# American Institute of Mathematical Sciences

September  2016, 11(3): 471-499. doi: 10.3934/nhm.2016005

## Osmosis for non-electrolyte solvents in permeable periodic porous media

 1 Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Göteborg, Sweden 2 Narvik University College, Postbox 385, 8505 Narvik, Norway and P.N. Lebedev Physical Institute RAS, Leninski ave., 53, Moscow 119991, Russian Federation

Received  April 2015 Revised  September 2015 Published  August 2016

The paper gives a rigorous description, based on mathematical homogenization theory, for flows of solvents with not charged solute particles under osmotic pressure for periodic porous media permeable for solute particles. The effective Darcy type equations for the flow under osmotic pressure distributed within the porous media are derived. The effective Darcy law contains an additional flux term representing the osmotic pressure. Coefficients in the effective homogenized equations are related to the values of the phenomenological coefficients in the Kedem-Katchalsky formulae (2).
Citation: Alexei Heintz, Andrey Piatnitski. Osmosis for non-electrolyte solvents in permeable periodic porous media. Networks and Heterogeneous Media, 2016, 11 (3) : 471-499. doi: 10.3934/nhm.2016005
##### References:
 [1] E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7. [2] G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymptotic Anal., 2 (1989), 203-222. [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [4] G. Allaire, Homogenization in porous media, CEA-EDF-INRIA School on Homogenization, 2 (2010), 1-30. [5] G. Allaire, R. Brizzi, J.-F. Dufrêche, A. Mikelić and A. Piatnitski, Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling, Phys. D, 282 (2014), 39-60. doi: 10.1016/j.physd.2014.05.007. [6] G. Allaire, A. Mikelić and A. Piatnitski, Homogenization of the linearized ionic transport equations in rigid periodic porous media, J. Math. Phys., 51 (2010), 123103, 18pp. doi: 10.1063/1.3521555. [7] J. L. Anderson, M. E. Lowell and D. C. Prieve, Motion of a particle generated by chemical gradients Part 1. Non-electrolytes, Journal of Fluid Mechanics, 117 (1982), 107-121. doi: 10.1017/S0022112082001542. [8] J. L. Anderson and D. M. Malone, Mechanism of osmotic flow in porous membranes, Biophysical Journal, 14 (1974), 957-982. doi: 10.1016/S0006-3495(74)85962-X. [9] A. Bandopadhyay, D. DasGupta, S. K. Mitra and S. Chakraborty, Electro-osmotic flows through topographically complicated porous media: Role of electropermeability tensor, Phys. Rev. E, 87 (2013), 033006. doi: 10.1103/PhysRevE.87.033006. [10] T. Y. Cath, A. E. Childress and M. Elimelech, Forward osmosis: Principles, applications, and recent developments, Journal of Membrane Science, 281 (2006), 70-87. doi: 10.1016/j.memsci.2006.05.048. [11] G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization, vol. 234 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2007, Methods and applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. [12] D. Coelho, M. Shapiro, J. Thovert and P. Adler, Electroosmotic phenomena in porous media, Journal of Colloid and Interface Science, 181 (1996), 169-190. doi: 10.1006/jcis.1996.0369. [13] C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.), 20 (1994), 279-318. [14] A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik, 322 (1905), 549-560. [15] H. Y. Elmoazzen, J. A. Elliott and L. E. McGann, Osmotic transport across cell membranes in nondilute solutions: A new nondilute solute transport equation, Biophysical Journal, 96 (2009), 2559-2571. doi: 10.1016/j.bpj.2008.12.3929. [16] D. Guell and H. Brenner, Physical mechanism of membrane osmotic phenomena, Industrial and Engineering Chemistry Research, 35 (1996), 3004-3014. doi: 10.1021/ie950787f. [17] D. Guell, The Physical Mechanism of Osmosis and Osmotic Pressure-a Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient, Massachusetts Institute of Technology, Department of Chemical Engineering, 1991, URL http://books.google.se/books?id=_U_7NwAACAAJ. [18] J. G. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 22 (1996), 325-352. doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. [19] K. H. Jensen, E. Rio, C. C. Rasmus Hansen and T. Bohr, Osmotically driven pipe flows and their relation to sugar transport in plants, Journal of Fluid Mechanics, 636 (2009), 371-396. doi: 10.1017/S002211200900799X. [20] O. Kedem and A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes, Biochimica et Biophysica Acta, 27 (1958), 229-246. doi: 10.1016/0006-3002(58)90330-5. [21] O. Kedem and A. Katchalsky, Thermodynamics of flow processes in biological systems, Biophysical Journal, 2 (1962), 53-78. [22] A. Kufner, Weighted Sobolev Spaces, vol. 31 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980, With German, French and Russian summaries. [23] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York, 1963. [24] B. E. Logan and M. Elimelech, Membrane-based processes for sustainable power generation using water, Nature, 488 (2012), 313-319. doi: 10.1038/nature11477. [25] J. R. Looker and S. L. Carnie, Homogenization of the ionic transport equations in periodic porous media, Transp. Porous Media, 65 (2006), 107-131. doi: 10.1007/s11242-005-6080-9. [26] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [27] B. Opic and A. Kufner, Hardy-type Inequalities, vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. [28] F. Reuss, Charge-induced flow, Proceedings of the Imperial Society of Naturalists of Moscow, 3 (1809), 327-344. [29] N. Scales and N. Tait, Modelling electroosmotic flow in porous media for microfluidic applications, in MEMS, NANO and Smart Systems, 2004. ICMENS 2004. Proceedings. 2004 International Conference on, 2004, 571-577. [30] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1015. doi: 10.1142/S0218202509003693. [31] M. Schmuck, Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach, Commun. Math. Sci., 9 (2011), 685-710. doi: 10.4310/CMS.2011.v9.n3.a3. [32] L. Tartar, Incompressible fluid flow through porous media. convergence of the homogenization process, in Nonhomogeneous media and vibration theory), vol. 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980, ix+398. [33] J. van't Hoff, The role of osmotic pressure in the analogy between solutions and gases, Zeitschrift fur physikalische Chemie, 1 (1887), 481-508. [34] M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Annalen der Physik, 326 (1906), 756-780. doi: 10.1002/andp.19063261405. [35] C. E. Wyman and M. D. Kostin, Anomalous osmosis: Solutions to the Nernst-Planck and Navier-Stokes equations, The Journal of Chemical Physics, 59 (1973), 3411-3413. doi: 10.1063/1.1680484. [36] Z.-Y. Yant, S. Weinbaum and R. Pfeffer, On the fine structure of osmosis including threedimensional pore entrance and exit behaviour, Journal of Fluid Mechanics, 162 (1986), 415-438. [37] X. Zhang, F.-R. Curry and S. Weinbaum, Mechanism of osmotic flow in a periodic fiber array, Am J Physiol Heart Circ Physiol, 290 (2006), H844-H852. doi: 10.1152/ajpheart.00695.2005. [38] S. Zhao, L. Zou, C. Y. Tang and D. Mulcahy, Recent developments in forward osmosis: Opportunities and challenges, Journal of Membrane Science, 396 (2012), 1-21. doi: 10.1016/j.memsci.2011.12.023. [39] V. V. Zhikov, On an extension and an application of the two-scale convergence method, Mat. Sb., 191 (2000), 31-72. doi: 10.1070/SM2000v191n07ABEH000491.

