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Adaptive mesh refinement for simulation of thin film flows

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Abstract

We present a robust and accurate numerical method for simulating gravity-driven, thin-film flow problems. The convection term in the governing equation is treated by a semi-implicit, essentially non-oscillatory scheme. The resulting nonlinear discrete equation is solved using a nonlinear full approximation storage multigrid algorithm with adaptive mesh refinement techniques. A set of representative numerical experiments are presented. We show that the use of adaptive mesh refinement reduces computational time and memory compared to the equivalent uniform mesh results. Our simulation results are consistent with previous experimental observations.

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References

  1. Thoroddsen ST, Mahadevan L (1997) Experimental study of coating flows in a partially-filled horizontally rotating cylinder. Exp Fluids 23:1–13

    Article  Google Scholar 

  2. Ruschak KJ (1985) Coating flows. Annu Rev Fluid Mech 17:65–89

    Article  ADS  Google Scholar 

  3. Bertozzi AL, Münch A, Fanton X, Cazabat AM (1998) Contact line stability and “undercompressive shocks” in driven thin film flow. Phys Rev Lett 81:5169–5172

    Article  ADS  Google Scholar 

  4. Sur J, Bertozzi AL, Behringer RP (2003) Reverse undercompressive shock structures in driven thin film flow. Phys Rev Lett 90:126105

    Article  ADS  Google Scholar 

  5. Magdy AE, Alla AE, Shereen ME (2013) Stokes’ first problem for a thermoelectric Newtonian fluid. Meccanica 48:1161–1175

    Article  MathSciNet  Google Scholar 

  6. Huppert HE (1982) Flow and instability of a viscous current down a slope. Nature 300:427–429

    Article  ADS  Google Scholar 

  7. Troian SM, Herbolzheimer E, Safran SA, Joanny JF (1989) Fingering instability of driven spreading films. Europhys Lett 10:25–30

    Article  ADS  Google Scholar 

  8. Spaid MA, Homsy GM (1996) Stability of Newtonian and viscoelastic dynamic contact lines. Phys Fluids 8:460–478

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Bertozzi AL, Brenner MP (1997) Linear stability and transient growth in driven contact lines. Phys Fluids 9:530–539

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Kataoka DE, Troian SM (1997) A theoretical study of instabilities at the advancing front of thermally driven coating films. J Colloid Interface Sci 15:350–362

    Article  Google Scholar 

  11. Diez JA, Kondic L (2001) Contact line instabilities of thin liquid films. Phys Rev Lett 86:632–635

    Article  ADS  Google Scholar 

  12. Goddard BD, Nold A, Savva N, Pavliotis GA, Kalliadasis S (2012) General dynamical density functional theory for classical fluids. Phys Rev Lett 109:120603

    Article  ADS  Google Scholar 

  13. Khayat RE, Kim KT, Delosquer S (2004) Influence of inertia, topography and gravity on transient axisymmetric thin-film flow. Int J Numer Methods Fluids 45:391–419

    Article  MATH  MathSciNet  Google Scholar 

  14. Aziz RC, Hashim L, Alomari AK (2011) Thin film flow and heat transfer on an unsteady stretching sheet with internal heating. Meccanica 46:349–357

    Article  MATH  MathSciNet  Google Scholar 

  15. Savva N, Kalliadasis S, Pavliotis A (2010) Two-dimensional droplet spreading over random topographical substrates. Phys Rev Lett 104:084501

    Article  ADS  Google Scholar 

  16. Savva N, Kalliadasis S (2012) Influence of gravity on the spreading of two-dimensional droplets over topographical substrates. J Eng Math 73:3–16

    Article  MathSciNet  Google Scholar 

  17. Savva N, Pavliotis A (2009) Two-dimensional droplet spreading over topographical substrates. Phys Fluids 21:092102

    Article  ADS  Google Scholar 

  18. Sibley DN, Savva N, Kalliadasis S (2012) Slip or not slip a methodical examination of the interface formation model using two-dimensional droplet spreading on a horizontal planar substrate as a prototype system. Phys Fluids 24:082105

    Article  ADS  Google Scholar 

  19. Christov CI, Pontes J, Walgraef D, Velarde MG (1997) Implicit time-splitting for fourth-order parabolic equations. Comput Methods Appl Mech Eng 148:209–224

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Karlsen KH, Lie KA (1999) An unconditionally stable splitting for a class of nonlinear parabolic equations. IMA J Numer Anal 19:609–635

    Article  MATH  MathSciNet  Google Scholar 

  21. Daniels N, Ehret P, Gaskell PH, Thompson HM, Decré M (2001) Multigrid methods for thin liquid film spreading flows. In: Proceedings of the first international conference on computational fluid dynamics, pp 279–284

    Google Scholar 

  22. Myers TG, Charpin JPF, Chapman SJ (2002) The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Phys Fluids 14:2788–2803

    Article  ADS  MathSciNet  Google Scholar 

  23. Momoniat E (2011) Numerical investigation of a third-order ODE from thin film flow. Meccanica 46:313–323

    Article  MATH  MathSciNet  Google Scholar 

  24. Kondic L (2003) Instabilities in gravity driven flow of thin fluid films. SIAM Rev 45:95–115

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Witelski TP, Bowen M (2003) ADI schemes for higher-order nonlinear diffusion equations. Appl Numer Math 45:331–351

    Article  MATH  MathSciNet  Google Scholar 

  26. Gaskell PH, Jimack PK, Sellier M, Thompson HM (2004) Efficient and accurate time adaptive multigrid simulations of droplet spreading. Int J Numer Methods Fluids 14:1161–1186

    Article  Google Scholar 

  27. Gaskell PH, Jimack PK, Sellier M, Thompson HM, Wilson MCT (2004) Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. J Fluid Mech 509:253–280

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Gaskell PH, Jimack PK, Sellier M, Thompson HM (2006) Flow of evaporating, gravity-driven thin liquid films over topography. Phys Fluids 18:031601

    Article  MathSciNet  Google Scholar 

  29. Kim J, Sur J (2005) A hybrid method for higher-order nonlinear diffusion equations. Commun Korean Math Soc 20:179–193

    Article  MATH  MathSciNet  Google Scholar 

  30. Myers TG, Lombe M (2006) The importance of the Coriolis force on axisymmetric horizontal rotating thin film flows. Chem Eng Process 45:90–98

    Article  Google Scholar 

  31. Kim J (2006) Adaptive mesh refinement for thin-film equations. J Korean Phys Soc 49:1903–1907

    Google Scholar 

  32. Lee YC, Thompson HM, Gaskell PH (2007) An efficient adaptive multigrid algorithm for predicting thin film flow on surfaces containing localised topographic features. Comput Fluids 37:838–855

    Article  Google Scholar 

  33. Vellingiri R, Savva N, Kalliadasis S (2011) Droplet spreading on chemically heterogeneous substrates. Phys Rev E 84:036305

    Article  ADS  Google Scholar 

  34. Sun P, Russell RD, Xu J (2007) A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem. J Comput Phys 224:1021–1048

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Ha Y, Kim YJ, Myers TG (2008) On the numerical solution of a driven thin film equation. J Comput Phys 227:7246–7263

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Lee YC, Thompson HM, Gaskell PH (2008) The efficient and accurate solution of continuous thin film flow over surface patterning and past occlusions. Int J Numer Methods Fluids 56:1375–1381

    Article  MATH  Google Scholar 

  37. Sellier M, Lee YC, Thompson HM, Gaskell PH (2009) Thin film flow on surfaces containing arbitrary occlusions. Comput Fluids 38:171–182

    Article  MATH  MathSciNet  Google Scholar 

  38. Wang G, Rothmayer AP (2009) Thin water films driven by air shear stress through roughness. Comput Fluids 38:235–246

    Article  MATH  Google Scholar 

  39. Sellier M, Panda S (2010) Beating capillarity in thin film flows. Int J Numer Methods Fluids 63:431–448

    MATH  Google Scholar 

  40. Veremieiev S, Thompson HM, Lee YC, Gaskell PH (2010) Inertial thin film flow on planar surfaces featuring topography. Comput Fluids 39:431–450

    Article  MATH  Google Scholar 

  41. Lee YC, Thompson HM, Gaskell PH (2011) Dynamics of thin film flow on flexible substrate. Chem Eng Process 50:525–530

    Article  Google Scholar 

  42. Li Y, Lee H-G, Yoon D, Hwang W, Shin S, Ha Y, Kim JS (2011) Numerical studies of the fingering phenomena for the thin film equation. Int J Numer Methods Fluids 67:1358–1372

    Article  MATH  MathSciNet  Google Scholar 

  43. Hu B, Kieweg SL (2012) The effect of surface tension on the gravity-driven thin film flow of Newtonian and power-law fluids. Comput Fluids 64:83–90

    Article  MathSciNet  Google Scholar 

  44. Shu CW, Osher S (1989) Efficient implementation of essentially non-oscillatory shock capturing schemes II. J Comput Phys 83:32–78

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. Almgren AS, Bell JB, Colella P, Howell LH, Welcome ML (1998) A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J Comput Phys 142:1–6

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Berger MJ, Rigoutsos I (1991) An algorithm for point clustering and grid generation. IEEE Trans Syst Man Cybern 21:1278–1286

    Article  Google Scholar 

  47. The Mathworks, Inc., Matlab. http://www.mathworks.com

  48. Trottenberg U, Oosterlee C, Schüller A (2001) Multigrid. Academic Press, London

    MATH  Google Scholar 

  49. Goodwin R, Homsy GM (1991) Viscous flow down a slope in the vicinity of a contact line. Phys Fluids A 3:515–528

    Article  ADS  MATH  Google Scholar 

  50. Lin TS, Kondic L (2010) Thin films flowing down inverted substrates: two dimensional flow. Phys Fluids 22:052105

    Article  ADS  Google Scholar 

  51. Caselles V, Catte F, Coll T, Dibos F (1993) A geometric model for active contours in image processing. Numer Math 66:1–31

    Article  MATH  MathSciNet  Google Scholar 

  52. Chan T, Vese L (2001) Active contours without edges. IEEE Trans Image Process 2:266–277

    Article  ADS  Google Scholar 

  53. Li Y, Kim J (2010) A fast and accurate numerical method for medical image segmentation. J Korea SIAM 14:201–210

    MathSciNet  Google Scholar 

  54. Li Y, Kim J (2011) Multiphase image segmentation with a phase-field model. Comput Math Appl 62:737–745

    Article  MATH  MathSciNet  Google Scholar 

  55. Li Y, Kim J (2012) An unconditionally stable numerical method for bimodal image segmentation. Appl Math Comput 219:3083–3090

    Article  MathSciNet  Google Scholar 

  56. Diez JA, Kondic L, Bertozzi A (2000) Global models for moving contact lines. Phys Rev E 63:011208

    Article  ADS  Google Scholar 

  57. Gratton R, Diez JA, Thomas LP, Marino B, Betelu S (1996) Quasi-self-similarity for wetting drops. Phys Rev E 53:3563–3572

    Article  ADS  Google Scholar 

  58. Cazabat AM, Stua MAC (1986) Dynamics of wetting: effects of surface roughness. J Phys Chem 90:5845–5849

    Article  Google Scholar 

  59. Mourik SV (2002) Numerical modelling of the dynamic contact angle. Master’s thesis, University of Groningen

  60. Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81:739–805

    Article  ADS  Google Scholar 

  61. Savva N, Kalliadasis S (2011) Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys Lett 94:64004

    Article  ADS  Google Scholar 

  62. Bertozzi AL, Shearer M (2000) Existence of undercompressive traveling waves in thin film equations. SIAM J Math Anal 32:194–213

    Article  MATH  MathSciNet  Google Scholar 

  63. Levy R, Shearer M (2004) Comparison of two dynamic contact line models for driven thin liquid films. Eur J Appl Math 15:625–642

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The corresponding author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0023794). The authors greatly appreciate the reviewers for their constructive comments and suggestions, which have improved the quality of this paper.

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  1. Department of Computational Science and Engineering, Yonsei University, Seoul, 120-749, Republic of Korea

    Yibao Li

  2. Department of Mathematics, Korea University, Seoul, 136-713, Republic of Korea

    Darae Jeong & Junseok Kim

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  1. Yibao Li
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  2. Darae Jeong
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Correspondence to Junseok Kim.

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Li, Y., Jeong, D. & Kim, J. Adaptive mesh refinement for simulation of thin film flows. Meccanica 49, 239–252 (2014). https://doi.org/10.1007/s11012-013-9788-6

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