.1016/S0001-8686(97)00003-1}, - urldate = {2024-01-08}, abstract = {Calculations of Hamaker constants using Lifshitz theory require the availability of accurate dielectric data, especially in the ultraviolet spectral region, and the use of a convenient and appropriate mathematical representation. In this review, a multiple oscillator model {\textemdash} the so-called Ninham-Parsegian (N-P) representation {\textemdash} has been used and spectral parameters for 31 different inorganic materials (including diamond) have been generated from critically evaluated optical data or collected from the literature. For most materials, a two-oscillator model (one UV and one IR term) was used but more detailed representations were included when available. The spectral parameters presented here can be combined with previous data, mainly focused on hydrocarbon and organic systems, to yield an extensive spectral data base for both solids and liquids enabling Lifshitz calculations of Hamaker constants for many materials combinations. Non-retarded Hamaker constants for symmetric material combinations across vacuum (A1v1) and water (A1w1) have been calculated for the different materials; these calculations were performed using the full Lifshitz theory. Asymmetric combinations, A1v3 and A1w3, against four commonly used materials in atomic force microscopy studies: silica, amorphous silicon nitride, sapphire, and muscovite mica, have also been covered. The use of a new dielectric representation for water resulted in significantly lower values of A1w1 compared to previous calculations. Analytical approximations to the full Lifshitz theory were evaluated and found to give surprisingly accurate results (the Tabor-Winterton approximation) for A1v1 when the IR contribution is of minor importance. An attempt to make the TW approximation more general by establishing some scaling relationship between n0 and {$\omega$}UV was met with little success; only the UV spectral parameters of the covalent oxides, sulphides and nitrides may be fitted to a simple power law relation. The Lifshitz calculations in this study were compared with an alternative method where a more detailed dielectric representation in the visible-ultraviolet spectral range was obtained through Kramers-Kronig (K-K) transformation of reflectivity data over a broad frequency range. Despite the difference in dielectric information, the two methods generally yield non-retarded Hamaker constants which do not differ significantly. This is not true for all materials, e.g. water, where a more detailed representation using either an N-P representation with several oscillators or the K-K representation must be used. It was shown that the omission of the static and low frequency contribution in the latter method may result in a significant underestimation of the value for A1w1 when the dispersive contribution becomes very small.}, keywords = {AFM,Hamaker constant,inorganic materials,interparticle forces,van der Waals interaction}, file = {C:\Users\zitz\Zotero\storage\6VTZS8DN\S0001868697000031.html} @@ -459,7 +454,6 @@ @book{koniglichegesellschaftderwissenschaftenzugottingenCommentationesSocietatis author = {{K\"onigliche Gesellschaft der Wissenschaften zu G\"ottingen}}, year = {1811}, publisher = {{G\"ottingen, Germany}}, - urldate = {2023-07-17}, abstract = {Preface to each volume contains report of the society.; Preface to each volume contains report of the society}, collaborator = {{University of Michigan}}, copyright = {http://creativecommons.org/publicdomain/mark/1.0/}, @@ -525,7 +519,6 @@ @techreport{grafColoradoRiverGrand1997 pages = {4}, institution = {{U.S. Geological Survey}}, doi = {10.3133/fs16897}, - urldate = {2023-07-19}, abstract = {Opening the jet tubes at Glen Canyon Dam on March 26, 1996, released from Lake Powell a controlled flood of water that traveled down the Colorado River in Grand Canyon. How fast did the water move? How long did it take for water to reach a particular point along the river? The answers to these questions are important because the speed of river water affects the amount of physical and chemical changes, such as warming by sunlight, that the water will undergo as it moves downstream. Also, very fine particles and substances dissolved in the water travel along with the water, and the speed of the water tells us how fast these move downstream.}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\RRH62RIM\\Graf - 1997 - The Colorado River in Grand Canyon how fast does .pdf} } @@ -864,7 +857,6 @@ @article{cardiffThirtyYearsFinite2021 number = {5}, pages = {3721--3780}, doi = {10.1007/s11831-020-09523-0}, - urldate = {2023-07-25}, abstract = {Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs. staggered vs.~vertex-centred), solution algorithm (implicit vs. explicit), and stabilisation strategy (Rhie\textendash Chow vs. Jameson\textendash Schmidt\textendash Turkel vs. Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finishes with a look towards future research directions and steps required for finite volume solid mechanics to achieve more widespread acceptance.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\V98DKZDL\\Cardiff and Demirdžić - 2021 - Thirty Years of the Finite Volume Method for Solid.pdf} } @@ -1112,7 +1104,6 @@ @article{hamakerLondonVanWaals1937 number = {10}, pages = {1058--1072}, doi = {10.1016/S0031-8914(37)80203-7}, - urldate = {2023-07-19}, abstract = {Frequently we experience the existance of adhesive forces between small particles. It seems natural to ascribe this adhesion for a large part to London-v.d. Waals forces. To obtain general information concerning their order of magnitude the London-v. d. Waals interaction between two spherical particles is computed as a function of the diameters and the distance separating them. A table is calculated which enables numerical application of the formulae derived. Besides approximations are added, which may be used when the distance between the particles is small. In a separate section it is investigated how the results must be modified, when both particles are immersed in a liquid. Here we are led to the important conclusion that even in that case London-v. d. Waals forces generally cause an attraction.}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\3BMCSG8U\\S0031891437802037.html} } @@ -1160,6 +1151,13 @@ @article{ducheminInviscidCoalescenceDrops2003 file = {C\:\\Users\\Zitzero\\Zotero\\storage\\Y2IDQ3LL\\Duchemin et al. - 2003 - Inviscid coalescence of drops.pdf} } +@article{verweyTheoryStabilityLyophobic, + title = {Theory of the {{Stability}} of {{Lyophobic Colloids}}.}, + author = {Verwey, E. J. W.}, + abstract = {The theory of the interaction of the double layers has been reviewed in relation to the stability of lyophobic colloids. It is concluded that the interaction must be associated with an increase of the free energy, leading to a repulsion between the particles. The repulsive potential calculated from a consideration of the free energy for certain special cases has been combined with the London-van der Waals attractive potential calculated by Hamaker to obtain curves of potential vs. distance. Predictions based on these curves appear to agree well with various experimental data. For example, the influence of electrolyte concentration and of the valencies of the ions on flocculation is satisfactorily explained in terms of the theory, although many complicated phenomena remain to be correlated with it.}, + file = {C\:\\Users\\Zitzero\\Zotero\\storage\\W2XXZWJT\\Verwey - Theory of the Stability of Lyophobic Colloids..pdf} +} + @book{dupratFluidStructureInteractionsLowReynoldsNumber2015, title = {Fluid-{{Structure Interactions}} in {{Low-Reynolds-Number Flows}}}, author = {Duprat, C. and Stone, H. A.}, @@ -1228,7 +1226,7 @@ @article{eggersCoalescenceLiquidDrops1999 volume = {401}, pages = {293--310}, publisher = {{Cambridge University Press}}, - doi = {10.1017/S002211209900662X}, + doi = {10.1017/ S002211209900662X}, abstract = {When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2{$\pi$}rm and width {$\Delta$}[Lt ]rm around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that {$\Delta\propto$}r3m and rm{$\sim$}(t{$\gamma$}/{$\pi\eta$}) ln [t{$\gamma$}({$\eta$}R)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A significantly different structure is found in which the outer-fluid forms a toroidal bubble of radius {$\Delta\propto$}r3/2m at the meniscus and rm{$\sim$}(t{$\gamma$}/4{$\pi\eta$}) ln [t{$\gamma$}/({$\eta$}R)]. This basic difference is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for rm(t)[Lt ]1 involves matching of lengthscales of order r2m, r3/2m, rm, 1 and probably r7/4m.}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\W96MUJUM\\Eggers et al. - 1999 - Coalescence of liquid drops.pdf} } @@ -1540,7 +1538,6 @@ @article{savvaViscousSheetRetraction2009 pages = {211--240}, publisher = {{Cambridge University Press}}, doi = {10.1017/S0022112009005795}, - urldate = {2023-07-24}, abstract = {We present the results of a combined theoretical and numerical investigation of the rim-driven retraction of flat fluid sheets in both planar and circular geometries. Particular attention is given to the influence of the fluid viscosity on the evolution of the sheet and its bounding rim. In both geometries, after a transient that depends on the sheet viscosity and geometry, the film edge eventually attains the Taylor\textendash Culick speed predicted on the basis of inviscid theory. The emergence of this result in the viscous limit is rationalized by consideration of both momentum and energy arguments. We first consider the planar geometry considered by Brenner \& Gueyffier (Phys. Fluids, vol. 11, 1999, p. 737) and deduce new analytical expressions for the speed of the film edge at the onset of rupture and the evolution of the maximum film thickness for viscous films. In order to consider the expansion of a circular hole, we develop an appropriate lubrication model that predicts the form of the early stage dynamics of film rupture. Simulations of a broad range of flow parameters confirm the importance of geometry on the dynamics, verifying the exponential hole growth reported in early experimental studies. We demonstrate the sensitivity of the initial retraction speed on the film profile, and so suggest that the anomalous rate of retraction reported in these experiments may be attributed in part to geometric details of the puncture process.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\NZJIQQUK\\Savva and Bush - 2009 - Viscous sheet retraction.pdf} } @@ -1697,7 +1694,6 @@ @article{rayleighInvestigationCharacterEquilibrium1882 number = {1}, pages = {170--177}, doi = {10.1112/plms/s1-14.1.170}, - urldate = {2023-08-10}, copyright = {\textcopyright{} 1882 London Mathematical Society}, file = {C\:\\Users\\zitz\\Zotero\\storage\\FHACBPN6\\Rayleigh - 1882 - Investigation of the Character of the Equilibrium .pdf;C\:\\Users\\zitz\\Zotero\\storage\\ZZJSFA6J\\s1-14.1.html} } @@ -1848,7 +1844,7 @@ @article{jabbarzadehWallSlipMolecular1999 @incollection{heilFlow2015, title = {Flow in {{Flexible}}/{{Collapsible}} Tubes}, - booktitle = {Fluid\textendash{{Structure}} Interactions in Low-Reynolds-Number Flows}, + booktitle = {Fluid- Structure Interactions in Low-Reynolds-Number Flows}, author = {Heil, M. and Hazel, A. L.}, year = {2015}, month = 11, @@ -1865,7 +1861,6 @@ @misc{breloffPlotsJl2021 year = {2021}, month = 6, doi = {10.5281/zenodo.4907285}, - urldate = {2023-08-09}, abstract = {Plots v1.16.3 Diff since v1.16.2 Closed issues: 0.7 Issue: UndefVarError: latexstring not defined error when trying to use log scale (\#1718) [DOC] Error in the "Input Data" Section on Documentation (Wrong Example Code) (\#3401) [BUG] Error when plotting small ({$<$}1e-15) but non-zero data (\#3549) Merged pull requests: fix setting marker = :none without seriestype (\#3550) (@BeastyBlacksmith) remove markers from fillranges (\#3553) (@BeastyBlacksmith)}, howpublished = {Zenodo}, file = {C\:\\Users\\zitz\\Zotero\\storage\\3GEZGI6P\\4907285.html} @@ -1998,8 +1993,7 @@ @article{xuUNSPLITTINGBGKTYPESCHEMES1999 month = 4, journal = {Int. J. Mod. Phys. C}, publisher = {{World Scientific Publishing Company}}, - doi = {10.1142/S0129183199000395}, - urldate = {2023-08-07}, + doi = {10.1142/ S0129183199000395}, abstract = {As a continuation of the research on BGK-type schemes, we present in this paper a second-order unsplitting method for the shallow water equations, where the source terms are included explicitly in ...}, file = {C\:\\Users\\zitz\\Zotero\\storage\\CZU47K8Q\\S0129183199000395.html} } @@ -2239,7 +2233,7 @@ @article{kingdonReactionflowLatticeBoltzmann1992 } @article{kloppDropletCoalescenceQuasiTwoDimensional2020, - title = {On {{Droplet Coalescence}} in {{Quasi-Two-Dimensional Fluids}}}, + title = {On {{Droplet Coalescence}} in {{Quasi-Two- Dimensional Fluids}}}, author = {Klopp, C. and Eremin, A.}, year = {2020}, month = 9, @@ -2451,7 +2445,6 @@ @article{wangMolecularDynamicsSimulations2015 pages = {7457--7462}, publisher = {{American Chemical Society}}, doi = {10.1021/acs.langmuir.5b01574}, - urldate = {2023-07-27}, abstract = {When an electric field with various strengths is applied to two adjacent conducting droplets, the droplets may completely coalesce, partially coalesce, or bounce off one another. To reveal an atom-scale mechanism of coalescence or non-coalescence, dynamic behaviors of two conducting nanodroplets at a homogeneous electric field are studied via molecular dynamics simulations in this work. The results show that there is a critical field strength and a critical cone angle above which the two droplets partially coalesce or bounce off. Charge transfer between the two droplets is observed when the droplets are brought into contact. The partial coalescence and the bounce-off of the two droplets at strong field strengths are found to be due to the high charge transfer rate, which leads to the breakup of the coalescing droplet at different locations.} } @@ -2523,7 +2516,6 @@ @article{carenzaLatticeBoltzmannMethods2019 number = {6}, pages = {81}, doi = {10.1140/epje/i2019-11843-6}, - urldate = {2023-08-10}, abstract = {We review the state of the art of active fluids with particular attention to hydrodynamic continuous models and to the use of Lattice Boltzmann Methods (LBM) in this field. We present the thermodynamics of active fluids, in terms of liquid crystals modelling adapted to describe large-scale organization of active systems, as well as other effective phenomenological models. We discuss how LBM can be implemented to solve the hydrodynamics of active matter, starting from the case of a simple fluid, for which we explicitly recover the continuous equations by means of Chapman-Enskog expansion. Going beyond this simple case, we summarize how LBM can be used to treat complex and active fluids. We then review recent developments concerning some relevant topics in active matter that have been studied by means of LBM: spontaneous flow, self-propelled droplets, active emulsions, rheology, active turbulence, and active colloids.}, keywords = {Soft Matter: Polymers and Polyelectrolytes}, file = {C\:\\Users\\zitz\\Zotero\\storage\\WHSSSXRB\\Carenza et al. - 2019 - Lattice Boltzmann methods and active fluids.pdf} @@ -2965,7 +2957,6 @@ @article{peschkaVariationalApproachDynamic2018 number = {8}, pages = {082115}, doi = {10.1063/1.5040985}, - urldate = {2023-07-24}, abstract = {This paper investigates a modeling approach for viscous flows with dynamic contact angles based on variational energy-dissipation principles. A corresponding Stokes free boundary problem is reduced to a thin-film equation and its variational structure is constructed. The usefulness of this abstract modeling approach is demonstrated by providing numerical schemes for the Stokes flow and the thin-film model and by computing numerical solutions for the problem of gravity-driven thin droplets. Some implications of the contact angle model and properties of the numerical scheme are highlighted in this setting.