Opuscula Mathematica
Opuscula Math. 43, no. 2 (2023), 199-220

Asymptotic analysis of the steady advection-diffusion problem in axial domains

Fernando A. Morales
Abstract. We present the asymptotic analysis of the steady advection-diffusion equation in a thin tube. The problem is modeled in a mixed-type variational formulation, in order to separate the phenomenon in the axial direction and a transverse one. Such formulation makes visible the natural separation of scales within the problem and permits a successful asymptotic analysis, delivering a limiting form, free from the initial geometric singularity and suitable for approximating the original one. Furthermore, it is shown that the limiting problem can be simplified to a significantly simpler structure.
Keywords: asymptotic analysis, mixed-type variational formulations, advection-diffusion.
Mathematics Subject Classification: 80A20, 35F15.
Full text (pdf)
References ▼
  1. [1] J. Auriault, P. Adler, Taylor dispersion in porous media: Analysis by multiple scale expansions, Advances in Water Resources 18 (1995), no. 4, 217-226. https://doi.org/10.1016/0309-1708(95)00011-7
  2. [2] J.-L. Auriault, C. Geindreau, C. Boutin, Filtration law in porous media with poor separation of scales, Transport in Porous Media 60 (2005), no. 1, 89-108. https://doi.org/10.1007/s11242-004-3649-7
  3. [3] F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
  4. [4] K.D. Cherednichenko, V.P. Smyshlyaev, On full two-scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher-order homogenised variational problems, Archive for Rational Mechanics and Analysis 174 (2004), no. 3, 385-442. https://doi.org/10.1007/s00205-004-0335-4
  5. [5] A. Fiori, I. Jankovic, On preferential flow, channeling and connectivity in heterogeneous porous formations, Mathematical Geosciences 44 (2012), no. 2, 133-145. https://doi.org/10.1007/s11004-011-9365-2
  6. [6] G. Gatica, A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, SpringerBriefs in Mathematics, Springer International Publishing, 2014.
  7. [7] U. Hornung, Miscible Displacement, Springer New York, New York, NY, 1997. https://doi.org/10.1007/978-1-4612-1920-0_6
  8. [8] R.G. McLaren, P.A. Forsyth, E.A. Sudicky, J.E. VanderKwaak, F.W. Schwartz, J.H. Kessler, Flow and transport in fractured tuff at Yucca Mountain: numerical experiments on fast preferential flow mechanisms, Journal of Contaminant Hydrology 43 (2000), no. 3, 211-238. https://doi.org/10.1016/S0169-7722(00)00085-1
  9. [9] F.A. Morales, J.M. Ramírez, Tangential fluid flow within 3D narrow fissures: Conservative velocity fields on associated triangulations and transport processes, Mathematical Methods in the Applied Sciences 40 (2017), no. 18, 6316-6331. https://doi.org/10.1002/mma.4459
  10. [10] M. Peszyńska, R. Showalter, S.-Y. Yi, Flow and transport when scales are not separated: Numerical analysis and simulations of micro- and macro-models, International Journal of Numerical Analysis and Modeling 12 (2015), no. 3, 476-515.
  11. [11] J.M. Ramirez, E.A. Thomann, E.C. Waymire, R. Haggerty, B. Wood, A generalized Taylor-Aris formula and skew diffusion, Multiscale Modeling & Simulation 5 (2006), no. 3, 786-801.
  12. [12] P. Royer, Low scale separation induces modification of apparent solute transport regime in porous media, Mechanics Research Communications 87 (2018), 29-34. https://doi.org/10.1016/j.mechrescom.2017.12.003
  13. [13] P. Royer, Advection-diffusion in porous media with low scale separation: Modelling via higher-order asymptotic homogenisation, Transport in Porous Media 128 (2019), no. 2, 511-551. https://doi.org/10.1007/s11242-019-01258-2
  14. [14] A.P.S. Selvadurai, On the advective-diffusive transport in porous media in the presence of time-dependent velocities, Geophysical Research Letters 31 (2004) 13. https://doi.org/10.1029/2004GL019646
  15. [15] R.E. Showalter, Hilbert Space Methods for Partial Differential Equations, volume 1 of Monographs and Studies in Mathematics, Pitman, London-San Francisco, CA-Melbourne, 1977.
  16. [16] L. Tartar, Correctors in Linear Homogenization, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-05195-1_13
  17. [17] S. Wang, Q. Feng, X. Han, A hybrid analytical/numerical model for the characterization of preferential flow path with non-darcy flow, PLOS ONE 8 (2014), no. 12, 1-16. https://doi.org/10.1371/journal.pone.0083536
About the authors ▼
Fernando A. Morales
https://orcid.org/0000-0002-5691-1247
Universidad Nacional de Colombia, Sede Medellín, Escuela de Matemáticas, Carrera 65 # 59A-110, Bloque 43, of 106, Medellín - Colombia
About the article ▼
Communicated by P.A. Cojuhari.
Received: 2023-01-24.
Revised: 2023-02-20.
Accepted: 2023-02-21.
Published online: 2023-03-27.
Cite this article as ▼
Fernando A. Morales, Asymptotic analysis of the steady advection-diffusion problem in axial domains, Opuscula Math. 43, no. 2 (2023), 199-220, https://doi.org/10.7494/OpMath.2023.43.2.199.
Download this article's citation as: