# American Institute of Mathematical Sciences

March  2020, 15(1): 87-110. doi: 10.3934/nhm.2020004

## Homogenization of Bingham flow in thin porous media

 1 Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Sevilla, 41012, Spain 2 Université de Lorraine, CNRS, IECL, Metz, 57000, France

* Corresponding author: Renata Bunoiu

Received  March 2019 Published  December 2019

Fund Project: María Anguiano is supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466.

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter $\varepsilon$ tends to zero, following the ratio between the height $\varepsilon$ of the porous medium and the relative dimension $a_\varepsilon$ of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow.

Citation: María Anguiano, Renata Bunoiu. Homogenization of Bingham flow in thin porous media. Networks & Heterogeneous Media, 2020, 15 (1) : 87-110. doi: 10.3934/nhm.2020004
##### References:
 [1] M. Anguiano, Darcy's laws for non-stationary viscous fluid flow in a thin porous medium, Math. Meth. Appl. Sci., 40 (2017), 2878-2895.  doi: 10.1002/mma.4204.  Google Scholar [2] M. Anguiano, On the non-stationary non-Newtonian flow through a thin porous medium, ZAMM-Z. Angew. Math. Mech., 97 (2017), 895-915.  doi: 10.1002/zamm.201600177.  Google Scholar [3] M. Anguiano and R. Bunoiu, On the flow of a viscoplastic fluid in a thin periodic domain,, in Integral Methods in Science and Engineering (eds. C. Constanda and P. Harris), Birkhäuser, Cham, (2019), 15–24. doi: 10.1007/978-3-030-16077-7_2.  Google Scholar [4] M. Anguiano and F. J. Suárez-Grau, Homogenization of an incompressible non-newtonian flow through a thin porous medium,, Z. Angew. Math. Phys., 68 (2017), Art. 45, 25 pp. doi: 10.1007/s00033-017-0790-z.  Google Scholar [5] M. Anguiano and F. J. Suárez-Grau, The transition between the Navier-Stokes equations to the Darcy equation in a thin porous medium, Mediterr. J. Math., 15 (2018), Art. 45, 21 pp. doi: 10.1007/s00009-018-1086-z.  Google Scholar [6] L. Berlyand, A. G. Kolpakov and A. Novikov, Introduction to the Network Approximation Method for Materials Modeling, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139235952.  Google Scholar [7] L. Berlyand and E. Khruslov, Homogenized non-newtonian viscoelastic rheology of a suspension of interacting particles in a viscous newtonian fluid,, SIAM Journal of Applied Mathematics, 64 (2004), 1002-1034.  doi: 10.1137/S0036139902403913.  Google Scholar [8] N. Bernabeu, P. Saramito and A. Harris, Laminar shallow viscoplastic fluid flowing through an array of vertical obstacles, J. Non-Newtonian Fluid Mech., 257 (2018), 59-70.  doi: 10.1016/j.jnnfm.2018.04.001.  Google Scholar [9] D. Bresch, E. D. Fernandez-Nieto, I. Ionescu and P. Vigneaux, Augmented lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches, in New Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics (eds. A.V. Fursikov, G.P. Galdi and V.V. Pukhnachev), Birkhäuser Basel, (2010), 57–89.  Google Scholar [10] A. Bourgeat and A. Mikelić, A note on homogenization of Bingham flow through a porous medium, J. Math. Pures Appl., 72 (1993), 405-414.   Google Scholar [11] R. Bunoiu and G. Cardone, Bingham flow in porous media with obstacles of different size, Math. Meth. Appl. Sci., 40 (2017), 4514-4528.  doi: 10.1002/mma.4322.  Google Scholar [12] R. Bunoiu, G. Cardone and C. Perugia, Unfolding method for the homogenization of Bingham flow, in Modelling and Simulation in Fluid Dynamics in Porous Media. Springer Proceedings in Mathematics & Statistics (eds. J. Ferreira, S. Barbeiro, G. Pena and M. Wheeler), Vol 28, Springer, New York, NY, (2013), 109–123. doi: 10.1007/978-1-4614-5055-9_7.  Google Scholar [13] R. Bunoiu, A. Gaudiello and A. Leopardi, Asymptotic analysis of a Bingham fluid in a thin T-like shaped structure, J. Math. Pures Appl., 123 (2019), 148-166.  doi: 10.1016/j.matpur.2018.01.001.  Google Scholar [14] R. Bunoiu and S. Kesavan, Fluide de Bingham dans une couche mince, Annals of University of Craiova, Math. Comp. Sci. Ser., 30 (2003), 71-77.   Google Scholar [15] R. Bunoiu and S. Kesavan, Asymptotic behaviour of a Bingham fluid in thin layers, Journal of Mathematical Analysis and Applications, 293 (2004), 405-418.  doi: 10.1016/j.jmaa.2003.10.049.  Google Scholar [16] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.  Google Scholar [17] D. Cioranescu, A. Damlamian and G. Griso, The Periodic Unfolding Method: Theory and Applications to Partial Differential Problems, Series in Contemporary Mathematics, 3, Springer, 2018. doi: 10.1007/978-981-13-3032-2.  Google Scholar [18] D. Cioranescu, V. Girault and K. R. Rajagopal, Mechanics and Mathematics of Fluids of the Differential Type, Advances in Mechanics and Mathematics, 35, Springer, 2016. doi: 10.1007/978-3-319-39330-8.  Google Scholar [19] G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, 21, Dunod, Paris, 1972.  Google Scholar [20] J. Fabricius, J. G. I. Hellström, T. S. Lundström, E. Miroshnikova and P. Wall, Darcy's law for flow in a periodic thin porous medium confined between two parallel plates, Transp. Porous Med., 115 (2016), 473-493.  doi: 10.1007/s11242-016-0702-2.  Google Scholar [21] E. D. Fernández-Nieto, P. Noble and J. P. Vila, Shallow water equations for power law and Bingham fluids, Science China Mathematics, 55 (2012), 277-283.  doi: 10.1007/s11425-011-4358-7.  Google Scholar [22] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar [23] G. Griso, Asymptotic behavior of a crane, C.R.Acad.Sci. Paris, Ser. Ⅰ, 338 (2004), 261-266.  doi: 10.1016/j.crma.2003.11.023.  Google Scholar [24] G. Griso and L. Merzougui, Junctions between two plates and a family of beams, Math. Meth. Appl. Sci., 41 (2018), 58-79.  doi: 10.1002/mma.4594.  Google Scholar [25] G. Griso, A. Migunova and J. Orlik, Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams, Journal of Elasticity, 128 (2017), 291-331.  doi: 10.1007/s10659-017-9628-3.  Google Scholar [26] I. R. Ionescu, Onset and dynamic shallow flow of a viscoplastic fluid on a plane slope, J. Non-Newtonian Fluid Mech., 165 (2010), 1328-1341.  doi: 10.1016/j.jnnfm.2010.06.016.  Google Scholar [27] I. R. Ionescu, Augmented lagrangian for shallow viscoplastic flow with topography, Journal of Computational Physics, 242 (2013), 544-560.  doi: 10.1016/j.jcp.2013.02.029.  Google Scholar [28] I. R. Ionescu, Viscoplastic shallow flow equations with topography, J. Non-Newtonian Fluid Mech., 193 (2013), 116-128.  doi: 10.1016/j.jnnfm.2012.09.009.  Google Scholar [29] J. L. Lions and E. Sánchez-Palencia, Écoulement d'un fluide viscoplastique de Bingham dans un milieu poreux, J. Math. Pures Appl., 60 (1981), 341-360.   Google Scholar [30] P. Lipman, J. Lockwood, R. Okamura, D. Swanson and K. Yamashita, Ground Deformation Associated with the 1975 Magnitude-7.2 Earthquake and Resulting Changes in Activity of Kilauea Volcano, Hawaii, Professional Paper, 1276, Technical report, US Goverment Printing Office, 1985. doi: 10.3133/pp1276.  Google Scholar [31] K. F. Liu and C. C. Mei, Approximate equations for the slow spreading of a thin sheet of Bingham plastic fluid, Physics of Fluids A: Fluid Dynamics, 2 (1990), 30-36.  doi: 10.1063/1.857821.  Google Scholar [32] S. E. Pastukhova, Asymptotic analysis in elasticity problems on thin periodic structures, Networks and Heterogeneous Media, 4 (2009), 577-604.  doi: 10.3934/nhm.2009.4.577.  Google Scholar [33] K. Vasilic, B. Meng, H. C. Kne and N. Roussel, Flow of fresh concrete through steel bars: A porous medium analogy, Cement and Concrete Research, 41 (2011), 496-503.  doi: 10.1016/j.cemconres.2011.01.013.  Google Scholar [34] Y. Zhengan and Z. Hongxing, Homogenization of a stationary Navier-Stokes flow in porous medium with thin film, Acta Mathematica Scientia, 28 (2008), 963-974.  doi: 10.1016/S0252-9602(08)60096-X.  Google Scholar

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##### References:
 [1] M. Anguiano, Darcy's laws for non-stationary viscous fluid flow in a thin porous medium, Math. Meth. Appl. Sci., 40 (2017), 2878-2895.  doi: 10.1002/mma.4204.  Google Scholar [2] M. Anguiano, On the non-stationary non-Newtonian flow through a thin porous medium, ZAMM-Z. Angew. Math. Mech., 97 (2017), 895-915.  doi: 10.1002/zamm.201600177.  Google Scholar [3] M. Anguiano and R. Bunoiu, On the flow of a viscoplastic fluid in a thin periodic domain,, in Integral Methods in Science and Engineering (eds. C. Constanda and P. Harris), Birkhäuser, Cham, (2019), 15–24. doi: 10.1007/978-3-030-16077-7_2.  Google Scholar [4] M. Anguiano and F. J. Suárez-Grau, Homogenization of an incompressible non-newtonian flow through a thin porous medium,, Z. Angew. Math. Phys., 68 (2017), Art. 45, 25 pp. doi: 10.1007/s00033-017-0790-z.  Google Scholar [5] M. Anguiano and F. J. Suárez-Grau, The transition between the Navier-Stokes equations to the Darcy equation in a thin porous medium, Mediterr. J. Math., 15 (2018), Art. 45, 21 pp. doi: 10.1007/s00009-018-1086-z.  Google Scholar [6] L. Berlyand, A. G. Kolpakov and A. Novikov, Introduction to the Network Approximation Method for Materials Modeling, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139235952.  Google Scholar [7] L. Berlyand and E. Khruslov, Homogenized non-newtonian viscoelastic rheology of a suspension of interacting particles in a viscous newtonian fluid,, SIAM Journal of Applied Mathematics, 64 (2004), 1002-1034.  doi: 10.1137/S0036139902403913.  Google Scholar [8] N. Bernabeu, P. Saramito and A. Harris, Laminar shallow viscoplastic fluid flowing through an array of vertical obstacles, J. Non-Newtonian Fluid Mech., 257 (2018), 59-70.  doi: 10.1016/j.jnnfm.2018.04.001.  Google Scholar [9] D. Bresch, E. D. Fernandez-Nieto, I. Ionescu and P. Vigneaux, Augmented lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches, in New Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics (eds. A.V. Fursikov, G.P. Galdi and V.V. Pukhnachev), Birkhäuser Basel, (2010), 57–89.  Google Scholar [10] A. Bourgeat and A. Mikelić, A note on homogenization of Bingham flow through a porous medium, J. Math. Pures Appl., 72 (1993), 405-414.   Google Scholar [11] R. Bunoiu and G. Cardone, Bingham flow in porous media with obstacles of different size, Math. Meth. Appl. Sci., 40 (2017), 4514-4528.  doi: 10.1002/mma.4322.  Google Scholar [12] R. Bunoiu, G. Cardone and C. Perugia, Unfolding method for the homogenization of Bingham flow, in Modelling and Simulation in Fluid Dynamics in Porous Media. Springer Proceedings in Mathematics & Statistics (eds. J. Ferreira, S. Barbeiro, G. Pena and M. Wheeler), Vol 28, Springer, New York, NY, (2013), 109–123. doi: 10.1007/978-1-4614-5055-9_7.  Google Scholar [13] R. Bunoiu, A. Gaudiello and A. Leopardi, Asymptotic analysis of a Bingham fluid in a thin T-like shaped structure, J. Math. Pures Appl., 123 (2019), 148-166.  doi: 10.1016/j.matpur.2018.01.001.  Google Scholar [14] R. Bunoiu and S. Kesavan, Fluide de Bingham dans une couche mince, Annals of University of Craiova, Math. Comp. Sci. Ser., 30 (2003), 71-77.   Google Scholar [15] R. Bunoiu and S. Kesavan, Asymptotic behaviour of a Bingham fluid in thin layers, Journal of Mathematical Analysis and Applications, 293 (2004), 405-418.  doi: 10.1016/j.jmaa.2003.10.049.  Google Scholar [16] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.  doi: 10.1137/080713148.  Google Scholar [17] D. Cioranescu, A. Damlamian and G. Griso, The Periodic Unfolding Method: Theory and Applications to Partial Differential Problems, Series in Contemporary Mathematics, 3, Springer, 2018. doi: 10.1007/978-981-13-3032-2.  Google Scholar [18] D. Cioranescu, V. Girault and K. R. Rajagopal, Mechanics and Mathematics of Fluids of the Differential Type, Advances in Mechanics and Mathematics, 35, Springer, 2016. doi: 10.1007/978-3-319-39330-8.  Google Scholar [19] G. Duvaut and J. L. Lions, Les Inéquations en Mécanique et en Physique, Travaux et Recherches Mathématiques, 21, Dunod, Paris, 1972.  Google Scholar [20] J. Fabricius, J. G. I. Hellström, T. S. Lundström, E. Miroshnikova and P. Wall, Darcy's law for flow in a periodic thin porous medium confined between two parallel plates, Transp. Porous Med., 115 (2016), 473-493.  doi: 10.1007/s11242-016-0702-2.  Google Scholar [21] E. D. Fernández-Nieto, P. Noble and J. P. Vila, Shallow water equations for power law and Bingham fluids, Science China Mathematics, 55 (2012), 277-283.  doi: 10.1007/s11425-011-4358-7.  Google Scholar [22] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.  Google Scholar [23] G. Griso, Asymptotic behavior of a crane, C.R.Acad.Sci. Paris, Ser. Ⅰ, 338 (2004), 261-266.  doi: 10.1016/j.crma.2003.11.023.  Google Scholar [24] G. Griso and L. Merzougui, Junctions between two plates and a family of beams, Math. Meth. Appl. Sci., 41 (2018), 58-79.  doi: 10.1002/mma.4594.  Google Scholar [25] G. Griso, A. Migunova and J. Orlik, Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams, Journal of Elasticity, 128 (2017), 291-331.  doi: 10.1007/s10659-017-9628-3.  Google Scholar [26] I. R. Ionescu, Onset and dynamic shallow flow of a viscoplastic fluid on a plane slope, J. Non-Newtonian Fluid Mech., 165 (2010), 1328-1341.  doi: 10.1016/j.jnnfm.2010.06.016.  Google Scholar [27] I. R. Ionescu, Augmented lagrangian for shallow viscoplastic flow with topography, Journal of Computational Physics, 242 (2013), 544-560.  doi: 10.1016/j.jcp.2013.02.029.  Google Scholar [28] I. R. Ionescu, Viscoplastic shallow flow equations with topography, J. Non-Newtonian Fluid Mech., 193 (2013), 116-128.  doi: 10.1016/j.jnnfm.2012.09.009.  Google Scholar [29] J. L. Lions and E. Sánchez-Palencia, Écoulement d'un fluide viscoplastique de Bingham dans un milieu poreux, J. Math. Pures Appl., 60 (1981), 341-360.   Google Scholar [30] P. Lipman, J. Lockwood, R. Okamura, D. Swanson and K. Yamashita, Ground Deformation Associated with the 1975 Magnitude-7.2 Earthquake and Resulting Changes in Activity of Kilauea Volcano, Hawaii, Professional Paper, 1276, Technical report, US Goverment Printing Office, 1985. doi: 10.3133/pp1276.  Google Scholar [31] K. F. Liu and C. C. Mei, Approximate equations for the slow spreading of a thin sheet of Bingham plastic fluid, Physics of Fluids A: Fluid Dynamics, 2 (1990), 30-36.  doi: 10.1063/1.857821.  Google Scholar [32] S. E. Pastukhova, Asymptotic analysis in elasticity problems on thin periodic structures, Networks and Heterogeneous Media, 4 (2009), 577-604.  doi: 10.3934/nhm.2009.4.577.  Google Scholar [33] K. Vasilic, B. Meng, H. C. Kne and N. Roussel, Flow of fresh concrete through steel bars: A porous medium analogy, Cement and Concrete Research, 41 (2011), 496-503.  doi: 10.1016/j.cemconres.2011.01.013.  Google Scholar [34] Y. Zhengan and Z. Hongxing, Homogenization of a stationary Navier-Stokes flow in porous medium with thin film, Acta Mathematica Scientia, 28 (2008), 963-974.  doi: 10.1016/S0252-9602(08)60096-X.  Google Scholar
View of the domain $\Omega_\varepsilon$
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