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\D8JSQKD9\\Peschka - 2018 - Variational approach to dynamic contact angles for.pdf;C\:\\Users\\zitz\\Zotero\\storage\\JRFRJZFE\\Variational-approach-to-dynamic-contact-angles-for.html} } @@ -3245,7 +3236,6 @@ @article{mcnamaraUseBoltzmannEquation1988 pages = {2332--2335}, publisher = {{American Physical Society}}, doi = {10.1103/PhysRevLett.61.2332}, - urldate = {2023-08-02}, abstract = {We discuss an alternative technique to the lattice-gas automata for the study of hydrodynamic properties, namely, we propose to model the lattice gas with a Boltzmann equation. This approach completely eliminates the statistical noise that plagues the usual lattice-gas simulations and therefore permits simulations that demand much less computer time. It is estimated to be more efficient than the lattice-gas automata for intermediate to low Reynolds number R{$\lessequivlnt$}100.}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\26FBXZRQ\\PhysRevLett.61.html} } @@ -3420,7 +3410,6 @@ @article{scheelViscousInertialCoalescence2023 pages = {074201}, publisher = {{American Physical Society}}, doi = {10.1103/PhysRevFluids.8.074201}, - urldate = {2024-01-05}, abstract = {Liquid lens coalescence is an important mechanism involved in many industrial and scientific applications. It has been investigated both theoretically and experimentally, yet it is numerically very challenging to obtain consistent results over the wide ranges of surface tension and viscosity values that are necessary to capture the asymptotic temporal behavior in the viscous and inertial limits. We report results of massively parallel simulations based on the color gradient lattice Boltzmann method, which overcome these limitations, and investigate the scaling laws of both regimes. For the two-dimensional case we find good agreement with the similarity solution of the thin-sheet equation, where in the viscous regime the connecting bridge grows linearly with time and in the inertial regime proportionally to t2/3. In three dimensions, the viscous growth of the bridge also exhibits a linear time dependence, while in the inertial regime the growth of both the bridge height and the bridge width is proportional to t1/2.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\ZVX65LBU\\Scheel et al. - 2023 - Viscous to inertial coalescence of liquid lenses .pdf;C\:\\Users\\zitz\\Zotero\\storage\\L7LTKMGY\\PhysRevFluids.8.html} } @@ -3624,7 +3613,7 @@ @article{shanKineticTheoryRepresentation2006 volume = {550}, pages = {413--441}, publisher = {{Cambridge University Press}}, - doi = {10.1017/S0022112005008153}, + doi = {10.1017/ S0022112005008153}, abstract = {We present in detail a theoretical framework for representing hydrodynamic systems through a systematic discretization of the Boltzmann kinetic equation. The work is an extension of a previously proposed formulation. Conventional lattice Boltzmann models can be shown to be directly derivable from this systematic approach. Furthermore, we provide here a clear and rigorous procedure for obtaining higher-order approximations to the continuum Boltzmann equation. The resulting macroscopic moment equations at each level of the systematic discretization give rise to the Navier\textendash Stokes hydrodynamics and those beyond. In addition, theoretical indications to the order of accuracy requirements are given for each discrete approximation, for thermohydrodynamic systems, and for fluid systems involving long-range interactions. All these are important for complex and micro-scale flows and are missing in the conventional Navier\textendash Stokes order descriptions. The resulting discrete Boltzmann models are based on a kinetic representation of the fluid dynamics, hence the drawbacks in conventional higher-order hydrodynamic formulations can be avoided.} } @@ -3739,7 +3728,6 @@ @article{quereFluidCoatingFiber1999 number = {1}, pages = {347--384}, doi = {10.1146/annurev.fluid.31.1.347}, - urldate = {2023-08-10}, abstract = {We discuss the thickness of the liquid layer entrained by a solid drawn out of a bath, focusing on the case where the solid is a fiber or a wire. Slow withdrawals out of a pure or a complex fluid are described as well as quick coatings. We specify the general laws of entrainment and stress the cases where the fiber curvature plays a role. We finally give an overview on the further evolution of the coated film.}, keywords = {boundary layers,surfactants,thin liquid films,wetting} } @@ -4012,7 +4000,6 @@ @book{ernTheoryPracticeFinite2004 publisher = {{Springer}}, address = {{New York, NY}}, doi = {10.1007/978-1-4757-4355-5}, - urldate = {2023-07-27}, isbn = {978-1-4419-1918-2}, keywords = {algorithms,calculus,finite element method,finite elements,Galerkin approximation,numerical analysis,partial differential equations,Sobolev space}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\2D8T3NLI\\Ern and Guermond - 2004 - Theory and Practice of Finite Elements.pdf} @@ -4159,13 +4146,6 @@ @article{vanthangStudy1DLattice2010 file = {C\:\\Users\\Zitzero\\Zotero\\storage\\89EXI383\\van Thang et al. - 2010 - Study of the 1D lattice Boltzmann shallow water eq.pdf;C\:\\Users\\Zitzero\\Zotero\\storage\\4K2AYJDF\\S0021999110003372.html} } -@article{verweyTheoryStabilityLyophobic, - title = {Theory of the {{Stability}} of {{Lyophobic Colloids}}.}, - author = {Verwey, E. J. W.}, - abstract = {The theory of the interaction of the double layers has been reviewed in relation to the stability of lyophobic colloids. It is concluded that the interaction must be associated with an increase of the free energy, leading to a repulsion between the particles. The repulsive potential calculated from a consideration of the free energy for certain special cases has been combined with the London-van der Waals attractive potential calculated by Hamaker to obtain curves of potential vs. distance. Predictions based on these curves appear to agree well with various experimental data. For example, the influence of electrolyte concentration and of the valencies of the ions on flocculation is satisfactorily explained in terms of the theory, although many complicated phenomena remain to be correlated with it.}, - file = {C\:\\Users\\Zitzero\\Zotero\\storage\\W2XXZWJT\\Verwey - Theory of the Stability of Lyophobic Colloids..pdf} -} - @article{vieyrasalasActiveControlEvaporative2012, title = {Active {{Control}} of {{Evaporative Solution Deposition}} by {{Modulated Infrared Illumination}}}, author = {Vieyra Salas, J. A. and {van der Veen}, J. M. and Michels, J. J. and Darhuber, A. A.}, @@ -4279,7 +4259,6 @@ @article{taylorInstabilityLiquidSurfaces1997 pages = {192--196}, publisher = {{Royal Society}}, doi = {10.1098/rspa.1950.0052}, - urldate = {2023-08-10}, abstract = {It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\NWDXNJ5M\\Taylor - 1997 - The instability of liquid surfaces when accelerate.pdf} } @@ -4295,7 +4274,6 @@ @article{bertrandDynamicsDewettingNanoscale2007 pages = {3774--3785}, publisher = {{American Chemical Society}}, doi = {10.1021/la062920m}, - urldate = {2023-07-27}, abstract = {Large-scale molecular dynamics simulations are used to model the dewetting of solid surfaces by partially wetting thin liquid films. Two levels of solid-liquid interaction are considered that give rise to large equilibrium contact angles. The initial length and thickness of the films are varied over a wide range at the nanoscale. Spontaneous dewetting is initiated by removing a band of molecules either from each end of the film or from its center. As observed experimentally and in previous simulations, the films recede at an initially constant speed, creating a growing rim of liquid with a constant receding dynamic contact angle. Consistent with the current understanding of wetting dynamics, film recession is faster on the more poorly wetted surface to an extent that cannot be explained solely by the increase in the surface tension driving force. In addition, the rates of recession of the thinnest films are found to increase with decreasing film thickness. These new results imply not only that the mobility of the liquid molecules adjacent to the solid increases with decreasing solid-liquid interactions, but also that the mobility adjacent to the free surface of the film is higher than in the bulk, so that the effective viscosity of the film decreases with thickness.}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\PXZEG88D\\la062920m.html} } @@ -4311,7 +4289,6 @@ @article{koplikPearlingInstabilityNanoscale2006 number = {3}, pages = {032104}, doi = {10.1063/1.2178786}, - urldate = {2023-07-27}, abstract = {We investigate the flow of a nanoscale incompressible ridge of low-volatility liquid along a ``chemical channel'': a long, straight, and completely wetting stripe embedded in a planar substrate, and sandwiched between two extended less wetting solid regions. Molecular dynamics simulations, a simple long-wavelength approximation, and a full stability analysis based on the Stokes equations are used, and give qualitatively consistent results. While thin liquid ridges are stable both statically and during flow, a (linear) pearling instability develops if the thickness of the ridge exceeds half of the width of the channel. In the flowing case, periodic bulges propagate along the channel and subsequently merge due to nonlinear effects. However, the ridge does not break up even when the flow is unstable, and the qualitative behavior is unchanged even when the fluid can spill over onto a partially wetting exterior solid region.}, file = {C\:\\Users\\Zitzero\\Zotero\\storage\\QB42QTV7\\Koplik et al. - 2006 - Pearling instability of nanoscale fluid flow confi.pdf;C\:\\Users\\Zitzero\\Zotero\\storage\\L2HF8ULJ\\Pearling-instability-of-nanoscale-fluid-flow.html} } @@ -4401,7 +4378,6 @@ @article{frischLatticeGasAutomataNavierStokes1986 pages = {1505--1508}, publisher = {{American Physical Society}}, doi = {10.1103/PhysRevLett.56.1505}, - urldate = {2023-07-25}, abstract = {We show that a class of deterministic lattice gases with discrete Boolean elements simulates the Navier-Stokes equation, and can be used to design simple, massively parallel computing machines.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\PW73V6GB\\Frisch et al. - 1986 - Lattice-Gas Automata for the Navier-Stokes Equatio.pdf;C\:\\Users\\zitz\\Zotero\\storage\\4W5EN67L\\PhysRevLett.56.html} } @@ -4689,7 +4665,6 @@ @article{zitzControllingDewettingMorphologies2023 pages = {L122001}, publisher = {{American Physical Society}}, doi = {10.1103/PhysRevFluids.8.L122001}, - urldate = {2024-01-08}, abstract = {Switchable and adaptive substrates emerged as valuable tools for controlling wetting and actuation of droplet motion. Here, we report a computational study of the dynamics of an unstable thin liquid film deposited on a switchable substrate, modeled with a space- and time-varying contact angle. For a sufficiently large rate of wettability variation, a topological transition appears. Instead of breaking up into droplets, as expected for a substrate with multiple wetting minima, a metastable rivulet state emerges. A criterion discriminating whether or not rivulets occur is identified in terms of a single dimensionless parameter. Finally, we show and derive theoretically how the film rupture times, droplet shape, and rivulet lifetime depend on the pattern wavelength and speed.}, file = {C\:\\Users\\zitz\\Zotero\\storage\\23T7UFPS\\Zitz et al. - 2023 - Controlling the dewetting morphologies of thin liq.pdf;C\:\\Users\\zitz\\Zotero\\storage\\KN29CM69\\PhysRevFluids.8.html} } diff --git a/chapters/Conclusion.tex b/chapters/Conclusion.tex index 30afb47..7441a60 100644 --- a/chapters/Conclusion.tex +++ b/chapters/Conclusion.tex @@ -17,7 +17,7 @@ \chapter{Conclusion and outlook} Numerical simulations based on the thin film equation may help to understand at least this part of the process. In the long run the found results should help to improve both the process itself as well as the efficiency of the final product. -To this end a novel approach for modelling the dynamics of thin liquid films has been developed, see Chap.~\ref{chapter:theory}-\ref{chapter:second_paper}. +To this end a novel approach for modelling the dynamics of thin liquid films has been developed, see Chap.~\ref{chapter:theory} and the first three publications. It is important to highlight that this model consists of two parts. There is the theoretical work, where we use shallow water theory and find matching conditions that allow us to solve the thin film equation. In Chap.~\ref{chapter:theory} we introduce both the shallow water theory and the thin film equation and discuss some overlap between them, we however like to point out that we are not the first ones to do so. @@ -38,7 +38,7 @@ \chapter{Conclusion and outlook} Finite volume methods are highly dependent on the quality of the mesh and developing self-consistent boundary conditions can be a tedious task, e.g., contact angle dynamics. Molecular dynamics simulations are inherently noisy and computationally demanding for large volumes, and yet all of those three methods can be used to simulate thin film problems. Each method can address different aspects of a problem and yet all have to satisfy the underlying physical boundary conditions, e.g., mass conservation. -We briefly addressed why our method fits well into the current landscape of thin film solvers in Sec~\ref{section:statement_software} of Chap.~\ref{chapter:fourth_paper}. +We briefly addressed why our method fits well into the current landscape of thin film solvers in the first publication. This statement of need covers only the positive sides of our method, but as written above all methods have their shortcomings and so does ours. All the numerical experiments conducted in this thesis do not rely on the lattice Boltzmann approach we developed. @@ -74,11 +74,11 @@ \chapter{Conclusion and outlook} It is becoming harder and harder to validate numerical simulations and argue about data, we think that transparency of our source code is at least a baby step into the right direction. The package approach, similar to e.g., Python, Matlab or R, makes it simple to share and collaborate with colleagues. Apart from these quality of life/usability features there are few things that make the derived model special and scientifically interesting. -In contrast to various numerical methods used in the literature of thin film dynamics, the here presented method is build upon the lattice Boltzmann method as described in Chaps~\ref{chapter:method}-\ref{chapter:first_paper}. +In contrast to various numerical methods used in the literature of thin film dynamics, the here presented method is build upon the lattice Boltzmann method as described in Chaps~\ref{chapter:method} and the first two publications. It is therefore not intended as a general tool to approximate differential equations, but only for the hydrodynamics of thin liquid films. While the term simplicity was used earlier as weakness, one of the nineteen guiding principles of python reads: ``Simple is better than complex''. A small codebase is maintainable, it is possible to write tests for every function and add documentation, all of this would be harder for general purpose solvers with much more lines of code. -In Chap.~\ref{chapter:first_paper} we show that our simple approach is in fact more than capable to simulate relevant physical problems. +In the second publication we show that our simple approach is in fact more than capable to simulate relevant physical problems. Multiple tests with analytical results have been used to validate the method, see Tanner's law and Cox-Voinov relation. Another benefit of the here presented approach is the dimensional reduction. While it is restricting on the one hand, e.g., contact angles smaller than $\pi/2$, it reduces the demand for computational resources quite significant. @@ -86,20 +86,20 @@ \chapter{Conclusion and outlook} All simulations presented in this thesis can be computed on a laptop with dedicated GPU, there is no real need for HPC resources. Working on initial conditions and setting up parameters can be a tedious task that can make or break a simulation. The here presented method turned out to be fairly robust and numerically stable in many regards. -To name just two, initial or ``boundary'' conditions with non-differentiable functions did not cause any issues, see e.g., Chap.~\ref{chapter:second_paper} Eq.