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##### References:
 [1] E. Acerbi, V. Chiadò Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Anal., 18 (1992), 481-496. doi: 10.1016/0362-546X(92)90015-7. [2] G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymptotic Anal., 2 (1989), 203-222. [3] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [4] G. Allaire, Homogenization in porous media, CEA-EDF-INRIA School on Homogenization, 2 (2010), 1-30. [5] G. Allaire, R. Brizzi, J.-F. Dufrêche, A. Mikelić and A. Piatnitski, Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling, Phys. D, 282 (2014), 39-60. doi: 10.1016/j.physd.2014.05.007. [6] G. Allaire, A. Mikelić and A. Piatnitski, Homogenization of the linearized ionic transport equations in rigid periodic porous media, J. Math. Phys., 51 (2010), 123103, 18pp. doi: 10.1063/1.3521555. [7] J. L. Anderson, M. E. Lowell and D. C. Prieve, Motion of a particle generated by chemical gradients Part 1. Non-electrolytes, Journal of Fluid Mechanics, 117 (1982), 107-121. doi: 10.1017/S0022112082001542. [8] J. L. Anderson and D. M. Malone, Mechanism of osmotic flow in porous membranes, Biophysical Journal, 14 (1974), 957-982. doi: 10.1016/S0006-3495(74)85962-X. [9] A. Bandopadhyay, D. DasGupta, S. K. Mitra and S. Chakraborty, Electro-osmotic flows through topographically complicated porous media: Role of electropermeability tensor, Phys. Rev. E, 87 (2013), 033006. doi: 10.1103/PhysRevE.87.033006. [10] T. Y. Cath, A. E. Childress and M. Elimelech, Forward osmosis: Principles, applications, and recent developments, Journal of Membrane Science, 281 (2006), 70-87. doi: 10.1016/j.memsci.2006.05.048. [11] G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization, vol. 234 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2007, Methods and applications, Translated from the 2007 Russian original by Tamara Rozhkovskaya. [12] D. Coelho, M. Shapiro, J. Thovert and P. Adler, Electroosmotic phenomena in porous media, Journal of Colloid and Interface Science, 181 (1996), 169-190. doi: 10.1006/jcis.1996.0369. [13] C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.), 20 (1994), 279-318. [14] A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik, 322 (1905), 549-560. [15] H. Y. Elmoazzen, J. A. Elliott and L. E. McGann, Osmotic transport across cell membranes in nondilute solutions: A new nondilute solute transport equation, Biophysical Journal, 96 (2009), 2559-2571. doi: 10.1016/j.bpj.2008.12.3929. [16] D. Guell and H. Brenner, Physical mechanism of membrane osmotic phenomena, Industrial and Engineering Chemistry Research, 35 (1996), 3004-3014. doi: 10.1021/ie950787f. [17] D. Guell, The Physical Mechanism of Osmosis and Osmotic Pressure-a Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient, Massachusetts Institute of Technology, Department of Chemical Engineering, 1991, URL http://books.google.se/books?id=_U_7NwAACAAJ. [18] J. G. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids, 22 (1996), 325-352. doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. [19] K. H. Jensen, E. Rio, C. C. Rasmus Hansen and T. Bohr, Osmotically driven pipe flows and their relation to sugar transport in plants, Journal of Fluid Mechanics, 636 (2009), 371-396. doi: 10.1017/S002211200900799X. [20] O. Kedem and A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes, Biochimica et Biophysica Acta, 27 (1958), 229-246. doi: 10.1016/0006-3002(58)90330-5. [21] O. Kedem and A. Katchalsky, Thermodynamics of flow processes in biological systems, Biophysical Journal, 2 (1962), 53-78. [22] A. Kufner, Weighted Sobolev Spaces, vol. 31 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980, With German, French and Russian summaries. [23] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York, 1963. [24] B. E. Logan and M. Elimelech, Membrane-based processes for sustainable power generation using water, Nature, 488 (2012), 313-319. doi: 10.1038/nature11477. [25] J. R. Looker and S. L. Carnie, Homogenization of the ionic transport equations in periodic porous media, Transp. Porous Media, 65 (2006), 107-131. doi: 10.1007/s11242-005-6080-9. [26] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [27] B. Opic and A. Kufner, Hardy-type Inequalities, vol. 219 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1990. [28] F. Reuss, Charge-induced flow, Proceedings of the Imperial Society of Naturalists of Moscow, 3 (1809), 327-344. [29] N. Scales and N. Tait, Modelling electroosmotic flow in porous media for microfluidic applications, in MEMS, NANO and Smart Systems, 2004. ICMENS 2004. Proceedings. 2004 International Conference on, 2004, 571-577. [30] M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system, Math. Models Methods Appl. Sci., 19 (2009), 993-1015. doi: 10.1142/S0218202509003693. [31] M. Schmuck, Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach, Commun. Math. Sci., 9 (2011), 685-710. doi: 10.4310/CMS.2011.v9.n3.a3. [32] L. Tartar, Incompressible fluid flow through porous media. convergence of the homogenization process, in Nonhomogeneous media and vibration theory), vol. 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980, ix+398. [33] J. van't Hoff, The role of osmotic pressure in the analogy between solutions and gases, Zeitschrift fur physikalische Chemie, 1 (1887), 481-508. [34] M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Annalen der Physik, 326 (1906), 756-780. doi: 10.1002/andp.19063261405. [35] C. E. Wyman and M. D. Kostin, Anomalous osmosis: Solutions to the Nernst-Planck and Navier-Stokes equations, The Journal of Chemical Physics, 59 (1973), 3411-3413. doi: 10.1063/1.1680484. [36] Z.-Y. Yant, S. Weinbaum and R. Pfeffer, On the fine structure of osmosis including threedimensional pore entrance and exit behaviour, Journal of Fluid Mechanics, 162 (1986), 415-438. [37] X. Zhang, F.-R. Curry and S. Weinbaum, Mechanism of osmotic flow in a periodic fiber array, Am J Physiol Heart Circ Physiol, 290 (2006), H844-H852. doi: 10.1152/ajpheart.00695.2005. [38] S. Zhao, L. Zou, C. Y. Tang and D. Mulcahy, Recent developments in forward osmosis: Opportunities and challenges, Journal of Membrane Science, 396 (2012), 1-21. doi: 10.1016/j.memsci.2011.12.023. [39] V. V. Zhikov, On an extension and an application of the two-scale convergence method, Mat. Sb., 191 (2000), 31-72. doi: 10.1070/SM2000v191n07ABEH000491.
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