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
[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. |