~(\ref{eq:sharp_contact_angle_spatial}). +To name just two, initial or ``boundary'' conditions with non-differentiable functions did not cause any issues, see e.g., third publication. Although the method approximates the thin film equation and as such is only strictly valid for small contact angles, we found that contact angle up to $70^{\circ}$ seemed to be within the validity of the method, at least if we assume that Cox-Voinov is applicable in the regime. Another feature that turned out to be of great value is the inclusion of hydrodynamic slip. Having a method that can be used in the no-slip regime while also be applicable in the intermediate to large slip regime is outstanding. -And as shown in Chap.~\ref{chapter:second_paper} slip does in fact play an important role in the dynamics of a dewetting thin film. -While the model does not account for e.g., surfactants, we supply a matching condition between our method and thin film like theories in Chaps.~\ref{chapter:first_paper}-\ref{chapter:second_paper}. +And as shown in the third publication slip does in fact play an important role in the dynamics of a dewetting thin film. +While the model does not account for e.g., surfactants, we supply a matching condition between our method and thin film like theories in the second and third publication. The seemingly simple ``Yes'' or ``No'' question could be discussed in even more detail, but it is a subjective open question. -Therefore we would like to have a look at Chaps.\ref{chapter:second_paper}-\ref{chapter:third_paper} and their findings. +Therefore we would like to have a look at third and last publication and their findings. -In Chap.~\ref{chapter:second_paper} we introduce thermal fluctuations to our model. +In the third publication we introduce thermal fluctuations to our model. These fluctuations lead to a constant excitement of the fluid air interface that is smoothed out by the surface tension, leading to the appearance of thermo-capillary waves. Their amplitude however is usually so small that they can be neglected in experimental analysis or numerical simulations. However having a dewetting thin film it can be assumed that the thickness of the film will be comparable to the amplitude of the fluctuations, at least in regions where the film ruptures. -As shown in Chap.~\ref{chapter:second_paper} these fluctuations can be added to the model using a matching condition for a fluctuating force term. +As shown in the third publication these fluctuations can be added to the model using a matching condition for a fluctuating force term. The fluctuating force term is then tested against the theory of capillary waves and does show in fact good agreement with the predicted spectrum. In the following it is shown that the stability difference, as measured in rupture times, between fluctuating and deterministic thin films~\footnote{$k_BT > 0$ fluctuating thin film, $k_BT = 0$ deterministic thin film.} does inversely depend on the logarithm of the wettability. By the virtue of the model a combination of a spatially varying wettability as well as the addition of thermal fluctuations is studied using a dewetting thin film. @@ -107,7 +107,7 @@ \chapter{Conclusion and outlook} At the beginning of this chapter we discuss the underlying theory and the modification to the thin film equation. The constructive approach we use to match our model with the stochastic thin film equation should in fact serve as a blueprint for other additions e.g., surfactant dynamics. -In Chap.~\ref{chapter:third_paper} a simple toy model is developed to study dewetting dynamics under the influence of ``switchable'' substrates. +In the last publication a simple toy model is developed to study dewetting dynamics under the influence of ``switchable'' substrates. The model is build on the idea that the wettability, e.g., the equilibrium contact angle $\theta_{\text{eq.}}$, can be changed based on external stimuli. We assume that the these stimuli can be modelled by an arbitrary function. In the here discussed case the arbitrary function is a trigonometric function ($\theta(x,t) \propto \sin(x t)$) that admits not only a spatial gradient but also a time dependent behaviour. diff --git a/chapters/Introduction.tex b/chapters/Introduction.tex index f3f3418..a5385f0 100644 --- a/chapters/Introduction.tex +++ b/chapters/Introduction.tex @@ -149,8 +149,7 @@ \section{Selected applications of the thin film equation} These results can be addressed as dimensional analysis, a rather fancy word for matching dimensions on both sides of the equation. In the end the height or thickness needs to be of the dimension of a length. They often support so-called self similar behaviour, meaning under change of one or more parameters it is possible to collapse data on a master curve. -Another dynamic problem that exhibits self similar behaviour is droplet coalescence, which is shortly discussed in Chapter~\ref{chapter:fourth_paper} and liquid lens coalescence~\cite{hackSelfSimilarLiquidLens2020, scheelViscousInertialCoalescence2023}. -% I am here +Another dynamic problem that exhibits self similar behaviour is droplet coalescence, which is shortly discussed in the first publication in this thesis and liquid lens coalescence~\cite{hackSelfSimilarLiquidLens2020, scheelViscousInertialCoalescence2023}. \section{Literature overview} \label{section:literature} \begin{figure} @@ -210,7 +209,7 @@ \section{Literature overview} The stochastic thin film equation is then, again, derived from the LLNS equation by integration along the minor dimension, usually considered the vertical dimension. This step however poses a hard problem, because the stochastic stress tensor in principle depends on the vertical dimension. Grün and coworkers found an elegant way to circumvent the problem with a single multiplicative Gaussian term~\cite{grunThinFilmFlowInfluenced2006, meckeThermalFluctuationsThin2005, fetzerThermalNoiseInfluences2007, zhangNanoscaleThinfilmFlows2020, zhangMolecularSimulationThin2019, nesicFullyNonlinearDynamics2015}. -This idea will be revisited in Chap.~\ref{chapter:second_paper}. +This idea will be revisited in the third publication. Around the same time Davidovitch et al., came up with a similar form for the stochastic thin film equation, studying the spreading of a droplet~\cite{davidovitchSpreadingViscousFluid2005}. The addition of a Gaussian noise term made it therefore possible to reproduce the observed capillary wave spectrum and to explain the difference in spreading behaviour. A further impact of this term was the reduction of rupture times for spinodal dewetting thin films~\cite{grunThinFilmFlowInfluenced2006, fetzerThermalNoiseInfluences2007}. @@ -252,7 +251,7 @@ \section{Literature overview} \label{fig:morph_transition} \end{figure} This opens up a new avenue for microfluidic devices. -In Chap.~\ref{chapter:third_paper} we discuss the influence a switchable wettability can have on the dewetting of a thin film. +In last publication we discuss the influence a switchable wettability can have on the dewetting of a thin film. In fact the switching can induce morphological transitions as depict in Fig.~\ref{fig:morph_transition}. \section{Scientific software} @@ -385,7 +384,7 @@ \section{Scientific software} Often however when a student starts a new software project, functionality or performance is the main point of interest. During the development step tests and even more important documentation are left aside. Ultimately this does not only open the gate for various bugs but to some extent it also renders the software useless for other researchers. -Needless to say, my very own CFD solver, see Chap.~\ref{chapter:fourth_paper}, generated correct results due to the cancellation of two bugs. +Needless to say, my very own CFD solver, see first publication, generated correct results due to the cancellation of two bugs. Defining a test suite from the beginning of the project would have spared me valuable time and effort. On the other hand, research became more than ever dependent on computational resources. @@ -401,7 +400,7 @@ \section{Scientific software} Now or never is the time to install protocols. Guides on how to deal with scientific software such that data can be reproduced or even more important reused. While this may not be of interest for most readers, to me it is an essential point. -That said, Chap.~\ref{chapter:fourth_paper} describes the software developed and used for this thesis. +That said, the first publication describes the software developed and used for this thesis. One can find the open source software repository that hosts the code, tests and documentation. Upon pull or merge requests an automated testing suite performs tests to ensure no bugs break necessary functionality. Furthermore by definition of version control it is possible to reproduce all results generated with that software, called \textit{Swalbe.jl}. @@ -430,12 +429,12 @@ \section{Outline} Not only is it possible to approximate the Navier-Stokes but also the shallow water equations, of course with a different set of constraints~\cite{salmonLatticeBoltzmannMethod1999, zhouLatticeBoltzmannMethods2004, vanthangStudy1DLattice2010, dellarNonhydrodynamicModesPriori2002}. Following that argumentation is the published article for the \textit{Journal of Open Source Software}. -Highlighting the implementation and development of the lattice Boltzmann solver called \textbf{Swalbe} (\textbf{s}hallow \textbf{wa}ter \textbf{l}attice \textbf{B}oltzmann solv\textbf{e}r) in Chap.~\ref{chapter:fourth_paper}. +Highlighting the implementation and development of the lattice Boltzmann solver called \textbf{Swalbe} (\textbf{s}hallow \textbf{wa}ter \textbf{l}attice \textbf{B}oltzmann solv\textbf{e}r) in the first publication. Key aspects are the automated continuous integration (CI) with a test suite and web-hosted documentation, as motivated in Sec.~\ref{section:statement_software}. The Julia package allows for either fast iterative model development in two dimensions or large scale simulations in three dimensions with GPU acceleration. It allows for prototyping and testing which is fast and easy to use, but also a framework for large three dimensional simulations with minor to no code changes. -Use cases and a derivation of the model can be found in Chap.~\ref{chapter:first_paper}. +Use cases and a derivation of the model can be found in the second publication. In this chapter the mandatory modelling assumptions are introduced that allow to match the shallow water system with the thin film equation. It further highlights which modifications are made to the shallow water lattice Boltzmann algorithm. With some emphasis on the numerical implementation of e.g., the computation of gradients and the Laplacian in agreement with Ref~\cite{junkDiscretizationsIncompressibleNavier2000, thampiIsotropicDiscreteLaplacian2013}. @@ -445,15 +444,15 @@ \section{Outline} The linear relation between Bond and Capillary number is validated with the sliding of a droplet. And for completeness a discussion on the performance of the algorithm and applicability for accelerated computing follows at the end of the chapter. -In the next chapter, Chap.~\ref{chapter:second_paper}, a functional extension to the lattice Boltzmann model is shown. +In the third publication, a functional extension to the lattice Boltzmann model is shown. Instead of approximating the deterministic thin film equation the stochastic thin film equation (STF) is the topic of interest in this chapter. Thermal fluctuations are often neglected in simulations while their existence in the experiment cannot be denied. For nanometric thin films fluctuations can accelerate the time scales of instabilities. Interestingly the time scales of these instabilities and resulting film ruptures are in fact contact angle dependent. Depending however on the substrate properties (patterning), it is still likely that fluctuations are only subdominant. -Dynamics during dewetting can either be generated due to forces, as shown in Chaps.~\ref{chapter:first_paper}-\ref{chapter:second_paper} or it can be induced due to time dependent potentials. -Giving the recent interest in dynamics of simple liquids on complex substrates and the dynamic wetting of flexible, adaptive and switchable surfaces, Chap.~\ref{chapter:fourth_paper} is dedicated to the question: ``What happens during dewetting with a spatio-temporal evolving wettability gradient?'' +Dynamics during dewetting can either be generated due to forces, as shown in the second and thrid publication or it can be induced due to time dependent potentials. +Giving the recent interest in dynamics of simple liquids on complex substrates and the dynamic wetting of flexible, adaptive and switchable surfaces, the last publication is dedicated to the question: ``What happens during dewetting with a spatio-temporal evolving wettability gradient?'' To study this problem a series of three dimensional simulations is performed with varying spatial as well as temporal evolution of the wettability. On the one hand the wettability dynamics has a small but measurable stabilizing effect on the spinodal dewetting of the thin film leading to a net increase in rupture times if the dynamic wetting is switched on. On the other hand after the rupture of the film a clear morphological transition is observable. diff --git a/chapters/Method.tex b/chapters/Method.tex index d34303e..325ee16 100644 --- a/chapters/Method.tex +++ b/chapters/Method.tex @@ -76,7 +76,7 @@ \subsection{Finite differences} At every point $x_i$ along the x-axis we know the value of the function $f(x_i)$. This however does not mean that we know the value of $f(x)$ at arbitrary $x$ values and even worse we do not know $\partial_x f(x_i)$ at $x_i$. For a continuous function, e.g., the film thickness $h(\mathbf{x},t)$, we can analytically compute the derivatives. -Here we are left with a discrete set of points and have yet to come up with an idea to calculate $\nabla h(\mathbf{x}_i, t)$, which is crucial to compute the film pressure, Eq.~(\ref{eq:pfilm}). +Here we are left with a discrete set of points and have yet to come up with an idea to calculate $\nabla h(\mathbf{x}_i, t)$, which is crucial to compute the film pressure. Discretization of a derivative is a long studied problem~\cite{booleTreatiseCalculusFinite1872, jordanCalculusFiniteDifferences1965}. The Taylor expansion gives us a tool to approximate the solution to a differential. @@ -139,7 +139,7 @@ \subsection{Finite differences} \subsection{Spectral methods} Another strategy to solve a differential equation is to perform a transformation. -One rather famous transformation is the Fourier Transformation (FT), which we have used to find the dispersion relation in Sec.~\ref{sec:thin_films} and will encounter again in Chap.~\ref{chapter:second_paper}. +One rather famous transformation is the Fourier Transformation (FT), which we have used to find the dispersion relation in Sec.~\ref{sec:thin_films} and will encounter again in the third publication. The set of solvers using a FT to solve a given differential equation is called spectral methods. Doing so has the advantage of interchanging the derivatives with simple multiplications. For example the term $\partial_x f(x)$ transforms to @@ -150,7 +150,7 @@ \subsection{Spectral methods} Instead of solving a differential equation one solves an algebraic equation, or a set of algebraic equations. While it is often straightforward to perform the transformation it can become problematic to perform the backward transformation. Usually the expressions are fairly complicated and often the backward transformation into real space is not easy to perform. -In Chap.~\ref{chapter:second_paper} we arrive at such an expression for the structure factor $\mathcal{S}(q, t)$. +In the third publication we arrive at such an expression for the structure factor $\mathcal{S}(q, t)$. A structure factor is a measure of the distribution of length scales, roughly speaking. In this specific case there is no need for a backward transformation because (scattering) experiments usually supply data in Fourier space. @@ -241,7 +241,7 @@ \section{The lattice Boltzmann method}\label{sec:LBM} Instead of dealing with every atom or molecule, which is displayed in the bottom left of the figure, it is also possible to use a more coarse approach which is shown in the centre of the figure where distributions over ensembles of particles are used~\cite{raabeOverviewLatticeBoltzmann2004}. Therefore instead of working in the microscopic world the lattice Boltzmann method works in a mesoscopic regime. Making it computationally more efficient than e.g., MD for a larger class of problems. -Of course, it is also possible to use a coarser approach and in fact a lot of the classical Navier-Stokes solvers operate in the upper right of Fig.~\ref{fig:scales_dummy}, see Sec.~\ref{sec:Joss_need} for some example solvers. +Of course, it is also possible to use a coarser approach and in fact a lot of the classical Navier-Stokes solvers operate in the upper right of Fig.~\ref{fig:scales_dummy}, see first publication for some example solvers. \subsection{The Boltzmann equation and macroscopic conservation equations} The dynamics of these distributions $f$ can be formalized and put into an evolution equation which is called the Boltzmann equation @@ -519,7 +519,7 @@ \subsection{From continuum to discrete lattice fields} So far however no assumptions on the lattice distributions are made and the discrete equilibrium distribution function has not been specified. Paving work by He and Luo as well as group theoretical arguments derived by Rubinstein and Luo showed that based on dimensionality just a few (discrete) velocities are sufficient~\cite{heTheoryLatticeBoltzmann1997, rubinsteinTheoryLatticeBoltzmann2008}. -In fact in Chap.~\ref{chapter:first_paper} all numerical experiments were performed on a D2Q9 lattice therefore $\mathbf{c}_{i}$ is +In fact in second publication all numerical experiments were performed on a D2Q9 lattice therefore $\mathbf{c}_{i}$ is \begin{equation}\label{eq:speeds_method} \mathbf{c}_{i} = \left\{ @@ -644,15 +644,15 @@ \subsection{The shallow water equations on the lattice}\label{susbsec:shallow_wa f^{eq}_i &= w_i h\left(\frac{3}{2}gh + 3 c_{i\alpha}u_{\alpha} + \frac{9}{2}(c_{i\alpha}u_{\alpha})^2 - \frac{3}{2}u^2\right), \quad i\neq 0 . \end{align} -In Chap.~\ref{chapter:first_paper} we will recap the main ideas of this derivation. +In the second publication we will recap the main ideas of this derivation. We will then discuss constructive matching conditions between the shallow water model and the thin film equation. The shallow water lattice Boltzmann model is not sufficient for thin film dynamics since we also have to introduce forces. These forces account for wetting dynamics, such as contact angles, and the velocity boundary condition at $h = 0$. The derivation of the lattice Boltzmann method for the Navier-Stokes equation as well as the shallow water system concludes this chapter. It follows a chapter about the sustainable approach I chose to keep the software first open source and maintainable for the future. -After the introduction of \textit{Swalbe.jl} in Chap.~\ref{chapter:fourth_paper} the method is thoroughly tested against problems such as the Rayleigh-Taylor instability in Chap.~\ref{chapter:first_paper}, while in Chap.~\ref{chapter:second_paper} an additional contribution due to thermal fluctuations is considered. +After the introduction of \textit{Swalbe.jl} in the first publication the method is thoroughly tested against problems such as the Rayleigh-Taylor instability in the second publication, while in the third publication an additional contribution due to thermal fluctuations is considered. Although the focus is solemnly on thermal fluctuations, this chapter should serve as a guideline for the inclusion of further dynamics to the thin film model. -Beyond additions to the thin film model we discuss a coupled problem in Chap.~\ref{chapter:third_paper}. +Beyond additions to the thin film model we discuss a coupled problem in the last publication. Coupled in the sense that we not only consider the dynamics of the film but allow for dynamical degrees of freedom for the substrate. \ No newline at end of file diff --git a/chapters/Theory.tex b/chapters/Theory.tex index 030fe63..e1b1f08 100644 --- a/chapters/Theory.tex +++ b/chapters/Theory.tex @@ -27,9 +27,9 @@ \section{The Navier-Stokes equation} Even the acceleration due to gravity when pointing the tube towards the ground does not exert enough force to generate a flow. In contrast to Newtonian liquids those liquids have a different response function to shear, they can be addressed as \textit{power law} fluids. While the phenomenology of these fluids, e.g., shear thinning or thickening, is very interesting and important in many industrial applications it is beyond the scope of this work. -That said, in Chap.~\ref{chapter:second_paper} an extension towards more complex flows is presented. +That said, in the third publication an extension towards more complex flows is presented. Instead of however changing the shear response we model thermal fluctuations to approximate the stochastic thin film equation (STF). -In fact, the step to derive an extension to our approach are nicely summarized in Chap.~\ref{chapter:second_paper}. +In fact, the step to derive an extension to our approach are nicely summarized in the third publication. It should serve as a guideline for further features, such as viscoelasticity for non-Newtonian liquids~\cite{bouchutNewModelShallow2013}. Knowing what a fluid is and taking into account the conservation of mass and momentum it is possible to derive the Navier-Stokes equation. @@ -149,7 +149,7 @@ \section{The Navier-Stokes equation} \section{Shallow water theory} \label{sec:theory_shallow_water} -Solving the Navier-Stokes equation for the problems discussed in Chaps.~\ref{chapter:fourth_paper}-\ref{chapter:third_paper} is not needed. +Solving the Navier-Stokes equation for the problems discussed in all publications is not needed. Looking at Figs.~\ref{fig:examples_intro}-\ref{fig:morph_transition} one can identify that the height or thickness of the fluid layer is much smaller than the horizontal extension of the substrate. This statement together with some more approximations allows to derive another system from the Navier-Stokes equation and the continuity equation which is called the shallow water or Saint-Venant equations~\cite{bTheorieMouvementNonpermanent1871}. @@ -198,7 +198,7 @@ \subsection{Definition} Third, the horizontal velocity of the fluid has to vanish at the ``solid'' ground, or in terms of Fig.~\ref{fig:shallow_water_drawing} looking down from the free surface at $-h(x,y)$. Although there is a viscosity associated with the fluid, friction with the bottom topography is the dominating dissipation channel. This kind of boundary condition has been introduced in Chap.~\ref{chapter:intro} is called \textbf{no-slip} boundary condition~\cite{richardsonNoslipBoundaryCondition1973, degennesFluidWallSlippage2002, salmonLatticeBoltzmannMethod1999}. -While this may seem counterintuitive because fluid has no problem to flow over yet dry surfaces, modelling wetting or the wet-dry transition is a complex task, as seen later in this chapter and in Chap.~\ref{chapter:first_paper}. +While this may seem counterintuitive because fluid has no problem to flow over yet dry surfaces, modelling wetting or the wet-dry transition is a complex task, as seen later in this chapter and in the second publication. After all, a theoretical model which is defined as a set of partial differential equations cannot account for the true complexity of nature~\cite{scrivenPhysicsApplicationsDIP1988}. We come back to the idea of the no-slip boundary condition in Sec.~\ref{sec:thin_films}. @@ -485,7 +485,7 @@ \subsection{Poiseuille flow} \end{align} where $\mathbf{u} = (u_x, u_y)$ and the only unknown $\mathcal{S}$ is a yet undefined source term. The system Eqs.~(\ref{eq:shallow_water_01}-\ref{eq:shallow_water_02}) concludes the shallow water model. -In Chap.~\ref{chapter:first_paper} we show how this translates into a numerical solver. +In second publication we show how this translates into a numerical solver. Following in this chapter are two more sections. The next section and the main point of interest of this work is a short and phenomenological oriented derivation of the thin film model. @@ -634,7 +634,7 @@ \subsection{Scaling and boundary conditions} For example the pressure scaling would change to $p = \rho U^2\tilde{p}$~\cite{crasterDynamicsStabilityThin2009}. In the shallow water section, we quickly reduced the problem to two dimensions (one horizontal and one vertical). -Here we keep the second horizontal dimension ($\tilde{y}, \tilde{u}_y$), because most numerical experiments in Chaps.