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An Introduction to Similarity

  • Richard Ghez
Chapter

Abstract

With iterated error functions we have already greatly expanded our vocabulary of solutions of the diffusion equation. The question is, however, whether or not any useful further expansion should be expected. The study of physical symmetries provides a partial answer, and that study is called similarity. In essence, we shall find that the linear diffusion equation in one dimension has two types of solutions: superpositions either of iterated error functions or of trigonometric functions, otherwise known as Fourier series. The strength of similarity methods lies in their applicability to nonlinear problems as well. This chapter continues with more physical examples, including exact solutions for the kinetics of certain first-order phase transformations and of diffusion problems where the diffusivity depends on the field. In contrast, Chaps. 7 and 8 describe deductive methods for the solution of diffusion problems.

Keywords

Diffusion Equation Similarity Solution Diffusion Problem Nonlinear Diffusion Stefan Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    H. Schlichting, Boundary-Layer Theory 6th ed. (McGraw-Hill, New York, 1968).MATHGoogle Scholar
  2. 2.
    G. Birkhoff, Hydrodynamics: A Study in Logic, Fact and Similitude, 2nd ed. (Princeton University Press, Princeton, NJ, 1960).MATHGoogle Scholar
  3. 3.
    N. N. Lebedev, Special Functions and their Applications (reprinted by Dover, New York, 1972).MATHGoogle Scholar
  4. 4.
    G. W. Bluman and J. D. Cole, Similarity Methods for Differential Equations (Springer-Verlag, Berlin, 1974).CrossRefMATHGoogle Scholar
  5. 5.
    W. F. Ames, Nonlinear Partial Differential Equations in Engineering, Vols. I and II (Academic Press, New York, 1965 and 1972).MATHGoogle Scholar
  6. 6.
    L. Dresner, Similarity Solutions of Nonlinear Partial Differential Equations (Pitman Advanced Publishing Program, Boston, 1983).MATHGoogle Scholar
  7. 7.
    L. Boltzmann, “Zur Integration der Diffusionsgleichung bei variabeln Diffusionscoefficienten, ” Ann. der Phys. 53, 959 (1894).ADSCrossRefMATHGoogle Scholar
  8. 8.
    G. Rosen, “Galilean Invariance and the General Co variance of Non-relativistic Laws, ” Amer. J. Phys. 40, 683 (1972).ADSCrossRefGoogle Scholar
  9. 9.
    G. W. Evans II, E. Isaacson, and J. K. L. MacDonald, “Stefan-like Problems, ” Quart. Appl. Math. 8, 312 (1950).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    B. Boley, “A General Starting Solution for Melting and Solidifying Slabs, ” Int. J. Engng. Sci. 6, 89 (1968).CrossRefMATHGoogle Scholar
  11. 11.
    M. B. Small and R. Ghez, “Growth and Dissolution Kinetics of III-V Heterostructures Formed by LPE, ” J. Appl. Phys. 50, 5322 (1979).ADSCrossRefGoogle Scholar
  12. 12.
    R. Ghez, “Expansions in Time for the Solution of One-Dimensional Stefan Problems of Crystal Growth, ” Int. J. Heat Mass Transfer 23, 425 (1980).CrossRefGoogle Scholar
  13. 13.
    F. C. Frank, “Radially Symmetric Phase Growth Controlled by Diffusion, ” Proc. Roy. Soc. 201A, 586 (1950).ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    C. Zener, “Theory of Growth of Spherical Precipitates from Solid Solution, ” J. Appl. Phys. 20, 950 (1949).ADSCrossRefGoogle Scholar
  15. 15.
    R. J. Schaeffer and M. E. Glicksman, “Fully Time-Dependent Theory for the Growth of Spherical Crystal Nuclei, ” J. Cryst. Growth 5, 44 (1969).ADSCrossRefGoogle Scholar
  16. 16.
    S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam, 1962). [Reprinted by Dover, New York.]MATHGoogle Scholar
  17. 17.
    L. S. Darken, “Diffusion, Mobility and their Interrelation through Free Energy in Binary Metallic Systems, ” Trans. AIME 175, 184 (1948).Google Scholar
  18. 18.
    J. R. Philip, “General Method of Exact Solution of the Concentration-Dependent Diffusion Equation, ” Australian J. Phys. 13, 1 (1960); see also ibid, p. 13.ADSMathSciNetMATHGoogle Scholar
  19. 19.
    B. Tuck, “Some Explicit Solutions to the Non-linear Diffusion Equation, ” J. Phys. D9, 1559 (1976).ADSGoogle Scholar
  20. 20.
    B. Tuck, Introduction to Diffusion in Semiconductors, pp. 199–203 (Peter Peregrinus, Stevenage, 1974).Google Scholar
  21. 21.
    M. Hillert, “Diffusion and Interface Control of Reactions in Alloys, ” Metall. Trans. 6A, 5 (1975).CrossRefGoogle Scholar
  22. 22.
    F. M. d’Heurle and P. Gas, “Kinetics of Formation of Silicides: A Review, ” J. Mater. Res. 1, 205 (1986).ADSCrossRefGoogle Scholar
  23. 23.
    L. L. Chang and A. Koma, “Interdiffusion between GaAs and AlAs, ” Appl. Phys. Lett. 29, 138 (1976).ADSCrossRefGoogle Scholar
  24. 24.
    K. S. Seo, P. K. Bhattacharya, G. P. Kothiyal, and S. Hong, “Interdiffusion and Wavelength Modification in In0.53Ga0.47As/In0.52Al0.48As Quantum Wells by Lamp Annealing, ” Appl. Phys. Lett. 49, 966 (1986).ADSCrossRefGoogle Scholar
  25. 25.
    G. V. Kidson, “Some Aspects of the Growth of Diffusion Layers in Binary Systems, ” J. Nucl. Mater. 3, 21 (1961).ADSCrossRefGoogle Scholar
  26. 26.
    C. Wagner, “The Evaluation of Data Obtained with Diffusion Couples of Binary Single-Phase and Multiphase Systems, ” Acta Met. 17, 99 (1969).CrossRefGoogle Scholar
  27. 27.
    D. S. Williams, R. A. Rapp, and J. P. Hirth, “Phase Suppression in the Transient Stages of Interdiffusion in Thin Films, ” Thin Solid Films 142, 65 (1986).CrossRefGoogle Scholar
  28. 28.
    R. E. Pattle, “Diffusion from an Instantaneous Point Source with a Concentration-Dependent Coefficient, ” Quart. J. Mech. and Appl. Math. 7, 407 (1959).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    R. H. Boyer, “On Some Solutions of a Non-linear Diffusion Equation, ” J. Math. & Phys. 40, 41 (1961).MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. II (Academic Press, New York, 1967).Google Scholar
  31. 31.
    Ya. B. Zel’dovich and G. I. Barenblatt, “The Asymptotic Properties of Self-Modelling Solutions of the Nonstationary Gas Filtration Equations, ” Dokl. Akad. Nauk SSSR 118, 4 (1958). [Engl. transl. in Sov. Phys. Doklady 3, 44 (1958).]Google Scholar

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© Springer Science+Business Media New York 2001

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  • Richard Ghez

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