- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Principal bifurcations and symmetries in the emergence of reaction-diffusion-advection patterns on finite domains

Arik Yochelis* and Moshe Sheintuch†

  • Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

  • *Present address: Landa Labs, P.O. Box 4060, Nes Ziona 74140, Israel. arik.yochelis@landalabs.com
  • †cermsll@technion.ac.il
PDF

Phys. Rev. E 80, 056201 – Published 2 November, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.056201

Abstract

Pattern formation mechanisms of a reaction-diffusion-advection system, with one diffusivity, differential advection, and (Robin) boundary conditions of Danckwerts type, are being studied. Pattern selection requires mapping the domains of coexistence and stability of propagating or stationary nonuniform solutions, which for the general case of far from instability onsets, is conducted using spatial dynamics and numerical continuations. The selection is determined by the boundary conditions which either preserve or destroy the translational symmetry of the model. Accordingly, we explain the criterion and the properties of stationary periodic states if the system is bounded and show that propagation of nonlinear waves (including solitary) against the advective flow corresponds to coexisting family that emerges nonlinearly from a distinct oscillatory Hopf instability. Consequently, the resulting pattern selection is qualitatively different from the symmetric finite wavenumber Turing or Hopf instabilities.

View figure in article

Article Text

References (63)

  1. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).
  2. R. Kapral and K. Showalter, Chemical Waves and Patterns (Kluwer, Dordrecht, 1995).
  3. P. K. Maini, K. J. Painter, and H. N. P. Chau, J. Chem. Soc., Faraday Trans. 93, 3601 (1997).
  4. I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos (Oxford University Press, New York, 1998).
  5. J. Keener and J. Sneyd, Mathematical Physiology (Springer-Verlag, New York, 1998).
  6. J. D. Murray, Mathematical Biology (Springer-Verlag, New York, 2002).
  7. L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics (Springer-Verlag, Berlin, 2006).
  8. R. B. Hoyle, Pattern Formation: An Introduction to Methods (Cambridge University Press, Cambridge, 2006).
  9. E. Knobloch, in Nonlinear Dynamics and Chaos: Where Do We Go from Here?, edited by S. J. Hogan (IOP, London, 2002).
  10. A. R. Champneys, Physica D 112, 158 (1998).
  11. P. D. Woods and A. R. Champneys, Physica D 129, 147 (1999).
  12. D. Gomila, A. J. Scroggie, and W. J. Firth, Physica D 227, 70 (2007).
  13. J. Burke and E. Knobloch, Chaos 17, 037102 (2007).
  14. J. Burke, A. Yochelis, and E. Knobloch, SIAM J. Appl. Dyn. Syst. 7, 651 (2008).
  15. J. H. P. Dawes, SIAM J. Appl. Dyn. Syst. 7, 186 (2008).
  16. A. Yochelis and A. Garfinkel, Phys. Rev. E 77, 035204(R) (2008).
  17. D. J. B. Lloyd, B. Sandstede, D. Avitabile, and A. R. Champneys, SIAM J. Appl. Dyn. Syst. 7, 1049 (2008).
  18. S. M. Houghton and E. Knobloch, Phys. Rev. E 80, 026210 (2009).
  19. E. Knobloch, Nonlinearity 21, T45 (2008), and references therein.
  20. A. B. Rovinsky and M. Menzinger, Phys. Rev. Lett. 70, 778 (1993).
  21. P. Huerre and P. A. Monkewitz, Annu. Rev. Fluid Mech. 22, 473 (1990).
  22. A. B. Rovinsky and M. Menzinger, Phys. Rev. Lett. 69, 1193 (1992).
  23. S. Ponce Dawson, A. Lawniczak, and R. Kapral, J. Chem. Phys. 100, 5211 (1994).
