A Discrete Model For Pattern Formation In Volatile Thin Films

M. Malik-Garbi; O. Agam

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 4, page 39-52
  • ISSN: 0973-5348

Abstract

top
We introduce a model, similar to diffusion limited aggregation (DLA), which serves as a discrete analog of the continuous dynamics of evaporation of thin liquid films. Within mean field approximation the dynamics of this model, averaged over many realizations of the growing cluster, reduces to that of the idealized evaporation model in which surface tension is neglected. However fluctuations beyond the mean field level play an important role, and we study their effect on the conserved quantities of the idealized evaporation model. Assuming the cluster to be a fractal, a heuristic approach is developed in order to explain the distinctive increase of the fractal dimension with the cluster size.

How to cite

top

Malik-Garbi, M., and Agam, O.. "A Discrete Model For Pattern Formation In Volatile Thin Films." Mathematical Modelling of Natural Phenomena 7.4 (2012): 39-52. <http://eudml.org/doc/222442>.

@article{Malik2012,
abstract = {We introduce a model, similar to diffusion limited aggregation (DLA), which serves as a discrete analog of the continuous dynamics of evaporation of thin liquid films. Within mean field approximation the dynamics of this model, averaged over many realizations of the growing cluster, reduces to that of the idealized evaporation model in which surface tension is neglected. However fluctuations beyond the mean field level play an important role, and we study their effect on the conserved quantities of the idealized evaporation model. Assuming the cluster to be a fractal, a heuristic approach is developed in order to explain the distinctive increase of the fractal dimension with the cluster size.},
author = {Malik-Garbi, M., Agam, O.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {viscous fingering; diffusion limited aggregation; reaction-diffusion},
language = {eng},
month = {7},
number = {4},
pages = {39-52},
publisher = {EDP Sciences},
title = {A Discrete Model For Pattern Formation In Volatile Thin Films},
url = {http://eudml.org/doc/222442},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Malik-Garbi, M.
AU - Agam, O.
TI - A Discrete Model For Pattern Formation In Volatile Thin Films
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/7//
PB - EDP Sciences
VL - 7
IS - 4
SP - 39
EP - 52
AB - We introduce a model, similar to diffusion limited aggregation (DLA), which serves as a discrete analog of the continuous dynamics of evaporation of thin liquid films. Within mean field approximation the dynamics of this model, averaged over many realizations of the growing cluster, reduces to that of the idealized evaporation model in which surface tension is neglected. However fluctuations beyond the mean field level play an important role, and we study their effect on the conserved quantities of the idealized evaporation model. Assuming the cluster to be a fractal, a heuristic approach is developed in order to explain the distinctive increase of the fractal dimension with the cluster size.
LA - eng
KW - viscous fingering; diffusion limited aggregation; reaction-diffusion
UR - http://eudml.org/doc/222442
ER -

References

top
  1. M. Elbaum, S.G. Lipson. How does a thin wetted film dry up?Phys. Rev. Lett., 72 (1994), 3562–3565.  
  2. S.G. Lipson. Pattern formation in drying water films. Physica Scripta, T67 (1996), 63–66.  
  3. I. Leizerson, S.G. Lipson, A.V. Lyushnin. Finger instability in wetting-dewetting phenomena. Langmuir, 20 (2004), 291–194.  
  4. S. G. Lipson. A thickness transition in evaporating water films. Phase Transitions, 77 (2004), 677–688.  
  5. I. Leizerson, S.G. Lipson. How does a thin volatile film move?Langmuir, 20 (2004), 8423–8425.  
  6. N. Samid-Merzel, S.G. Lipson, D.S. Tannhauser. Pattern formation in drying water films. Phys. Rev., E 57 (1998), 2906–2913.  
  7. G. Taylor, P.G. Saffman. A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Quart. J. Mech. Appl. Math., 12 (1959), 265–279.  Zbl0091.18502
  8. T.A. Witten, L.M. Sander. Diffusin-limited aggragation. Phys. Rev., B 27 (1983), 5686–5697.  
  9. J. Mathiesen, I. Procaccia, H. L. Swinney, M. Thrasher. The universality class of diffusion-limited aggregation and viscous-limited aggregation. Europhys. Lett., 76 (2006) No. 2, 257–263.  
  10. A. Arneodo, Y. Couder, G. Grasseau, V. Hakim, M. Rabaud. Uncovering the analytical Saffman-Taylor finger in unstable viscous fingering and diffusion-limited aggregation. Phys. Rev. Lett., 63 (1989), 984–987.  
  11. A. Arneodo, J. Elezgaray, M. Tabard, F. Tallet. Statistical analysis of off-lattice diffusion-limited aggregates in channeland sector geometries. Phys. Rev., E 53 (1996), 6200–6223.  
  12. E. Somfai, R.C. Ball, J.P. DeVita, L.M. Sander. Diffusion-limited aggregation in channel geometry. Phys. Rev., E 68 (2003), 020401 
  13. M.B. Hastings, L.S. Levitov. Laplacian growth as one-dimensional turbulence. Physica, D116 (1998), 244–252.  Zbl0962.76542
  14. O. Agam. Viscous fingering in volatile thin films. Phys. Rev., E 79 (2009), 021603.  
  15. H. Diamant, O. Agam. Localized Rayleigh instability in evaporation fronts. Phys. Rev. Lett., 104 (2010), 047801.  
  16. V.M. Entov, P.I. Étingof. Some exact sdolutions of the thin-sheet stamping problem. Fluid Dyn., 27 (1992), 169–176.  Zbl0793.76035
  17. M. Doi. 2nd quantization representation for classical many-particle system. J. Phys., A 9 (1976), 1465–1477.  
  18. L. Peliti. Path integral approach to birth-death processes on a lattice. J. Physique, 46 (1985), 1469–1483.  
  19. B. P. Lee. Renormalization-group calculation for the reaction kA → ∅. Phys., A27 (1994), 2633–2652.  
  20. J. Cardy, U. C. Tauber. Theory of branching and annihilating random walksPhys. Rev Lett., 77 (1996), 4780–4783.  
  21. Here the Hamiltonian which defines the evolution does not account for the constraint that A particle cannot be born on a site ocuupied by B particle. This constraint can be taken into account by replcing the term with , where Θ(x) is the haviside function and ϵ is a positive infintesimal number. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.