Research article Special Issues

Effective governing equations for heterogenous porous media subject to inhomogeneous body forces

  • Received: 03 June 2020 Accepted: 20 July 2020 Published: 21 August 2020
  • We derive a new homogenized model for heterogeneous porous media driven by inhomogeneous body forces. We assume that the fine scale, characterizing the heterogeneities in the medium, is larger than the pore scale, but nonetheless much smaller than the size of the material (the coarse scale). We decouple spatial variations and assume periodicity on the fine scale. Fine scale variations are formally reflected in a locally unbounded source for the arising system of partial differential equations. We apply the asymptotic homogenization technique to obtain a well-defined coarse scale Darcy-type model. The resulting problem is driven by an effective source which comprises both the coarse scale divergence of the average body force, and additional contributions which are to be computed solving a well-defined diffusion-type cell problem which is driven solely by fine scale variations of the given force. The present model can be used to predict the effect of externally applied magnetic (or electric) fields on ferrofluids (or electrolytes) flowing in porous media. This work can, in perspective, pave the way for investigations of the effect of applied forces on complex and heterogeneous hierarchical materials, such as systems of fractures or cancerous biological tissues.

    Citation: Raimondo Penta, Ariel Ramírez-Torres, José Merodio, Reinaldo Rodríguez-Ramos. Effective governing equations for heterogenous porous media subject to inhomogeneous body forces[J]. Mathematics in Engineering, 2021, 3(4): 1-17. doi: 10.3934/mine.2021033

    Related Papers:

  • We derive a new homogenized model for heterogeneous porous media driven by inhomogeneous body forces. We assume that the fine scale, characterizing the heterogeneities in the medium, is larger than the pore scale, but nonetheless much smaller than the size of the material (the coarse scale). We decouple spatial variations and assume periodicity on the fine scale. Fine scale variations are formally reflected in a locally unbounded source for the arising system of partial differential equations. We apply the asymptotic homogenization technique to obtain a well-defined coarse scale Darcy-type model. The resulting problem is driven by an effective source which comprises both the coarse scale divergence of the average body force, and additional contributions which are to be computed solving a well-defined diffusion-type cell problem which is driven solely by fine scale variations of the given force. The present model can be used to predict the effect of externally applied magnetic (or electric) fields on ferrofluids (or electrolytes) flowing in porous media. This work can, in perspective, pave the way for investigations of the effect of applied forces on complex and heterogeneous hierarchical materials, such as systems of fractures or cancerous biological tissues.


    加载中


    [1] Allaire G (1989) Homogenization of the stokes flow in a connected porous medium. Asymptotic Anal 2: 203-222.
    [2] Allaire G, Briane M (1996) Multiscale convergence and reiterated homogenisation. P Roy Soc Edinb A 126: 297-342.
    [3] Arbogast T, Lehr HL (2006) Homogenization of a darcy-stokes system modeling vuggy porous media. Computat Geosci 10: 291-302.
    [4] Bakhvalov N, Panasenko G (1989) Homogenisation Averaging Processes in Periodic Media, Springer.
    [5] Bendel P, Bernardo M, Dunsmuir JH, et al. (2003) Electric field driven flow in natural porous media. Magn Reson Imaging 21: 321-332.
    [6] Burridge R, Keller J (1981) Poroelasticity equations derived from microstructure. J Acoust Soc Am 70: 1140-1146.
    [7] Cioranescu D, Donato P (1999) An Introduction to Homogenization, Oxford University Press.
    [8] Collis J, Hubbard ME, O'Dea RD (2017) A multi-scale analysis of drug transport and response for a multi-phase tumour model. Eur J Appl Math 28: 499-534.
    [9] Collis J, Hubbard ME, O'Dea R (2016) Computational modelling of multiscale, multiphase fluid mixtures with application to tumour growth. Comput Method Appl M 309: 554-578.
    [10] El-Amin MF, Brahimi T (2017) Numerical modeling of magnetic nanoparticles transport in a two-phase flow in porous media, In: SPE Reservoir Characterisation and Simulation Conference and Exhibition, Society of Petroleum Engineers.
    [11] Holmes M (1995) Introduction to Perturbation Method, Springer-Verlag.
    [12] Hornung U (1997) Homogenization and Porous Media, Springer.
    [13] Irons L, Collis J, O'Dea RD (2017) Microstructural influences on growth and transport in biological tissue-a multiscale description, In: Modeling of Microscale Transport in Biological Processes, Elsevier, 311-334.
    [14] Lukkassen D, Milton GW (2002) On hierarchical structures and reiterated homogenization, In: Function Spaces, Interpolation Theory and Related Topics (Lund, 2000), 355-368.
    [15] Mascheroni P, Penta R (2017) The role of the microvascular network structure on diffusion and consumption of anticancer drugs. Int J Numer Method Biomed Eng, 33: 10.1002/cnm.2857.
    [16] Nabil M, Zunino P (2016) A computational study of cancer hyperthermia based on vascular magnetic nanoconstructs. Roy Soc Open Sci 3: 160287.
    [17] Oldenburg CM, Borglin SE, Moridis GJ (2000) Numerical simulation of ferrofluid flow for subsurface environmental engineering applications. Transport Porous Med 38: 319-344.
    [18] Pankhurst QA, Connolly J, Jones SK, et al. (2003) Applications of magnetic nanoparticles in biomedicine. J Phys D Appl Phys 36: R167.
    [19] Papanicolau G, Bensoussan A, Lions JL (1978) Asymptotic Analysis for Periodic Structures, Elsevier.
    [20] Penta R, Ambrosi D (2015) The role of microvascular tortuosity in tumor transport phenomena. J Theor Bio 364: 80-97.
    [21] Penta R, Ambrosi D, Quarteroni A (2015) Multiscale homogenization for fluid and drug transport in vascularized malignant tissues. Math Mod Meth Appl S 25: 79-108.
    [22] Penta R, Ambrosi D, Shipley RJ (2014) Effective governing equations for poroelastic growing media. Q J Mech Appl Math 67: 69-91.
    [23] Penta R, Gerisch A (2015) Investigation of the potential of asymptotic homogenization for elastic composites via a three-dimensional computational study. Comput Vis Sci 17: 185-201.
    [24] Penta R, Merodio J (2017) Homogenized modeling for vascularized poroelastic materials. Meccanica 52: 3321-3343.
    [25] Penta R, Raum K, Grimal Q, et al. (2016) Can a continuous mineral foam explain the stiffening of aged bone tissue? a micromechanical approach to mineral fusion in musculoskeletal tissues. Bioinspir Biomim 11: 035004.
    [26] Penta R, Gerisch A (2017) The asymptotic homogenization elasticity tensor properties for composites with material discontinuities. Continuum Mech Therm 29: 187-206.
    [27] Preziosi L, Farina A (2002) On darcy's law for growing porous media. Int J NonLin Mech 37: 485-491.
    [28] Raj K, Moskowitz R (1990) Commercial applications of ferrofluids. J Magn Magn Mater 85: 233-245.
    [29] Rajagopal KR (2007) On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math Mod Meth Appl S 17: 215-252.
    [30] Ramírez-Torres A, Rodríguez-Ramos R, Merodio J, et al. (2015) Action of body forces in tumor growth. Int J Eng Sci 89: 18-34.
    [31] Ramírez-Torres A, Penta R, Rodríguez-Ramos R, et al. (2019) Effective properties of hierarchical fiber-reinforced composites via a three-scale asymptotic homogenization approach. Math Mech Solids 24: 3554-3574.
    [32] Ramírez-Torres A, Penta R, Rodríguez-Ramos R, et al. (2018) Homogenized out-of-plane shear response of three-scale fiber-reinforced composites. Comput Vis Sci 20: 85-93.
    [33] Ramírez-Torres A, Penta R, Rodríguez-Ramos R, et al. (2018) Three scales asymptotic homogenization and its application to layered hierarchical hard tissues. Int J Solids Struct 130: 190-198.
    [34] Rosensweig RE (2013) Ferrohydrodynamics, Courier Corporation.
    [35] Penta R, Ramírez-Torres A, Merodio J, et al. (2017) Effective balance equations for elastic composites subject to inhomogeneous potentials. Continuum Mech Therm 30: 145-163.
    [36] Sanchez-Palencia E (1980) Non-Homogeneous Media and Vibration Theory, Springer-Verlag.
    [37] Shipley RJ, Chapman J (2010) Multiscale modelling of fluid and drug transport in vascular tumors. B Math Bio 72: 1464-1491.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4508) PDF downloads(645) Cited by(5)

Article outline

Figures and Tables

Figures(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog