THERMAL SCIENCE

International Scientific Journal

Zhaoyang Wang

,

Liancun Zheng

,

Lianxi Ma

,

Goong Chen

ANOMALOUS DIFFUSION AND HEAT TRANSFER ON COMB STRUCTURE WITH ANISOTROPIC RELAXATION IN FRACTAL POROUS MEDIA

ABSTRACT
A kind of anomalous diffusion and heat transfer on a comb structure with anisotropic relaxation are studied, which can be used to model many problems in bio-logic and nature in fractal porous media. The Hausdorff derivative is introduced and new governing equations is formulated in view of fractal dimension. Numerical solutions are obtained and the Fox H-function analytical solutions is given for special cases. The particles spatial-temporal evolution and the mean square displacement vs. time are presented. The effects of backbone and finger relaxation parameters, and the time fractal parameter are discussed. Results show that the mean square displacement decreases with the increase of backbone parameter or the decrease of finger relaxation parameter in a short of time, but they have little effect on mean square displacement in a long period. Particularly, the mean square displacement has time dependence in the form of tα/2 (0 < α ≤ 1)when t>τ, which indicates that the diffusion is an anomalous sub-diffusion and heat transfer.
KEYWORDS
Hausdorff derivative, anomalous diffusion and heat transfer, anisotropic relaxation, particles transport, MSD
PAPER SUBMITTED: 2020-01-13
PAPER REVISED: 2020-03-02
PAPER ACCEPTED: 2020-03-12
PUBLISHED ONLINE: 2020-04-04
DOI REFERENCE: https://doi.org/10.2298/TSCI200113153W
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Issue 1, PAGES [733 - 742]
REFERENCES
  1. Iomin, A., M´endez, V., Horsthemke, W.: Fractional Dynamics in Comb-Like Structures., Singapore, World Scientific (2018)
  2. Baskin, E., Iomin, A.: Superdiffusion on a comb structure. Phys. Rev. Lett. 93(2009), pp.120603.
  3. Iomin, A.: Superdiffusion of cancer on a comb structure. J. Phys. Conf. Ser. 7(2005), pp. 57-67.
  4. Iomin, A.: Toy model of fractional transport of cancer cells due to self-entrapping. Phys. Rev. E 73(2006), pp. 061918.
  5. Iomin, A., Zaburdaev, V., Pfohl, T.: Reaction front propagation of actin polymerization in a comb-reaction system. Chaos Soliton Fractals 92 (2016), pp. 115-122.
  6. Iomin, A.: Richardson diffusion in neurons. Phys. Rev. E 100 (2019), pp. 010104.
  7. Milovanov, A. V., Iomin, A.: Subdiffusive Lévy flights in quantum nonlinear Schrödinger lattices with algebraic power nonlinearity. Phys. Rev. E 99(2019), pp.052223.
  8. Iomin, A.: Subdiffusion on a fractal comb. Phys. Rev. E 83 (2011), pp. 052106.
  9. Sandev, T., Iomin, A., Kantz, H.: Fractional diffusion on a fractal grid comb. Phys. Rev. E 91 (2015), pp.032108.
  10. Korabel, N., Barkai, E.: Paradoxes of subdiffusive infiltration in disordered systems. Phys. Rev. Lett 104 (2010), pp.170603.
  11. Arkhincheev, V.E., Kunnen, E., Baklanov, M.R.: Active species in porous media: Random walk and capture in traps. Microelectronic Engineering 88 (2011), pp. 694-696.
  12. Chen, W., Liang, Y.J.: New methodologies in fractional and fractal derivatives modeling. Chaos Solitons Fractals 102 (2017), pp. 72-77.
  13. Cai, W., Chen, W., Wang, F.J.: Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media. Ther. Sci 22(2018), pp.1-6.
  14. Liang, Y.J., Chen, W., Xu, W., Sun, H.G.: Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media. Commun. Nonlinear Sci. Numer. Simul. 70 (2019), pp. 384-393.
  15. Yang, X., Liang, Y.J., Chen, W.: A local structural derivative PDE model for ultraslow creep. Comput. Math. Appl 76 (2018), pp.1713-1718.
  16. Qi, H.T., Guo, X.W.: Transient fractional heat conduction with generalized Cattaneo model. Int. J Heat Mass Transfer 76 (2014), pp. 535-539.
  17. Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J Phys A 30(1997), pp. 7277-7289.
  18. Cattaneo, C.: Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3(1948), pp. 83-101.
  19. Fan, Y., Liu, L., Zheng, L.C.: Anomalous subdiffusion in angular and radial direction on a circular comb-like structure with nonisotropic relaxation. Appl. Math. Model 64(2018), pp. 615-623.
  20. Podlubny, I.: Fractional Differential Equation. Academic Press, New York (1999).
  21. Sandev, T., Schulz, A., Kantz, H., Iomin, A.: Heterogeneous diffusion in comb and fractal grid structures. Chaos. Solitons Fractals 114(2018), pp.551-555.
  22. Mathai, A.M., Saxena, R.K., Haubold, H.J.: The H-function: theory and applications. Springer Science Business Media (2009).
  23. Brzezinski, D.W., Ostalczyk, P.: Numerical calculations accuracy comparison of the inverse Laplace transform algorithms for solutions of fractional order differential equations. Nonlinear Dyn. 84(2016), pp. 65-77.
  24. Liu, L., Zheng, L.C., Fan, Y., Chen, Y.P., Liu, F.W.: Comb model for the anomalous diffusion with dual-phase-lag constitutive relation. Commun. Nonlinear Sci. Numer. Simul. 63(2018), pp. 135-144.
PDF VERSION [DOWNLOAD]
Anomalous diffusion and heat transfer on comb structure with anisotropic relaxation in fractal porous media

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence