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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Anomalous Heat Diffusion in a Boundary Layer: A Fractionary Blasius's System

Journal of Applied Nonlinear Dynamics 12(2) (2023) 245--256 | DOI:10.5890/JAND.2023.06.004

Thiago Bissiatte Monteiro, Alexandre C. L. Almeida, Ad\'{e}lcio C. Oliveira

Departamento de Estatística, F'{i}sica e Matem'{a}tica, Universidade Federal de S~{a}o Jo~{a}o del Rei, Ouro Branco, C.P. 131, MG, Brazil

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Abstract

A model of an anomalous boundary layer over a flat plate was presented through fractional calculus. It is a generalization of Blasius's system, including a fractional derivative order in the heat equation. A nonlinear differential system with contour condition is usually difficult to solve, in this case, the system has a fractional differential term, therefore a semi-analytical approximation method was used combined with the initial value problem approximation by sequential parameter optimization method. Besides, it was shown that the derivative order has a significant influence on the boundary layer thermal shape and causes a bigger change in the temperature gradient curve.

Acknowledgments

The authors acknowledge José Eloy Ottoni for the helpful comments and suggestions. The authors gratefully acknowledge the support of Brazilian agency Fundação de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG) through grant No. APQ-01366-16. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and by National Council for Scientific and Technological Development – CNPq.

References

  1. [1]  Einstein, A. (1905), Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annals of Physics, 17, 549.
  2. [2]  Brown, R. (1828), ``A brief account of microscopical observations made on the particles contained in the pollen of plants" in London and Edinburgh Philosophical Magazine and Journal of Science, 4, 161-173.
  3. [3]  Thiele, T.N. (1880), French version: Sur la Compensation de quelques Erreurs quasi-systematiques par la Methode des moindres Carres. Reitzel, Copenhague.
  4. [4]  Thiele, T.N. (1903), Theory of Observations. Dayton, London. Reprinted in Annals of Mathematical Statistics, 2, 165-308, 1931.
  5. [5]  Bachelier, L. (1900), Theorie de la speculation, Annales Scientifiques de l'Ecole Normale Supérieuré, Sér, 3,17, 21-86
  6. [6]  Torres Jr, M.S. and Figueiredo, J.M.A. (2003), Probability amplitude structure of Fokker–Plank equation, Physica A: Statistical Mechanics and its Applications, 329, 68-80.
  7. [7]  Oliveira, A.C. and Amarante-Segundo, G.S. (2009), Tunnel effect as a hidden variable theory test, Physica A: Statistical Mechanics and its Applications, 388(8), 1413-1418.
  8. [8]  Oliveira, A.C., Peixoto de Faria, J.G., and Nemes, M.C. (2006), Quantum-classical transition of the open quartic oscillator: the role of the environment, Physical Review E, 73, 046207.
  9. [9]  Angelo, R.M. (2008), Low-resolution measurements induced classicality, arXiv0809.4616.
  10. [10]  Oliveira, A.C., Junior, Z.T.O., and Correia, N.S. (2013), Complementarity and classical limit of quantum mechanics: Energy measurement aspects, arXiv preprint, arXiv:1307.0528.
  11. [11]  Sornette, D. (2003), Why Stock Markets Crash: Critical Events in Complex Financial System, Princeton University Press.
  12. [12]  Tsallis, C. and Lenzi, E. K. (2002), Anomalous diffusion: nonlinear fractional Fokker–Planck equation, Chemical Physics, 284, 341-347.
  13. [13]  Tsallis, C. and Bukman, D.J. (1996), Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis, Physical Review E, 54, R2197(R).
  14. [14]  Combe, G., Richefeu, V., Stasiak, M., and Atman, A.F. (2015), Experimental validation of nonextensive scaling law in confined granular media, Physical Review Letters, 115, 238301.
  15. [15]  Klafter, J. and Zumofen, G. (1994), Scale-invariant motion in intermittent chaotic systems, Physical Review E, 49, 4873.
  16. [16]  Muskat, M. (1937), The flow of homogenous fluids through porous media, International Series in Physics, New York: McGraw- Hill.
  17. [17]  Peng, C.K., Mietus, J., Hausdorff, J.M., Havlin, S., Stanley, H.E., and Goldberger, A.L. (1993), Long-range anticorrelations and non-Gaussian behavior of the heartbeat, Physical Review Letters, 70, 1343-1346.
  18. [18]  Miramontes, O., DeSouza, O., Paiva, L.R., Marins, A., and Orozco, S. (2014), Lévy Flights and Self-Similar Exploratory Behaviour of Termite Workers: Beyond Model Fitting, Plos One, 9, e111183.
  19. [19]  Paiva, L.R , Silva, H.S., Ferreira, S.C., and Martins, M.L. (2013), Multiscale model for the effects of adaptive immunity suppression on the viral therapy of cancer, Physical Biology (Online), 10, 025005.
  20. [20]  Brito, R.Y., Fabrino, D.L., and Oliveira, A.C. (2016), Estudo de Vibraçoes Mecânicas em Microtúbulos Celulares, in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 10020.
  21. [21]  Paiva, L.R., Binny, C., Ferreira Jr., S.C., and Martins, M.L. (2009), A multiscale mathematical model for oncolytic virotherapy, Cancer Research, 69(3), 1205.
  22. [22]  Hidalgo, A. and Tello, L. (2017), Computers and Fluids, doi:10.1016/j.compfluid.2017.07.019.
  23. [23]  Wegler, U. (2004), Diffusion of seismic waves in a thick layer: Theory and application to Vesuvius volcano, Journal of Geophysical Research, 109, B07303, doi:10.1029/2004JB003048.
  24. [24]  Hughes, A.L.H. and Armitage, P.J. (2010), Particle transport in evolving protoplanetary disks: implications for results from stardust, The Astrophysical Journal, 719(2), p.1633.
  25. [25]  Machado, J.T. and Galhano, A.M. (2018), A fractional calculus perspective of distributed propeller design, Communication Nonlinear Science and Numerical Simulation, 55 174-182
  26. [26]  Carpinteri, A., Chiaia, B., and Cornetti, P. (2001), On the mechanics of quasi-brittle materials with a fractal microstructure, Computer Methods in Applied Mechanics and Engineering, 191, 3-19.
  27. [27]  MYERS, T.G. (1998), Thin films with high surface tension, Society for Industrial and Applied Mathematics, 40(3), pp. 441-462.
  28. [28]  Myers, T.G. (1996), Surface tension driven thin film flows, in The Mechanics of Thin Film Coatings, Wiley.
  29. [29]  Almeida, A.C.L. and Oliveira, A.C. (2018), Diffusion processes modeled with higher order differential terms, Far East Journal of Applied Mathematics, 100, 1-10.
  30. [30] Bagley, R.L. and Torvik, P.J. (1986), On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 3, 133.
  31. [31]  Machado, J.A.T. (1997), Analysis and design of fractional-order digital control systems, Journal Systems Analysis, Modelling, Simulation, 27, 107.
  32. [32]  Machado, J.A.T. and Azenha, A. (1998), IEEE International Workshop on Advanced Motion Control, Coimbra, 548-553.
  33. [33]  Machado, J.A.T. (2002), System modeling and control through fractional-order algorithms, Nonlinear Dynamics, Chaos, Control and Their Applications to Engineering Sciences, 4, 99.
  34. [34]  Ozaktas, H.M., Arikan, O., Kutay, M.A., and Bozdagi, G. (1996), Digital computation of the fractional Fourier transform, IEEE Transactions on Signal Processing, 44, 2141.
  35. [35]  Oustaloup, A. (1981), Fractional order sinusoidal oscillators: Optimization and their use in highly linear FM modulation, IEEE Transactions on Circuits and Systems, 28, 1007.
  36. [36]  Mathieu, B., Lay, L.L., and Oustaloup, A. (1996), Identification of non integer order systems in the time domain, IEEESMC/ IMACS Symposium on Control, Optimization and Supervision, Lille, 952-956.
  37. [37]  Loverro, A. (2004), Fractional Calculus: History, Definitions and Applications for the Engineer, Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A., May 8.
  38. [38]  Evangelista, L. and Lenzi, E. (2018), Fractional Diffusion Equations and Anomalous Diffusion, Cambridge: Cambridge University Press. DOI:10.1017/9781316534649.
  39. [39]  Du, M., Wang, Z., and Zu, H. (2013), Measuring memory with the order of fractional derivative, Scientific Reports, 3, 431.
  40. [40]  Tateishi, A.A., Ribeiro, H.V., and Lenzi, E.K. (2017), The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5.
  41. [41]  Fu, H., Wu, G.C., Yang, G., and Huang, L.L. (2021), Continuous time random walk to a general fractional Fokker–Planck equation on fractal media, The European Physical Journal Special Topics, 230, pages3927–3933.
  42. [42]  Fan, Q., Wu, G.C., and Fu, H. (2021), A note on function space and boundedness of the general fractional integral in continuous time random walk, Journal of Nonlinear Mathematical Physics, 29, 95–102.
  43. [43]  Ertürk, V.S. and Momani, S. (2008), Numerical Solutions of two forms of Blasius equation on a half-infinite domain, Journal of Algorithms and Computational Technology, 2, 359–370.
  44. [44]  Yang, W., Chen, X., Meng, Y., Zhang, X., and Shiyun, M. (2020), Numerical solutions of unsteady boundary layer flow with a time-space fractional constitutive relationship, Symmetry, 12, 1446.
  45. [45]  Blasius, H. (1908), Grenzschichten in Flüssigkeiten mit kleiner Reibung, Zeitschrift für Mathematik und Physik, 56, 1-37; 60, 397-398. English translation in NACA TM 1256.
  46. [46]  Carvalho, M.D. and Ottoni, J.E. (2018), Introdução ao Cálculo Fracionário com aplicações, Revista de Matemática da Universidade Federal de Ouro Preto, 5, p. 32-56.
  47. [47]  Teodoro, G.S., Oliveira, D.S., and Oliveira, E.C. (2018), Sobre derivadas fracionárias, Revista Brasileira de Ensino de F´ısica, 40(2), e2307.
  48. [48]  Oliveira, A.C. and Almeida, A.C.L. (2019), Numerical solution of boundary layer equations based on optimization: the Ostrach and Blasius models, International Journal of Non-Linear Mechanics, 113, 103-111.
  49. [49]  W.H., et al. (2007), ``Section 18.1. The Shooting Method", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press. ISBN 978-0-521-88068-8.
  50. [50]  Oliveira, A.C. (2019), Using the parameter optimization method for solving differential equations with contour conditions: The nonlinear Euler-Bernoulli beam, Discontinuity, Nonlinearity, and Complexity, 8(4) 447-458.
  51. [51]  Oliveira, A.C. and Almeida, A.C.L. (2018), Solução numérica da equação de Landau-Levich para alturas de filmes finos, Anais do XXI ENMC – e IX ECTM, Buzios, RJ.
  52. [52]  Landau, L. and Levich, B. (1942), Dragging of a liquid by a moving plate, Acta Physicochimica URSS, 17(42), 42-54.
  53. [53]  Bergman, T.L., Incropera, F.P., DeWitt, D.P., and Lavine, A.S. (2011), Fundamentals of Heat and Mass Transfer, 7th Edition, John Wiley \& Sons Inc.
  54. [54]  Schlichting, H. and Gersten, K. (2000), Boundary-Layer Theory. 8th Edition, Springer-Verlag, Berlin.
  55. [55]  Ostrach, S. (1953), An analysis of Laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, NACA Report 1111.
  56. [56]  Riley, N. (1975), Unsteady laminar boundary layers, SIAM Review, 17(2), 274-297.
  57. [57]  Stewartson, K. (1960), The theory of unsteady laminar boundary layers Advances in Applied Mechanics, 6, 1-37.
  58. [58]  Oldham, K.B. and Spanier, J. (2006), The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications.
  59. [59]  Zaslavsky, G.M. (2002), Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371, 461-580.
  60. [60]  Metzler, R. and Klafter, J. (2000), The random walk's guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339, 1-77.
  61. [61]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego.
  62. [62]  Aslan, E., Kürkçü, Ö.K., and Sezer, M. (2019), A fast numerical method for fractional partial differential equations, Advances in Difference Equations, 452.
  63. [63]  Morgado, M.L. , Ford, N.J., and Lima, P.M. (2013), Analysis and numerical methods for fractional differential equations with delay, Journal of Computational and Applied Mathematics, 252, 159-168.
  64. [64]  Holmes, W.R. (2019), Subdiffusive dynamics lead to depleted particle densities near cellular borders, Biophysical Journal, 116(8), 1538-1546.
  65. [65]  Kozima, H. (2020), The ``anomalous heat effect” is a normal event in the cold fusion phenomenon, International Journal of Hydrogen Energy, 45, 38, 19577-19582.
  66. [66]  Kitamura, A., Takahashi, A., Takahashi, K., Seto, R., Hatano, T., Iwamura, Y., Itoh, T., Kasagi, J., Nakamura, M., Uchimura, M., and Takahashi, H. (2018), Excess heat evolution from nanocomposite samples under exposure to hydrogen isotope gases, International Journal of Hydrogen Energy, 43, 16,187–16,200.
  67. [67]  Chamon, C. and Nayak, C. (2002), Anomalous quantum diffusion at the superfluid-insulator transition, Physics Review B, 66, 094506.
  68. [68] Shor, Y., Yahel, E., and Makov, G. (2012), Short range order in anomalous liquid metals, Journal of Non-Crystalline Solids, 358, Issues 18–19, 2012, 2687-2694.
  69. [69]  Giordanengo, B., Benazzi, N., Vinckel, J., Gasser, J.G., and Roubi, L. (1999), Thermal conductivity of liquid metals and metallic alloys, Journal of Non-Crystalline Solids, 250–252, Part 1, 377-383, ISSN 0022-3093.
  70. [70]  Belashchenko, D.K. (2009), Application of the embedded atom model to liquid metals: liquid sodium, High Temperature, 47, 494–507. https://doi.org/10.1134/S0018151X09040063
  71. [71]  Ernst, D., Köhler, J., and Weiss, M. (2004), Probing the type of anomalous diffusion with single-particle tracking, Physical Chemistry Chemical Physics, 16, (17).
  72. [72]  WWeiss, M., Elsner, M., Kartberg, F., and Nilsson, T. (2004), Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells, Biophysical Journal, 87(5), 3518-3524.

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