  24. S. P. Kuznetsov, E. Mosekilde, G. Dewel, and P. Borckmans, J. Chem. Phys. 106, 7609 (1997).
  25. M. Sheintuch, Physica D 102, 125 (1997).
  26. R. A. Satnoianu, J. H. Merkin, and S. K. Scott, Physica D 124, 345 (1998).
  27. P. Andresén, M. Bache, E. Mosekilde, G. Dewel, and P. Borckmanns, Phys. Rev. E 60, 297 (1999).
  28. J. R. Bamforth, S. Kalliadasis, J. H. Merkin, and S. K. Scott, Phys. Chem. Chem. Phys. 2, 4013 (2000).
  29. O. A. Nekhamkina, A. A. Nepomnyashchy, B. Y. Rubinstein, and M. Sheintuch, Phys. Rev. E 61, 2436 (2000).
  30. R. A. Satnoianu and M. Menzinger, Phys. Rev. E 62, 113 (2000).
  31. J. R. Bamforth, S. Kalliadasis, J. H. Merkin, and S. K. Scott, Phys. Chem. Chem. Phys. 3, 1435 (2001).
  32. O. Nekhamkina and M. Sheintuch, Phys. Rev. E 66, 016204 (2002).
  33. O. Nekhamkina and M. Sheintuch, Phys. Rev. E 68, 036207 (2003).
  34. I. Kiss, J. H. Merkin, S. K. Scott, and P. L. Simon, Q. J. Mech. Appl. Math. 57, 467 (2004).
  35. E. H. Flach, S. Schnell, and J. Norbury, Phys. Rev. E 76, 036216 (2007).
  36. D. A. Vasquez, J. Meyer, and H. Suedhoff, Phys. Rev. E 78, 036109 (2008).
  37. M. C. Cross, Phys. Rev. Lett. 57, 2935 (1986).
  38. M. C. Cross and E. Y. Kuo, Physica D 59, 90 (1992).
  39. J. M. Chomaz, Phys. Rev. Lett. 69, 1931 (1992).
  40. S. M. Tobias, M. R. E. Proctor, and E. Knobloch, Physica D 113, 43 (1998).
  41. H. Sevčíková, M. Marek, and S. C. Müller, Science 257, 951 (1992).
  42. O. Nekhamkina, B. Y. Rubinstein, and M. Sheintuch, AIChE J. 46, 1632 (2000).
  43. M. Kærn, M. Menzinger, R. Satnoianu, and A. Hunding, Faraday Discuss. 120, 295 (2002).
  44. H. Nagahara, Y. Ma, Y. Takenaka, R. Kageyama, and K. Yoshikawa, Phys. Rev. E 80, 021906 (2009).
  45. H. Yamada, T. Nakagaki, R. E. Baker, and P. K. Maini, J. Math. Biol. 54, 745 (2007).
  46. Y. Khazan and L. M. Pismen, Phys. Rev. Lett. 75, 4318 (1995).
  47. F. Borgogno, P. D’Odorico, F. Laio, and L. Ridolfi, Rev. Geophys. 47, RG1005 (2009).
  48. A. Uppal, W. H. Ray, and A. B. Poore, Chem. Eng. Sci. 29, 967 (1974).
  49. A. Yochelis, Y. Tintut, L. L. Demer, and A. Garfinkel, New J. Phys. 10, 055002 (2008).
  50. J. D. Crawford and E. Knobloch, Annu. Rev. Fluid Mech. 23, 341 (1991).
  51. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
  52. Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer-Verlag, New York, 1995).
  53. L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II (World Scientific, Singapore, 2001).
  54. E. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. E. Oldeman, B. Sandstede, and X. Wang, AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HOMCONT), http://indy.cs.concordia.ca/auto/
  55. R. A. Satnoianu, P. K. Maini, and M. Menzinger, Physica D 160, 79 (2001).
  56. A. Yochelis, E. Knobloch, Y. Xie, Z. Qu, and A. Garfinkel, EPL 83, 64005 (2008).
  57. J. Kosek, H. Ševčíková, and M. Marek, J. Phys. Chem. 99, 6889 (1995).
  58. M. Kærn and M. Menzinger, Phys. Rev. E 65, 046202 (2002).
  59. A. Yochelis and M. Sheintuch, Phys. Chem. Chem. Phys. 11, 9210 (2009).
  60. F. Zhang, M. Mangold, and A. Kienle, Chem. Eng. Sci. 61, 7161 (2006).
  61. D. A. Smith and R. M. Simmons, Biophys. J. 80, 45 (2001).
  62. G. A. Truskey, F. Yuan, and D. F. Katz, Transport Phenomena in Biological Systems (Pearson Prentice-Hall, NJ, 2004).
  63. A. V. Kuznetsov and K. Hooman, Int. J. Heat Mass Transfer 51, 5695 (2008).

Outline

Information

Phys. Rev. E 80, 056201– Published 2 November, 2009
  • Received 27 October 2008

© 2009 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation