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One-dimensional slow invariant manifolds for spatially homogenous reactive systems

J. Chem. Phys. 131, 024118 (2009); doi:10.1063/1.3171613

Published 14 July 2009

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Ashraf N. Al-Khateeb,1 Joseph M. Powers,1,2 Samuel Paolucci,1 Andrew J. Sommese,2 Jeffrey A. Diller,2 Jonathan D. Hauenstein,2 and Joshua D. Mengers1
1Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana 46556-5637, USA
2Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 465556-4618, USA
A reactive system's slow dynamic behavior is approximated well by evolution on manifolds of dimension lower than that of the full composition space. This work addresses the construction of one-dimensional slow invariant manifolds for dynamical systems arising from modeling unsteady spatially homogeneous closed reactive systems. Additionally, the relation between the systems' slow dynamics, described by the constructed manifolds, and thermodynamics is clarified. It is shown that other than identifying the equilibrium state, traditional equilibrium thermodynamic potentials provide no guidance in constructing the systems' actual slow invariant manifolds. The construction technique is based on analyzing the composition space of the reactive system. The system's finite and infinite equilibria are calculated using a homotopy continuation method. The slow invariant manifolds are constructed by calculating attractive heteroclinic orbits which connect appropriate equilibria to the unique stable physical equilibrium point. Application of the method to several realistic reactive systems, including a detailed hydrogen-air kinetics model, reveals that constructing the actual slow invariant manifolds can be computationally efficient and algorithmically easy. ©2009 American Institute of Physics
History: Received 9 March 2009; accepted 17 June 2009; published 14 July 2009
Permalink: http://link.aip.org/link/?JCPSA6/131/024118/1
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KEYWORDS and PACS

Keywords
PACS
  • 82.20.Fd
    Collision theories and trajectory models of chemical kinetics
  • 82.60.-s
    Chemical thermodynamics
  • YEAR: 2009

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ISSN:
0021-9606 (print)   1089-7690 (online)
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REFERENCES (56)
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For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. J. H. Seinfeld and S. N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (Wiley, New York, 1998).
  2. A. L. Kuharsky and A. L. Fogelson, Biophys. J. 80, 1050 (2001).
  3. J. F. Griffiths, Prog. Energy Combust. Sci. 21, 25 (1995).
  4. J. M. Powers, J. Propul. Power 22, 1217 (2006).
  5. Z. Ren and S. B. Pope, Combust. Flame 147, 243 (2006).
  6. T. Turanyi, T. Berces, and S. Vajda, Int. J. Chem. Kinet. 21, 83 (1989).
  7. G. Li, A. Tomlin, H. Rabitz, and J. Toth, J. Chem. Phys. 101, 1172 (1994).
  8. L. Petzold and W. Zhu, AIChE J. 45, 869 (1999).
  9. T. Lu and C. K. Law, Proc. Combust. Inst. 30, 1333 (2005).
  10. U. Maas and S. B. Pope, Combust. Flame 88, 239 (1992).
  11. V. Bykov, V. Gol'dshtein, and U. Maas, Combust. Theory Modell. 12, 389 (2008).
  12. S. H. Lam and D. A. Goussis, Proc. Combust. Inst. 22, 931 (1988).
  13. S. H. Lam, Combust. Sci. Technol. 89, 375 (1993).
  14. S. H. Lam and D. A. Goussis, Int. J. Chem. Kinet. 26, 461 (1994).
  15. M. R. Roussel and S. J. Fraser, J. Chem. Phys. 93, 1072 (1990).
  16. M. R. Roussel and S. J. Fraser, J. Chem. Phys. 94, 7106 (1991).
  17. M. J. Davis and R. T. Skodje, J. Chem. Phys. 111, 859 (1999).
  18. A. N. Gorban, I. V. Karlin, and A. Y. Zinovyev, Phys. Rep. 396, 197 (2004).
  19. D. Lebiedz, J. Chem. Phys. 120, 6890 (2004).
  20. A. N. Gorban and I. V. Karlin, Chem. Eng. Sci. 58, 4751 (2003).
  21. Z. Ren, S. B. Pope, A. Vladimirsky, and J. M. Guckenheimer, J. Chem. Phys. 124, 114111 (2006).
  22. F. Creta, A. Adrover, S. Cerbelli, M. Valorani, and M. Giona, J. Phys. Chem. A 110, 13447 (2006).
  23. F. Creta, A. Adrover, S. Cerbelli, M. Valorani, and M. Giona, J. Phys. Chem. A 110, 13463 (2006).
  24. A. Zagaris, H. G. Kaper, and T. J. Kaper, J. Nonlinear Sci. 14, 59 (2004).
  25. D. A. Goussis, M. Valorani, F. Creta, and H. N. Najm, Prog. Comput. Fluid Dyn. 5, 316 (2005).
  26. S. Singh, J. M. Powers, and S. Paolucci, J. Chem. Phys. 117, 1482 (2002).
  27. J. Warnatz, U. Maas, and R. W. Dibble, Combustion (Springer, Berlin, 1999).
  28. V. Reinhardt, M. Winckler, and D. Lebiedz, J. Phys. Chem. A 112, 1712 (2008).
  29. J. C. Keck and D. Gillespie, Combust. Flame 17, 237 (1971).
  30. S. Ugarte, Y. Gao, and H. Metghalchi, Int. J. Thermodyn. 8, 43 (2005).
  31. Z. Ren, S. B. Pope, A. Vladimirsky, J. M. Guckenheimer, and M. John, Proc. Combust. Inst. 31, 473 (2007).
  32. H. G. Kaper and T. J. Kaper, Physica D 165, 66 (2002).
  33. M. Valorani, F. Creta, D. A. Goussis, J. C. Lee, and H. N. Najm, Combust. Flame 146, 29 (2006).
  34. S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1984).
  35. I. Müller and W. Weiss, Entropy and Energy: A Universal Competition (Springer, Berlin, 2005).
  36. I. Prigogine and R. Defay, Chemical Thermodynamics (Longmans, London, 1954).
  37. W. J. Vincenti and C. H. Kruger, Introduction to Physical Gas Dynamics (Wiley, New York, 1965).
  38. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, 1985).
  39. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes (Interscience, New York, 1967).
  40. J. M. Powers and S. Paolucci, Am. J. Phys. 76, 848 (2008).
  41. L. Perko, Differential Equations and Dynamical Systems (Springer, New York, 2001).
  42. A. J. Sommese and C. W. Wampler, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science (World Scientific, Hackensack, NJ, 2005).
  43. S. J. Fraser, J. Chem. Phys. 116, 1277 (2002).
  44. D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, BERTINI, software for numerical algebraic geometry, available at http://www.nd.edu/~sommese/bertini.
  45. R. J. Kee, F. M. Rupley, and J. A. Miller, Sandia National Laboratories Report No. SAND87–8215B, 1992.
  46. A. Morgan, Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems (Prentice Hall, Englewood Cliffs, NJ, 1987).
  47. J. Guckenheimer and P. Holmes, Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
  48. D. B. Shear, J. Chem. Phys. 48, 4144 (1968).
  49. A. A. Andronov, Qualitative Theory of Second Order Dynamical Systems (Wiley, New York, 1973).
  50. D. L. Baulch, C. T. Bowman, C. J. Cobos, R. A. Cox, T. Just, J. A. Kerr, M. J. Pilling, D. Stocker, J. Troe, W. Tsang, R. W. Walker, and J. Warnatz, J. Phys. Chem. Ref. Data 34, 757 (2005).
  51. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer-Verlag, New York, 1992).
  52. S. Lefschetz, Differential Equations: Geometric Theory (Interscience, New York, 1963).
  53. J. A. Miller, R. E. Mitchell, M. D. Smooke, and R. J. Kee, Proc. Combust. Inst. 19, 181 (1982).
  54. M. D. Smooke, J. A. Miller, and R. J. Kee, Combust. Sci. Technol. 34, 79 (1983).
  55. F. Dabireau, B. Cuenot, O. Vermorel, and T. Poinsot, Combust. Flame 135, 123 (2003).
  56. J. M. Powers and S. Paolucci, AIAA J. 43, 1088 (2005).

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