Nonlinear dynamics of dewetting thin films

Thomas P. Witelski; Thomas P. Witelski

doi:10.3934/math.2020270

AIMS Mathematics

2020, Volume 5, Issue 5: 4229-4259. doi: 10.3934/math.2020270

Previous Article Next Article

Review Topical Sections

Nonlinear dynamics of dewetting thin films

Thomas P. Witelski ^,

Department of Mathematics, Duke University, Box 90320, Durham NC 27708-0320, USA

Received: 31 January 2020 Accepted: 19 March 2020 Published: 05 May 2020
MSC : 76A20, 76D08, 76E17, 35K25, 35Q35

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called "dewetting". These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields coarsening, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

Keywords:

nonlinear parabolic partial differential equations,
fluid dynamics,
thin film equation,
dewetting,
finite-time singularities,
long-time asymptotics

Citation: Thomas P. Witelski. Nonlinear dynamics of dewetting thin films[J]. AIMS Mathematics, 2020, 5(5): 4229-4259. doi: 10.3934/math.2020270

Related Papers:

[1]	Zairong Wang, Xuan Tang, Haohuai Liu, Lingxi Peng . Artificial immune intelligence-inspired dynamic real-time computer forensics model. Mathematical Biosciences and Engineering, 2020, 17(6): 7221-7233. doi: 10.3934/mbe.2020370
[2]	Hui Yao, Yuhan Wu, Shuo Liu, Yanhao Liu, Hua Xie . A pavement crack synthesis method based on conditional generative adversarial networks. Mathematical Biosciences and Engineering, 2024, 21(1): 903-923. doi: 10.3934/mbe.2024038
[3]	Jiajia Jiao, Xiao Xiao, Zhiyu Li . dm-GAN: Distributed multi-latent code inversion enhanced GAN for fast and accurate breast X-ray image automatic generation. Mathematical Biosciences and Engineering, 2023, 20(11): 19485-19503. doi: 10.3934/mbe.2023863
[4]	Hao Wang, Guangmin Sun, Kun Zheng, Hui Li, Jie Liu, Yu Bai . Privacy protection generalization with adversarial fusion. Mathematical Biosciences and Engineering, 2022, 19(7): 7314-7336. doi: 10.3934/mbe.2022345
[5]	Jinhua Zeng, Xiulian Qiu, Shaopei Shi . Image processing effects on the deep face recognition system. Mathematical Biosciences and Engineering, 2021, 18(2): 1187-1200. doi: 10.3934/mbe.2021064
[6]	Song Wan, Guozheng Yang, Lanlan Qi, Longlong Li , Xuehu Yan, Yuliang Lu . Multiple security anti-counterfeit applications to QR code payment based on visual secret sharing and QR code. Mathematical Biosciences and Engineering, 2019, 16(6): 6367-6385. doi: 10.3934/mbe.2019318
[7]	Dehua Feng, Xi Chen, Xiaoyu Wang, Xuanqin Mou, Ling Bai, Shu Zhang, Zhiguo Zhou . Predicting effectiveness of anti-VEGF injection through self-supervised learning in OCT images. Mathematical Biosciences and Engineering, 2023, 20(2): 2439-2458. doi: 10.3934/mbe.2023114
[8]	Si Li, Limei Peng, Fenghuan Li, Zengguo Liang . Low-dose sinogram restoration enabled by conditional GAN with cross-domain regularization in SPECT imaging. Mathematical Biosciences and Engineering, 2023, 20(6): 9728-9758. doi: 10.3934/mbe.2023427
[9]	Xiao Wang, Jianbiao Zhang, Ai Zhang, Jinchang Ren . TKRD: Trusted kernel rootkit detection for cybersecurity of VMs based on machine learning and memory forensic analysis. Mathematical Biosciences and Engineering, 2019, 16(4): 2650-2667. doi: 10.3934/mbe.2019132
[10]	Sonam Saluja, Munesh Chandra Trivedi, Ashim Saha . Deep CNNs for glioma grading on conventional MRIs: Performance analysis, challenges, and future directions. Mathematical Biosciences and Engineering, 2024, 21(4): 5250-5282. doi: 10.3934/mbe.2024232

Abstract

References

[1]	D. J. Acheson, Elementary Fluid Dynamics, The Clarendon Press Oxford University Press, New York, 1990.
[2]	V. S. Ajaev, Evolution of dry patches in evaporating liquid films, Phys. Rev. E, 72 (2005), 031605.
[3]	V. S. Ajaev, Spreading of thin volatile liquid droplets on uniformly heated surfaces, J. Fluid Mech., 528 (2005), 279-296. doi: 10.1017/S0022112005003320
[4]	V. S. Ajaev, Interfacial Fluid Mechanics, Springer, New York, 2012.
[5]	V. S. Ajaev, E. Y. Gatapova, O. A. Kabov, Stability and break-up of thin liquid films on patterned and structured surfaces, Adv. Colloid Interface Sci., 228 (2016), 92-104. doi: 10.1016/j.cis.2015.11.011
[6]	V. S. Ajaev and G. M. Homsy, Steady vapor bubbles in rectangular microchannels, J. Colloid Interface Sci., 240 (2001), 259-271. doi: 10.1006/jcis.2001.7562
[7]	V. S. Ajaev, J. Klentzman, T. Gambaryan-Roisman, et al. Fingering instability of partially wetting evaporating liquids, J. Eng. Math., 73 (2012), 31-38. doi: 10.1007/s10665-010-9448-y
[8]	D. M. Anderson, M. K. Gupta, A. A. Voevodin, et al. Using amphiphilic nanostructures to enable long-range ensemble coalescence and surface rejuvenation in dropwise condensation, ACS Nano, 6 (2012), 3262-3268. doi: 10.1021/nn300183d
[9]	D. G. Aronson, The porous medium equation. In Nonlinear diffusion problems (Montecatini Terme, 1985), volume 1224 of Lecture Notes in Math., pages 1-46. Springer, Berlin, 1986.
[10]	M. Asgari and A. Moosavi, Coarsening dynamics of dewetting nanodroplets on chemically patterned substrates, Phys. Rev. E, 86 (2012), 016303.
[11]	G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics, Cambridge University Press, New York, 1996.
[12]	G. I. Barenblatt, Scaling, Cambridge University Press, New York, 2003.
[13]	J. Becker, G. Grün, R. Seemann, et al. Complex dewetting scenarios captured by thin-film models, Nat. Mater., 2 (2003), 59-63. doi: 10.1038/nmat788
[14]	P. Beltrame and U. Thiele, Time integration and steady-state continuation for 2d lubrication equations, SIAM J. Appl. Dyn. Syst., 9 (2010), 484-518. doi: 10.1137/080718619
[15]	F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equ., 1 (1996), 337-368.
[16]	F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differ. Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y
[17]	F. Bernis, J. Hulshof, J. R. King, Dipoles and similarity solutions of the thin film equation in the half-line, Nonlinearity, 13 (2000), 413-439. doi: 10.1088/0951-7715/13/2/305
[18]	A. L. Bertozzi, Symmetric singularity formation in lubrication-type equations for interface motion, SIAM J. Appl. Math., 56 (1996), 681-714. doi: 10.1137/S0036139994271972
[19]	A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices of the American Mathematical Society, 45 (1998), 689-697.
[20]	A. L. Bertozzi, M. P. Brenner, T. F. Dupont, et al. Singularities and similarities in interface flows. In Trends and perspectives in applied mathematics, pages 155-208. Springer, New York, 1994.
[21]	A. L. Bertozzi, G. Grün, T. P. Witelski, Dewetting films: bifurcations and concentrations, Nonlinearity, 14 (2001), 1569-1592. doi: 10.1088/0951-7715/14/6/309
[22]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: the moving contact line with a "porous media" cut-off of van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. doi: 10.1088/0951-7715/7/6/002
[23]	A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions, Commun. Pure Appl. Math., 49 (1996), 85-123. doi: 10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2
[24]	A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Commun. Pure Appl. Math., 51 (1998), 625-661. doi: 10.1002/(SICI)1097-0312(199806)51:6<625::AID-CPA3>3.0.CO;2-9
[25]	A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), 1323-1366.
[26]	M. Bertsch, R. Dal Passo, H. Garcke, et al. The thin viscous flow equation in higher space dimensions, Adv. Differ. Equations, 3 (1998), 417-440.
[27]	J. Bischof, D. Scherer, S. Herminghaus, et al. Dewetting modes of thin metallic films: Nucleation of holes and spinodal dewetting, Phys. Rev. Lett., 77 (1996), 1536-1539. doi: 10.1103/PhysRevLett.77.1536
[28]	S. Boatto, L. P. Kadanoff, P. Olla, Traveling-wave solutions to thin-film equations, Phys. Rev. E, 48 (1993), 4423-4431.
[29]	D. Bonn, J. Eggers, J. Indekeu, et al. Wetting and spreading, Rev. Mod. Phys., 81 (2009), 739-805. doi: 10.1103/RevModPhys.81.739
[30]	M. Bowen, J. Hulshof, J. R. King, Anomalous exponents and dipole solutions for the thin film equation, SIAM J. Appl. Math., 62 (2001), 149-179. doi: 10.1137/S0036139900366936
[31]	R. J. Braun, Dynamics of the tear film, Annu. Rev. Fluid Mech., 44 (2012), 267-297. doi: 10.1146/annurev-fluid-120710-101042
[32]	L. Brusch, H. Kühne, U. Thiele, et al. Dewetting of thin films on heterogeneous substrates: Pinning versus coarsening, Phys. Rev. E, 66 (2002), 011602.
[33]	L. N. Brush and S. H. Davis, A new law of thinning in foam dynamics, J. Fluid Mech., 534 (2005), 227-236. doi: 10.1017/S0022112005004763
[34]	J. P. Burelbach, S. G. Bankoff, S. H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech., 195 (1988), 463-494. doi: 10.1017/S0022112088002484
[35]	J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Commun. Math. Phys., 225 (2002), 551-571. doi: 10.1007/s002200100591
[36]	A.-M. Cazabat and G. Guena, Evaporation of macroscopic sessile droplets, Soft Matter, 6 (2010), 2591-2612. doi: 10.1039/b924477h
[37]	S. J. Chapman, P. H. Trinh, T. P. Witelski, Exponential asymptotics for thin film rupture, SIAM J. Appl. Math., 73 (2013), 232-253. doi: 10.1137/120872012
[38]	K.-S. Chou and S.-Z. Du, Estimates on the Hausdorff dimension of the rupture set of a thin film, SIAM J. Math. Anal., 40 (2008), 790-823. doi: 10.1137/070685348
[39]	K.-S. Chou and Y.-C. Kwong, Finite time rupture for thin films under van der Waals forces, Nonlinearity, 20 (2007), 299-317. doi: 10.1088/0951-7715/20/2/004
[40]	P. Constantin, T. F. Dupont, R. E. Goldstein, et al. Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E, 47 (1993), 4169-4181. doi: 10.1103/PhysRevE.47.4169
[41]	P. Constantin, T. Elgindi, H. Nguyen, et al. On singularity formation in a Hele-Shaw model, Commun. Math. Phys., 363 (2018), 139-171. doi: 10.1007/s00220-018-3241-6
[42]	R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Mod. Phys., 81(2009), 1131-1198. doi: 10.1103/RevModPhys.81.1131
[43]	M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112. doi: 10.1103/RevModPhys.65.851
[44]	B. Dai, L. G. Leal, A. Redondo, Disjoining pressure for nonuniform thin films, Phys. Rev. E, 78 (2008), 061602.
[45]	S. B. Dai, On a mean field model for 1D thin film droplet coarsening, Nonlinearity, 23 (2010), 325-340. doi: 10.1088/0951-7715/23/2/006
[46]	S. B. Dai, On the Ostwald ripening of thin liquid films, Commun. Math. Sci., 9 (2011), 143-160. doi: 10.4310/CMS.2011.v9.n1.a7
[47]	S. B. Dai and R. L. Pego, Universal bounds on coarsening rates for mean-field models of phase transitions, SIAM J. Math. Anal., 37 (2005), 347-371. doi: 10.1137/040618047
[48]	R. Dal Passo, H. Garcke, G. Grün, On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., 29 (1998), 321-342. doi: 10.1137/S0036141096306170
[49]	R. Dal Passo, L. Giacomelli, A. Shishkov, The thin film equation with nonlinear diffusion, Commun. Part. Diff. Eq., 26 (2001), 1509-1557. doi: 10.1081/PDE-100107451
[50]	M. C. Dallaston, M. A. Fontelos, D. Tseluiko, et al. Discrete self-similarity in interfacial hydrodynamics and the formation of iterated structures, Phys. Rev. Lett., 120 (2018), 34505.
[51]	M. C. Dallaston, D. Tseluiko, Z. Zheng, et al. Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows, Nonlinearity, 30 (2017), 2647-2666. doi: 10.1088/1361-6544/aa6eb3
[52]	A. A. Darhuber and S. M. Troian, Principles of microfluidic actuation by modulation of surface stresses, Annu. Rev. Fluid Mech., 37 (2005), 425-455. doi: 10.1146/annurev.fluid.36.050802.122052
[53]	A. A. Darhuber, S. M. Troian, S. M. Miller, et al. Morphology of liquid microstructures on chemically patterned surfaces, J. Appl. Phys., 87 (2000), 7768-7775. doi: 10.1063/1.373452
[54]	A. A. Darhuber, S. M. Troian, W. W. Reisner, Dynamics of capillary spreading along hydrophilic microstripes, Phys. Rev. E, 64 (2001), 031603.
[55]	B. Davidovitch, E. Moro, H. A. Stone, Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations, Phys. Rev. Lett., 95 (2005), 244505.
[56]	P. G. de Gennes, Wetting - statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863. doi: 10.1103/RevModPhys.57.827
[57]	P. G. de Gennes, F. Brochard-Wyart, D. Quere, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, Springer Verlag, New York, 2003.
[58]	M. di Bernardo, C. J. Budd, A. R. Champneys, et al. Piecewise-smooth Dynamical Systems, volume 163 of Applied Mathematical Sciences, Springer-Verlag London, Ltd., London, 2008.
[59]	J. A. Diez, A. G. Gonzalez, L. Kondic, On the breakup of fluid rivulets, Phys. Fluids, 21 (2009), 082105.
[60]	J. A. Diez and L. Kondic, On the breakup of fluid films of finite and infinite extent, Phys. Fluids, 19 (2007), 072107.
[61]	M. A. Durán-Olivencia, R. S. Gvalani, S. Kalliadasis, et al. Instability, rupture and fluctuations in thin liquid films: Theory and computations, J. Stat. Phys., 174 (2019), 579-604. doi: 10.1007/s10955-018-2200-0
[62]	M. Dziwnik, M. Korzec, A. Münch, et al. Stability analysis of unsteady, nonuniform base states in thin film equations, Multiscale Model. Sim., 12 (2014), 755-780. doi: 10.1137/130943352
[63]	E. Weinan, Principles of Multiscale Modeling, Cambridge University Press, Cambridge, 2011.
[64]	J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Rev. Mod. Phys., 69 (1997), 865-929. doi: 10.1103/RevModPhys.69.865
[65]	J. Eggers and M. A. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity, 22 (2009), R1-R44.
[66]	J. Eggers and M. A. Fontelos, Singularities: Formation, Structure, and Propagation, Cambridge University Press, 2015.
[67]	J. Eggers and L. Pismen, Nonlocal description of evaporating drops, Phys. Fluids, 22 (2010), 112101.
[68]	C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662
[69]	S. Engelnkemper, S. V. Gurevich, H. Uecker, et al. Continuation for thin film hydrodynamics and related scalar problems. In Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, pages 459-501. Springer, 2019.
[70]	R. Enright, N. Miljkovic, J. L. Alvarado, et al. Dropwise condensation on micro-and nanostructured surfaces, Nanosc. Microsc. Therm., 18 (2014), 223-250. doi: 10.1080/15567265.2013.862889
[71]	P. L. Evans, J. R. King, A. Münch, Intermediate-asymptotic structure of a dewetting rim with strong slip, Applied Mathematics Research Express, 2006 (2006), 25262.
[72]	P. L. Evans, J. R. King, A. Münch, The structure of a dewetting rim with strong slip: the long-time evolution, Multiscale Model. Sim., 16 (2018), 1365-1391. doi: 10.1137/15M1051221
[73]	R. Fetzer, K. Jacobs, A. Munch, et al. New slip regimes and the shape of dewetting thin liquid films, Phys. Rev. Lett., 95 (2005), 127801.
[74]	J. Fischer and G. Grün, Existence of positive solutions to stochastic thin-film equations, SIAM J. Math. Anal., 50 (2018), 411-455. doi: 10.1137/16M1098796
[75]	L. S. Fisher and A. A. Golovin, Nonlinear stability analysis of a two-layer thin liquid film: Dewetting and autophobic behavior, J. Colloid Interf. Sci., 291 (2005), 515-528. doi: 10.1016/j.jcis.2005.05.024
[76]	Y. Gao, H. Ji, J.-G. Liu, et al. A vicinal surface model for epitaxial growth with logarithmic free energy, Discrete & Continuous Dynamical Systems-B, 23 (2018), 4433-4453.
[77]	Y. Gao, J.-G. Liu, X. Y. Lu, Gradient flow approach to an exponential thin film equation: global existence and latent singularity, ESAIM: Control, Optimisation and Calculus of Variations, 25 (2019), 49.
[78]	A. Ghatak, R. Khanna, A. Sharma, Dynamics and morphology of holes in dewetting thin films, J. Colloid Interf. Sci., 212 (1999), 483-494. doi: 10.1006/jcis.1998.6052
[79]	L. Giacomelli, A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane, Appl. Math. Lett., 12 (1999), 107-111.
[80]	L. Giacomelli, M. V. Gnann, F. Otto, Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3, Eur. J. Appl. Math., 24 (2013), 735-760. doi: 10.1017/S0956792513000156
[81]	L. Giacomelli and F. Otto, Rigorous lubrication approximation, Interfaces Free Bound., 5 (2003), 483-529.
[82]	M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions, Birkhäuser Boston, Ltd., Boston, MA, 2010.
[83]	K. Glasner and S. Orizaga, Improving the accuracy of convexity splitting methods for gradient flow equations, J. Comput. Phys., 315 (2016), 52-64. doi: 10.1016/j.jcp.2016.03.042
[84]	K. Glasner, F. Otto, T. Rump, et al. Ostwald ripening of droplets: The role of migration, Eur. J. Appl. Math., 20 (2009), 1-67. doi: 10.1017/S0956792508007559
[85]	K. B. Glasner, Spreading of droplets under the influence of intermolecular forces, Phys. Fluids, 15 (2003), 1837-1842. doi: 10.1063/1.1578076
[86]	K. B. Glasner, Ostwald ripening in thin film equations, SIAM J. Appl. Math., 6 (2008), 473-493.
[87]	K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), 016302.
[88]	K. B. Glasner and T. P. Witelski, Collision versus collapse of droplets in coarsening of dewetting thin films, Physica D, 209 (2005), 80-104. doi: 10.1016/j.physd.2005.06.010
[89]	M. V. Gnann and M. Petrache, The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness, J. Differ. Equations, 265 (2018), 5832-5958. doi: 10.1016/j.jde.2018.07.015
[90]	R. E. Goldstein, A. I. Pesci, M. J. Shelley, Attracting manifold for a viscous topology transition, Phys. Rev. Lett., 75 (1995), 3665-3668. doi: 10.1103/PhysRevLett.75.3665
[91]	C. P. Grant, Spinodal decomposition for the Cahn-Hilliard equation, Commun. Part. Diff. Eq., 18 (1993), 453-490. doi: 10.1080/03605309308820937
[92]	M. B. Gratton and T. P. Witelski, Transient and self-similar dynamics in thin film coarsening, Physica D, 238 (2009), 2380-2394. doi: 10.1016/j.physd.2009.09.015
[93]	H. P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84 (1978), 125-143. doi: 10.1017/S0022112078000075
[94]	G. Grün, K. Mecke, M. Rauscher, Thin-film flow influenced by thermal noise, J. Stat. Phys., 122 (2006), 1261-1291. doi: 10.1007/s10955-006-9028-8
[95]	G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math., 87 (2000), 113-152. doi: 10.1007/s002110000197
[96]	G. Grün and M. Rumpf, Simulation of singularities and instabilities arising in thin film flow, Eur. J. Appl. Math., 12 (2001), 293-320. doi: 10.1017/S0956792501004429
[97]	E. K. O. Hellen and J. Krug, Coarsening of sand ripples in mass transfer models, Phys. Rev. E, 66 (2002), 011304.
[98]	D. Herde, U. Thiele, S. Herminghaus, et al. Driven large contact angle droplets on chemically heterogeneous substrates, EPL, 100 (2012), 16002.
[99]	S. Herminghaus, M. Brinkmann, R. Seemann, Wetting and dewetting of complex surface geometries, Annu. Rev. Mater. Res., 38 (2008), 101-121. doi: 10.1146/annurev.matsci.38.060407.130335
[100]	S. Herminghaus and F. Brochard, Dewetting though nucleation, C. R. Phys., 7 (2006), 1073-1081. doi: 10.1016/j.crhy.2006.10.021
[101]	S. Herminghaus, K. Jacobs, K. Mecke, et al. Spinodal dewetting in liquid crystal and liquid metal films, Science, 282 (1998), 916-919.
[102]	L. M. Hocking, The influence of intermolecular forces on thin fluid layers, Phys. Fluids, 5 (1993), 793-798. doi: 10.1063/1.858627
[103]	C. Huh and L. E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interf. Sci., 35 (1971), 85-101. doi: 10.1016/0021-9797(71)90188-3
[104]	H. J. Hwang and T. P. Witelski, Short-time pattern formation in thin film equations, Discrete and Continuous Dynamical Systems. Series A, 23 (2009), 867-885. doi: 10.3934/dcds.2009.23.867
[105]	J. N. Israelachvili, Intermolecular and Surface Forces, Academic Press, New York, 1992.
[106]	K. Jacobs, R. Seemann, S. Herminghaus, Stability and dewetting of thin liquid films. In Polymer thin films, pages 243-265. World Scientific, 2008.
[107]	H. Ji and T. P. Witelski, Finite-time thin film rupture driven by modified evaporative loss, Physica D, 342 (2017), 1-15. doi: 10.1016/j.physd.2016.10.002
[108]	H. Ji and T. P. Witelski, Instability and dynamics of volatile thin films, Phys. Rev. Fluids, 3 (2018), 024001.
[109]	H. Ji and T. P. Witelski, Steady states and dynamics of a thin-film-type equation with nonconserved mass, Eur. J. Appl. Math., (2019), 1-34.
[110]	H. Jiang and W.-M. Ni, On steady states of van der Waals force driven thin film equations, Eur. J. Appl. Math., 18 (2007), 153-180. doi: 10.1017/S0956792507006936
[111]	S. Kalliadasis, C. Ruyer-Quil, B. Scheid, et al. Falling Liquid Films, volume 176 of Applied Mathematical Sciences, Springer, London, 2012.
[112]	H. S. Kheshgi and L. E. Scriven, Dewetting - nucleation and growth of dry regions, Chem. Eng. Sci., 46 (1991), 519-526. doi: 10.1016/0009-2509(91)80012-N
[113]	J. R. King, Two generalisations of the thin film equation, Math. Comput. Model., 34 (2001), 737-756. doi: 10.1016/S0895-7177(01)00095-4
[114]	J. R. King and M. Bowen, Moving boundary problems and non-uniqueness for the thin film equation, Eur. J. Appl. Math., 12 (2001), 321-356. doi: 10.1017/S0956792501004405
[115]	J. R. King, A. Münch, B. Wagner, Linear stability of a ridge, Nonlinearity, 19 (2006), 2813-2831. doi: 10.1088/0951-7715/19/12/005
[116]	J. R. King, A. Münch, B. A. Wagner, Linear stability analysis of a sharp-interface model for dewetting thin films, J. Eng. Math., 63 (2009), 177-195. doi: 10.1007/s10665-008-9242-2
[117]	G. Kitavtsev, L. Recke, B. Wagner, Centre manifold reduction approach for the lubrication equation, Nonlinearity, 24 (2011), 2347-2369. doi: 10.1088/0951-7715/24/8/010
[118]	G. Kitavtsev, L. Recke, B. Wagner, Asymptotics for the spectrum of a thin film equation in a singular limit, SIAM J. Appl. Dyn. Syst., 11 (2012), 1425-1457. doi: 10.1137/100813488
[119]	R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Commun. Math. Phys., 229 (2002), 275-295.
[120]	R. Konnur, K. Kargupta, A. Sharma, Instability and morphology of thin liquid films on chemically heterogeneous substrates, Phys. Rev. Lett., 84 (2000), 931-934. doi: 10.1103/PhysRevLett.84.931
[121]	M.-A. Y.-H. Lam, L. J. Cummings, L. Kondic, Computing dynamics of thin films via large scale GPU-based simulations, Journal of Computational Physics: X, 2 (2019), 100001.
[122]	E. Lauga, M. P. Brenner, H. A. Stone, Microfluidics: The no-slip boundary condition. In J. Foss, C. Tropea, and A. Yarin, editors, Handbook of Experimental Fluid Dynamics, chapter 19, pages 1-27. Springer, 2007.
[123]	R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. An., 154 (2000), 3-51. doi: 10.1007/PL00004234
[124]	R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, Eur. J. Appl. Math., 11 (2000), 293-351. doi: 10.1017/S0956792599003794
[125]	R. S. Laugesen and M. C. Pugh, Energy levels of steady states for thin-film-type equations, J. Differ. Equations, 182 (2002), 377-415. doi: 10.1006/jdeq.2001.4108
[126]	R. S. Laugesen and M. C. Pugh, Heteroclinic orbits, mobility parameters and stability for thin film type equations, Electron. J. Differ. Eq., 2002 (2002), 1-29.
[127]	R. N. Leach, F. Stevens, S. C. Langford, et al. Dropwise condensation: Experiments and simulations of nucleation and growth of water drops in a cooling system, Langmuir, 22 (2006), 8864-8872. doi: 10.1021/la061901+
[128]	L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes, Cambridge University Press, 2007.
[129]	I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids, 19 (1961), 35-50. doi: 10.1016/0022-3697(61)90054-3
[130]	R. Limary and P. F. Green, Dewetting instabilities in thin block copolymer films: nucleation and growth, Langmuir, 15 (1999), 5617-5622. doi: 10.1021/la981693o
[131]	R. Limary and P. F. Green, Dynamics of droplets on the surface of a structured fluid film: latestage coarsening, Langmuir, 19 (2003), 2419-2424. doi: 10.1021/la026560o
[132]	F. Liu, G. Ghigliotti, J. J. Feng, et al. Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces, J. Fluid Mech., 752 (2014), 39-65. doi: 10.1017/jfm.2014.320
[133]	W. Liu and T. P. Witelski, Steady-states of thin film droplets on chemically heterogeneous substrates, preprint, 2020.
[134]	A. M. Macner, S. Daniel, P. H. Steen, Condensation on surface energy gradient shifts drop size distribution toward small drops, Langmuir, 30 (2014), 1788-1798. doi: 10.1021/la404057g
[135]	J. A. Marqusee and J. Ross, Kinetics of phase transitions: Theory of Ostwald ripening, J. Chem. Phys., 79 (1983), 373-378. doi: 10.1063/1.445532
[136]	L. C. Mayo, S. W. McCue, T. J. Moroney, et al. Simulating droplet motion on virtual leaf surfaces, Roy. Soc. Open Sci., 2 (2015), 140528.
[137]	K. Mecke and M. Rauscher, On thermal fluctuations in thin film flow, Journal of Physics: Condensed Matter, 17 (2005), S3515.
[138]	N. Miljkovic, R. Enright, E. N. Wang, Effect of droplet morphology on growth dynamics and heat transfer during condensation on superhydrophobic nanostructured surfaces, ACS Nano, 6 (2012), 1776-1785. doi: 10.1021/nn205052a
[139]	A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Mathematics, 2 (2017), 479-544. doi: 10.3934/Math.2017.2.479
[140]	V. Mitlin, Dewetting revisited: New asymptotics of the film stability diagram and the metastable regime of nucleation and growth of dry zones, J. Colloid Interf. Sci., 227 (2000), 371-379. doi: 10.1006/jcis.2000.6792
[141]	V. S. Mitlin, Dewetting of a solid surface: analogy with spinodal decomposition, J. Colloid Interf. Sci., 156 (1993), 491-497. doi: 10.1006/jcis.1993.1142
[142]	V. S. Mitlin and N. V. Petviashvili, Nonlinear dynamics of dewetting: kinetically stable structures, Phys. Lett. A, 192 (1994), 323-326. doi: 10.1016/0375-9601(94)90213-5
[143]	R. Mukherjee and A. Sharma, Instability, self-organization and pattern formation in thin soft films, Soft matter, 11 (2015), 8717-8740. doi: 10.1039/C5SM01724F
[144]	A. Münch and B. Wagner, Contact-line instability of dewetting thin films, Physica D, 209 (2005), 178-190. doi: 10.1016/j.physd.2005.06.027
[145]	A. Münch, B. Wagner, T. P. Witelski, Lubrication models with small to large slip lengths, J. Eng. Math., 53 (2005), 359-383. doi: 10.1007/s10665-005-9020-3
[146]	N. Murisic and L. Kondic, On evaporation of sessile drops with moving contact lines, J. Fluid Mech., 679 (2011), 219-246. doi: 10.1017/jfm.2011.133
[147]	T. G. Myers, Thin films with high surface tension, SIAM Review, 40 (1998), 441-462. doi: 10.1137/S003614459529284X
[148]	C. Neto, K. Jacobs, R. Seemann, et al. Correlated dewetting patterns in thin polystyrene films, Journal of Physics: Condensed Matter, 15 (2003), S421-S426.
[149]	C. Neto, K. Jacobs, R. Seemann, et al. Satellite hole formation during dewetting: experiment and simulation, Journal of Physics: Condensed Matter, 15 (2003), 3355-3366. doi: 10.1088/0953-8984/15/19/334
[150]	B. Niethammer, Derivation of the LSW-theory for Ostwald ripening by homogenization methods, Arch. Ration. Mech. An., 147 (1999), 119-178. doi: 10.1007/s002050050147
[151]	B. Niethammer, The mathematics of Ostwald ripening. In Geometric analysis and nonlinear partial differential equations, pages 649-663. Springer, Berlin, 2003.
[152]	B. Niethammer and R. L. Pego, Non-self-similar behavior in the LSW theory of Ostwald ripening, J. Stat. Phys., 95 (1999), 867-902. doi: 10.1023/A:1004546215920
[153]	B. Niethammer and R. L. Pego, On the initial-value problem in the Lifshitz-Slyozov-Wagner theory of Ostwald ripening, SIAM J. Math. Anal., 31 (2000), 467-485. doi: 10.1137/S0036141098338211
[154]	B. Niethammer and R. L. Pego, Well-posedness for measure transport in a family of nonlocal domain coarsening models, Indiana U. Math. J., 54 (2005), 499-530. doi: 10.1512/iumj.2005.54.2598
[155]	B. Niethammer and J. J. L. Velázquez, On the convergence to the smooth self-similar solution in the LSW model, Indiana U. Math. J., 55 (2006), 761-794. doi: 10.1512/iumj.2006.55.2854
[156]	A. Novick-Cohen, The Cahn-Hilliard equation. In Handbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., pages 201-228. Elsevier/North-Holland, Amsterdam, 2008.
[157]	A. Novick-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5
[158]	J. R. Ockendon and H. Ockendon, Viscous Flow, Cambridge University, Cambridge, 1995.
[159]	A. Oron and S. G. Bankoff, Dewetting of a heated surface by an evaporating liquid film under conjoining/disjoining pressures, J. Colloid Interf. Sci., 218 (1999), 152-166. doi: 10.1006/jcis.1999.6390
[160]	A. Oron and S. G. Bankoff, Dynamics of a condensing liquid film under conjoining/disjoining pressures, Phys. Fluids, 13 (2001), 1107-1117. doi: 10.1063/1.1355022
[161]	A. Oron, S. H. Davis, S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931
[162]	F. Otto, T. Rump, D. Slepcev, Coarsening rates for a droplet model: rigorous upper bounds, SIAM J. Math. Anal., 38 (2006), 503-529. doi: 10.1137/050630192
[163]	S. B. G. O'Brien and L. W. Schwartz, Theory and Modeling of Thin Film Flows, In Encyclopedia of Surface and Colloid Science, pages 5283-5297. Marcel Dekker, 2002.
[164]	A. A. Pahlavan, L. Cueto-Felgueroso, A. E. Hosoi, et al. Thin films in partial wetting: Stability, dewetting and coarsening, J. Fluid Mech., 845 (2018), 642-681. doi: 10.1017/jfm.2018.255
[165]	A. A. Pahlavan, L. Cueto-Felgueroso, G. H. McKinley, et al. Thin films in partial wetting: internal selection of contact-line dynamics, Phys. Rev. Lett., 115 (2015), 034502.
[166]	D. Peschka, S. Haefner, L. Marquant, et al. Signatures of slip in dewetting polymer films, P. Natl. Acad. Sci. USA, 116 (2019), 9275-9284. doi: 10.1073/pnas.1820487116
[167]	L. M. Pismen, Spinodal dewetting in a volatile liquid film, Phys. Rev. E, 70 (2004), 021601.
[168]	L. M. Pismen and Y. Pomeau, Mobility and interactions of weakly nonwetting droplets, Phys. Fluids, 16 (2004), 2604-2612. doi: 10.1063/1.1758911
[169]	A. Pototsky, M. Bestehorn, D. Merkt, et al. Alternative pathways of dewetting for a thin liquid two-layer film, Phys. Rev. E, 70 (2004), 025201.
[170]	C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1992.
[171]	M. Rauscher and S. Dietrich, Wetting phenomena in nanofluidics, Annu. Rev. Mater. Res., 38 (2008), 143-172. doi: 10.1146/annurev.matsci.38.060407.132451
[172]	G. Reiter, Dewetting of thin polymer films, Phys. Rev. Lett., 68 (1992), 75-78. doi: 10.1103/PhysRevLett.68.75
[173]	S. N. Reznik and A. L. Yarin, Spreading of a viscous drop due to gravity and capillarity on a horizontal or an inclined dry wall, Phys. Fluids, 14 (2002), 118-132. doi: 10.1063/1.1426388
[174]	A. J. Roberts and Z. Li, An accurate and comprehensive model of thin fluid flows with inertia on curved substrates, J. Fluid Mech., 553 (2006), 33-73. doi: 10.1017/S0022112006008640
[175]	N. O. Rojas, M. Argentina, E. Cerda, et al. Inertial lubrication theory, Phys. Rev. Lett., 104 (2010), 187801.
[176]	J. W. Rose, On the mechanism of dropwise condensation, Int. J. Heat Mass Tran., 10 (1967), 755-762. doi: 10.1016/0017-9310(67)90135-4
[177]	J. W. Rose, Dropwise condensation theory and experiment: A review, Journal of Power Energy, 216 (2012), 115-128.
[178]	R. V. Roy, A. J. Roberts, M. E. Simpson, A lubrication model of coating flows over a curved substrate in space, J. Fluid Mech., 454 (2002), 235-261. doi: 10.1017/S0022112001007133
[179]	E. Ruckenstein and R. K. Jain, Spontaneous rupture of thin liquid films, Journal of the Chemical Society-Faraday Transactions II, 70 (1974), 132-147. doi: 10.1039/f29747000132
[180]	K. Rykaczewski, A. T. Paxson, M. Staymates, et al. Dropwise condensation of low surface tension fluids on omniphobic surfaces, Scientific reports, 4 (2015), 4158.
[181]	E. Sander and T. Wanner, Monte Carlo simulations for spinodal decomposition, J. Stat. Phys., 95 (1999), 925-948. doi: 10.1023/A:1004550416829
[182]	E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM J. Appl. Math., 60 (2000), 2182-2202. doi: 10.1137/S0036139999352225
[183]	L. W. Schwartz, Unsteady simulation of viscous thin-layer flows. In P. A. Tyvand, editor, Free surface flows with viscosity, pages 203-233. Computational Mechanics Publications, Boston, 1997.
[184]	L. W. Schwartz, R. V. Roy, R. R. Eley, et al. Dewetting patterns in a drying liquid film, J. Colloid Interf. Sci., 234 (2001), 363-374. doi: 10.1006/jcis.2000.7312
[185]	L. W. Schwartz and D. E. Weidner, Modeling of coating flows on curved surfaces, J. Eng. Math., 29 (1995), 91-103. doi: 10.1007/BF00046385
[186]	R. Seemann, S. Herminghaus, K. Jacobs, Dewetting patterns and molecular forces: a reconciliation, Phys. Rev. Lett., 86 (2001), 5534-5537. doi: 10.1103/PhysRevLett.86.5534
[187]	A. Sharma, Many paths to dewetting of thin films, Eur. Phys. J. E, 12 (2003), 397-407. doi: 10.1140/epje/e2004-00008-5
[188]	A. Sharma and R. Khanna, Pattern formation in unstable thin liquid films, Phys. Rev. Lett., 81 (1998), 3463-3466. doi: 10.1103/PhysRevLett.81.3463
[189]	A. Sharma and G. Reiter, Instability of thin polymer films on coated substrates: rupture, dewetting, and drop formation, J. Colloid Interf. Sci., 178 (1996), 383-399. doi: 10.1006/jcis.1996.0133
[190]	A. Sharma and R. Verma, Pattern formation and dewetting in thin films of liquids showing complete macroscale wetting: From "pancakes" to "swiss cheese", Langmuir, 20 (2004), 10337-10345. doi: 10.1021/la048669x
[191]	D. N. Sibley, A. Nold, N. Savva, et al. A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading, J. Eng. Math., 94 (2015), 19-41.
[192]	V. M. Starov, M. G. Velarde, C. J. Radke, Wetting and Spreading Dynamics, CRC Press, Boca Raton Florida, 2007.
[193]	P. S. Stewart and S. H. Davis, Dynamics and stability of metallic foams: Network modeling, J. Rheol., 56 (2012), 543-574. doi: 10.1122/1.3695029
[194]	P. S. Stewart and S. H. Davis, Self-similar coalescence of clean foams, J. Fluid Mech., 722 (2013), 645-664. doi: 10.1017/jfm.2013.145
[195]	H. A. Stone, A. D. Stroock, A. Ajdari, Engineering flows in small devices: microfluidics toward a lab-on-a-chip, Annu. Rev. Fluid Mech., 36 (2004), 381-411. doi: 10.1146/annurev.fluid.36.050802.122124
[196]	U. Thiele, Open questions and promising new fields in dewetting, Eur. Phys. J. E, 12 (2003), 409-414. doi: 10.1140/epje/e2004-00009-4
[197]	U. Thiele, Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth, Journal of Physics: Condensed Matter, 22 (2010), 084019.
[198]	U. Thiele, Patterned deposition at moving contact lines, Adv. Colloid Interfac., 206 (2014), 399-413. doi: 10.1016/j.cis.2013.11.002
[199]	U. Thiele, Recent advances in and future challenges for mesoscopic hydrodynamic modelling of complex wetting, Colloids and Surfaces A, 553 (2018), 487-495. doi: 10.1016/j.colsurfa.2018.05.049
[200]	U. Thiele, A. J. Archer, L. M. Pismen, Gradient dynamics models for liquid films with soluble surfactant, Phys. Rev. Fluids, 1 (2016), 083903.
[201]	U. Thiele, A. J. Archer, M. Plapp, Thermodynamically consistent description of the hydrodynamics of free surfaces covered by insoluble surfactants of high concentration, Phys. Fluids, 24 (2012), 102107.
[202]	U. Thiele and E. Knobloch, Driven drops on heterogeneous substrates: Onset of sliding motion, Phys. Rev. Lett., 97 (2006), 204501.
[203]	U. Thiele, M. Mertig, W. Pompe, Dewetting of an evaporating thin liquid film: Heterogeneous nucleation and surface instability, Phys. Rev. Lett., 80 (1998), 2869-2872. doi: 10.1103/PhysRevLett.80.2869
[204]	U. Thiele, M. G. Velarde, K. Neuffer, et al. Film rupture in the diffuse interface model coupled to hydrodynamics, Phys. Rev. E, 64 (2001), 031602.
[205]	D. Tseluiko and D. T. Papageorgiou, Nonlinear dynamics of electrified thin liquid films, SIAM J. Appl. Math., 67 (2007), 1310-1329. doi: 10.1137/060663532
[206]	D. Tseluiko, J. Baxter, U. Thiele, A homotopy continuation approach for analysing finite-time singularities in thin liquid films, IMA J. Appl. Math., 78 (2013), 762-776. doi: 10.1093/imamat/hxt021
[207]	H. B. van Lengerich, M. J. Vogel, P. H. Steen, Coarsening of capillary drops coupled by conduit networks, Phys. Rev. E, 82 (2010), 66312.
[208]	F. Vandenbrouck, M. P. Valignat, A. M. Cazabat, Thin nematic films: metastability and spinodal dewetting, Phys. Rev. Lett., 82 (1999), 2693-2696. doi: 10.1103/PhysRevLett.82.2693
[209]	S. J. VanHook, M. F. Schatz, W. D. McCormick, et al. Long-wavelength surface-tension-driven Bénard convection: experiment and theory, J. Fluid Mech., 345 (1997), 45-78. doi: 10.1017/S0022112097006101
[210]	J. L. Vázquez, The Porous Medium Equation, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.
[211]	A. Vrij, Possible mechanism for spontaneous rupture of thin free liquid films, Discussions of the Faraday Society, 42 (1966), 23-33. doi: 10.1039/df9664200023
[212]	C. Wagner, Theorie der alterung von niedershlagen durch umlosen (Ostwald-Reifung), Z. Elektrochem, 65 (1961), 581-591.
[213]	M. H. Ward, Interfacial thin films rupture and self-similarity, Phys. Fluids, 23 (2011), 062105.
[214]	S. J. Watson, F. Otto, B. Y. Rubinstein, et al. Coarsening dynamics of the convective Cahn-Hilliard equation, Physica D, 178 (2003), 127-148. doi: 10.1016/S0167-2789(03)00048-4
[215]	T. Wei and F. Duan, Interfacial stability and self-similar rupture of evaporating liquid layers under vapor recoil, Phys. Fluids, 28 (2016), 124106.
[216]	G. M. Whitesides, The origins and the future of microfluidics, Nature, 442 (2006), 368-373. doi: 10.1038/nature05058
[217]	M. B. Williams and S. H. Davis, Nonlinear theory of film rupture, J. Colloid Interf. Sci., 90 (1982), 220-228. doi: 10.1016/0021-9797(82)90415-5
[218]	T. P. Witelski, Computing finite-time singularities in interfacial flows. In G. Sabidussi, editor, Modern Methods in Scientific Computing and Applications, NATO ASI series proceedings, pages 451-487. Kluwer, 2002.
[219]	T. P. Witelski and A. J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids, 11 (1999), 2443-2445. doi: 10.1063/1.870138
[220]	T. P. Witelski and A. J. Bernoff, Dynamics of three-dimensional thin film rupture, Physica D, 147 (2000), 155-176. doi: 10.1016/S0167-2789(00)00165-2
[221]	T. P. Witelski, A. J. Bernoff, A. L. Bertozzi, Blowup and dissipation in a critical-case unstable thin film equation, Eur. J. Appl. Math., 15 (2004), 223-256. doi: 10.1017/S0956792504005418
[222]	T. P. Witelski and M. Bowen, ADI schemes for higher-order nonlinear diffusion equations, Appl. Numer. Math., 45 (2003), 331-351. doi: 10.1016/S0168-9274(02)00194-0
[223]	Q. Wu and H. Wong, A slope-dependent disjoining pressure for non-zero contact angles, J. Fluid Mech., 506 (2004), 157-185. doi: 10.1017/S0022112004008420
[224]	W. W. Zhang and J. R. Lister, Similarity solutions for van der Waals rupture of a thin film on a solid substrate, Phys. Fluids, 11 (1999), 2454-2462. doi: 10.1063/1.870110
[225]	L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal., 37 (2000), 523-555.