~\ref{chapter:fourth_paper}-\ref{chapter:third_paper} are computed in a two dimensional domain. +Here we keep the second horizontal dimension ($\tilde{y}, \tilde{u}_y$), because most numerical experiments in the last and second to last publication are computed in a two-dimensional domain. The first step in the derivation is to insert the dimensionless variables into the continuity equation, Eq.~(\ref{eq:incomp}) \begin{equation}\label{eq:cont_thin_film_1} @@ -770,8 +770,8 @@ \subsection{Vertical integration} \partial_t h(\mathbf{x},t) = \nabla\cdot\left(\frac{h^3}{3\mu}\nabla p\right), \end{equation} which completes the task to derive the thin film equation from the Navier-Stokes equation. -To sum up the approach in one sentence, we non dimensionalized the variables, collected terms in orders of $\varepsilon$, used our boundary conditions and performed an integration along the vertical dimension to average out the vertical dynamics. -Of course this derivation comes short if there are additional forces to consider such as thermal fluctuations, that said in Chap.~\ref{chapter:second_paper} we discuss the effect of thermal fluctuations on dewetting and show how the thin film equation has to be modified. +To sum up the approach in one sentence, we non-dimensionalized the variables, collected terms in orders of $\varepsilon$, used our boundary conditions and performed an integration along the vertical dimension to average out the vertical dynamics. +Of course this derivation comes short if there are additional forces to consider such as thermal fluctuations, that said in the third publication we discuss the effect of thermal fluctuations on dewetting and show how the thin film equation has to be modified. Because the thin film equation is the main theoretical framework for the following Chaps. it is instructive to derive some more phenomenology of Eq.~(\ref{eq:thin_final}). One helpful tool is the so called linear stability analysis. @@ -846,17 +846,17 @@ \subsection{Linear stability analysis} t_0 = \frac{3\mu}{\gamma h_0^3 q_0^4}, \end{equation} with is used to normalize the time in Fig.~\ref{fig:dispertion_1}. -Because the concept of linear stability is such an important one, we encounter it again in Chap.~\ref{chapter:first_paper} (see Fig.~\ref{fig:RTI}) as well as in Chap.~\ref{chapter:second_paper} (see Figs.~\ref{fig:theory_simulation_structure_factor}, \ref{fig:spectral_analysis_deter_sine8}, \ref{fig:spectral_analysis_stoch_sine8}, \ref{fig:square_wave_both}). -Chap.~\ref{chapter:second_paper} focuses only on the addition of a fluctuating term and the so called stochastic thin film equation. +Because the concept of linear stability is such an important one, we encounter it again in the second publication as well as in the third publication. +The third publication focuses only on the addition of a fluctuating term and the so called stochastic thin film equation. The new term due to thermal fluctuations offers some interesting features on both the structure factor as well as the dispersion. \begin{figure} \centering \includegraphics[width=0.65\textwidth]{graphics/spectratheta20.pdf} \caption{Dispersion relation of a spinodally dewetting thin film with and without thermal fluctuations. - Where $S(q)$ is the structure factor depending on the wavemode $q$. + Where $S(q)$ is the structure factor depending on the wave mode $q$. The dots and triangles are generated using the here presented numerical method. - Full and dashed lines are the theoretical prediction based on Eq.~(\ref{eq:simple_thin_film}) and its fluctuating extension, Eq.~(\ref{eq:structure_factor_2}). + Full and dashed lines are the theoretical prediction and its fluctuating extension. } \label{fig:dispertion_1} \end{figure} @@ -921,4 +921,4 @@ \section{Differences and overlap} Although this question seems odd at first it is actually of great significance to show that problems can be solved with different approaches. There are several numerical methods to iteratively solve the thin film equation. There are as well several different numerical methods to approximate the Navier-Stokes equation. -The important contribution to our understanding of CFD is that methods build to solve the shallow water equations can actually be used to solve the thin film problem, as discussed in Chap.~\ref{chapter:first_paper} \ No newline at end of file +The important contribution to our understanding of CFD is that methods build to solve the shallow water equations can actually be used to solve the thin film problem, as discussed in the second publication. \ No newline at end of file diff --git a/chapters/chapter_1_heading.tex b/chapters/chapter_1_heading.tex new file mode 100644 index 0000000..c9dfe08 --- /dev/null +++ b/chapters/chapter_1_heading.tex @@ -0,0 +1,11 @@ +\chapter{Publications} +\label{chapter:fourth_paper} +\epigraph{\textit{Du wirst nicht enttäuscht, wenn du nie etwas erwartest und bevor du etwas falsch machst, dann mach mal lieber gar nichts.}}{Kraftklub} + +The effort and work during my time as doctoral candidate has lead to four peer-reviewed articles. +In this chapter we add these articles as they appeared in their respective journal. + +The first article is published in the Journal of Open Source software and discusses the code and partly the method. +The second article appeared in Physical Review E and introduces the method and the derivation of our thin film model. +The third article appeared as well in Physical Review E and discusses an extension to the thin film equation, namely a stochastic term that accounts for thermal fluctuations. +The fourth articles appeared as a letter in Physical Review Fluids and introduces the idea of a modeled switchable substrate coupled to a dewetting thin film. \ No newline at end of file diff --git a/main.sty b/main.sty index 9072850..ed484a9 100644 --- a/main.sty +++ b/main.sty @@ -25,6 +25,7 @@ \usepackage{minted} \usemintedstyle{pastie} \usepackage[babel, german=quotes]{csquotes} +\usepackage{pdfpages} \newcommand{\subsubsubsection}[1]{\paragraph{#1} ~ \vspace{1.7ex plus 0.2ex}} \newcommand*\diff{\mathop{}\!\mathrm{d}} diff --git a/main.tex b/main.tex index b60c5c9..b837eb0 100644 --- a/main.tex +++ b/main.tex @@ -47,11 +47,21 @@ \include{chapters/Introduction}% \include{chapters/Theory} \include{chapters/Method} -\include{chapters/Paper_four} -\include{chapters/Paper_one} -\include{chapters/Paper_two} -\include{chapters/Paper_three} \include{chapters/Conclusion} +\include{chapters/chapter_1_heading} +\includepdf[pages=-, addtotoc={1, section, 1, Swalbe.jl: A lattice Boltzmann solver for thin film hydrodynamics, paper_4}]{papers/Joss_paper.pdf} +%\include{chapters/chapter_2_heading} +\includepdf[pages=-, addtotoc={1, section, 1, Lattice Boltzmann method for thin-liquid-film hydrodynamics, paper_1}]{papers/paper_1.pdf} +%\include{chapters/chapter_3_heading} +\includepdf[pages=-, addtotoc={1, section, 1, Lattice Boltzmann simulations of stochastic thin film dewetting, paper_2}]{papers/paper_2.pdf} +%\include{chapters/chapter_4_heading} +\includepdf[pages=-, addtotoc={1, section, 1, Controlling the dewetting morphologies of thin liquid films by switchable substrates, paper_3}]{papers/paper_3.pdf} +\includepdf[pages=-]{papers/SuppMat.pdf} +%\include{chapters/Paper_four} +%\include{chapters/Paper_one} +%\include{chapters/Paper_two} +%\include{chapters/Paper_three} + % \cleardoublepage% \nocite{*}% @@ -65,7 +75,7 @@ \cleardoublepage% % \setcounter{page}{1}% -\include{appendix/List_of_publications} +%\include{appendix/List_of_publications} \include{appendix/acknowledgements} %\include{appendix/example}% % diff --git a/papers/Joss_paper.pdf b/papers/Joss_paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f6716aa15e56e4764480acc188213d90286bc0d2 GIT binary patch literal 421712 zcmaI7V~`|m6eZj?rfu7{ZClf}ZM%Egw!5co+tapf+y3g^-H6>E`)x&3R7O=~#*aT_andX@ zaZUjTv>ai8q&yX+1NDATB@Lmq7%FVELH{*`mg=Ib 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