This article has been cited by:

1.	Yangjin Kim, Hyunji Kang, Gibin Powathil, Hyeongi Kim, Dumitru Trucu, Wanho Lee, Sean Lawler, Mark Chaplain, Dominik Wodarz, Role of extracellular matrix and microenvironment in regulation of tumor growth and LAR-mediated invasion in glioblastoma, 2018, 13, 1932-6203, e0204865, 10.1371/journal.pone.0204865
2.	Yangjin Kim, Junho Lee, Donggu Lee, Hans Othmer, Synergistic Effects of Bortezomib-OV Therapy and Anti-Invasive Strategies in Glioblastoma: A Mathematical Model, 2019, 11, 2072-6694, 215, 10.3390/cancers11020215
3.	Christian Engwer, Christian Stinner, Christina Surulescu, On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, 2017, 27, 0218-2025, 1355, 10.1142/S0218202517400188
4.	Markos Antonopoulos, Dimitra Dionysiou, Georgios Stamatakos, Nikolaos Uzunoglu, Three-dimensional tumor growth in time-varying chemical fields: a modeling framework and theoretical study, 2019, 20, 1471-2105, 10.1186/s12859-019-2997-9
5.	Junho Lee, Donggu Lee, Sean Lawler, Yangjin Kim, Stacey Finley, Role of neutrophil extracellular traps in regulation of lung cancer invasion and metastasis: Structural insights from a computational model, 2021, 17, 1553-7358, e1008257, 10.1371/journal.pcbi.1008257
6.	Stefaan W. Verbruggen, Laoise M. McNamara, 2018, 9780128129524, 157, 10.1016/B978-0-12-812952-4.00006-4
7.	Thomas Hillen, Kevin J. Painter, Magdalena A. Stolarska, Chuan Xue, Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer, 2020, 80, 0303-6812, 275, 10.1007/s00285-020-01473-2
8.	Min-Jhe Lu, Chun Liu, John Lowengrub, Shuwang Li, Complex Far-Field Geometries Determine the Stability of Solid Tumor Growth with Chemotaxis, 2020, 82, 0092-8240, 10.1007/s11538-020-00716-z
9.	Yangjin Kim, Donggu Lee, Junho Lee, Seongwon Lee, Sean Lawler, Eugene Demidenko, Role of tumor-associated neutrophils in regulation of tumor growth in lung cancer development: A mathematical model, 2019, 14, 1932-6203, e0211041, 10.1371/journal.pone.0211041
10.	Raluca Eftimie, Joseph J. Gillard, Doreen A. Cantrell, Mathematical Models for Immunology: Current State of the Art and Future Research Directions, 2016, 78, 0092-8240, 2091, 10.1007/s11538-016-0214-9
11.	S. L. Waters, L. J. Schumacher, A. J. El Haj, Regenerative medicine meets mathematical modelling: developing symbiotic relationships, 2021, 6, 2057-3995, 10.1038/s41536-021-00134-2
12.	Vladimir Simic, Miljan Milosevic, Vladimir Milicevic, Nenad Filipovic, Milos Kojic, A novel composite smeared finite element for mechanics (CSFEM): Some applications, 2022, 09287329, 1, 10.3233/THC-220414
13.	Miloš Kojić, Miljan Milošević, Arturas Ziemys, 2023, 9780323884723, 65, 10.1016/B978-0-323-88472-3.00002-5
14.	Gabriella Bretti, Adele De Ninno, Roberto Natalini, Daniele Peri, Nicole Roselli, Estimation Algorithm for a Hybrid PDE–ODE Model Inspired by Immunocompetent Cancer-on-Chip Experiment, 2021, 10, 2075-1680, 243, 10.3390/axioms10040243
15.	Dimitrios G. Patsatzis, Algorithmic asymptotic analysis: Extending the arsenal of cancer immunology modeling, 2022, 534, 00225193, 110975, 10.1016/j.jtbi.2021.110975
16.	Joseph D. Butner, Prashant Dogra, Vittorio Cristini, Thomas S. Deisboeck, Zhihui Wang, 2023, 9780128216248, 251, 10.1016/B978-0-12-821618-7.00244-3
17.	Jonggul Lee, Donggu Lee, Yangjin Kim, Mathematical model of STAT signalling pathways in cancer development and optimal control approaches, 2021, 8, 2054-5703, 10.1098/rsos.210594
18.	Junho Lee, Jin Su Kim, Yangjin Kim, Stacey Finley, Atorvastatin-mediated rescue of cancer-related cognitive changes in combined anticancer therapies, 2021, 17, 1553-7358, e1009457, 10.1371/journal.pcbi.1009457
19.	Aurelio A. de los Reyes, Yangjin Kim, Optimal regulation of tumour-associated neutrophils in cancer progression, 2022, 9, 2054-5703, 10.1098/rsos.210705
20.	Min-Jhe Lu, Wenrui Hao, Chun Liu, John Lowengrub, Shuwang Li, Nonlinear simulation of vascular tumor growth with chemotaxis and the control of necrosis, 2022, 459, 00219991, 111153, 10.1016/j.jcp.2022.111153
21.	Tingzhe Sun, Multi-scale modeling of hippo signaling identifies homeostatic control by YAP-LATS negative feedback, 2021, 208, 03032647, 104475, 10.1016/j.biosystems.2021.104475
22.	Yangjin Kim, Junho Lee, Chaeyoung Lee, Sean Lawler, Role of senescent tumor cells in building a cytokine shield in the tumor microenvironment: mathematical modeling, 2023, 86, 0303-6812, 10.1007/s00285-022-01850-z
23.	Rebecca M. Crossley, Philip K. Maini, Tommaso Lorenzi, Ruth E. Baker, Traveling waves in a coarse‐grained model of volume‐filling cell invasion: Simulations and comparisons, 2023, 0022-2526, 10.1111/sapm.12635
24.	Anneke S.K. Verbruggen, Elan C. McCarthy, Roisin M. Dwyer, Laoise M. McNamara, Mechanobiological cues to bone cells during early metastasis drive later osteolysis: a computational mechanoregulation framework prediction, 2024, 29499070, 100100, 10.1016/j.mbm.2024.100100

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)