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 & 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. doi: 10.1016/0362-546X(92)90015-7. Google Scholar

[2]

G. Allaire, Homogenization of the Stokes flow in a connected porous medium,, Asymptotic Anal., 2 (1989), 203. Google Scholar

[3]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[4]

G. Allaire, Homogenization in porous media,, CEA-EDF-INRIA School on Homogenization, 2 (2010), 1. Google Scholar

[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. doi: 10.1016/j.physd.2014.05.007. Google Scholar

[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). doi: 10.1063/1.3521555. Google Scholar

[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. doi: 10.1017/S0022112082001542. Google Scholar

[8]

J. L. Anderson and D. M. Malone, Mechanism of osmotic flow in porous membranes,, Biophysical Journal, 14 (1974), 957. doi: 10.1016/S0006-3495(74)85962-X. Google Scholar

[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). doi: 10.1103/PhysRevE.87.033006. Google Scholar

[10]

T. Y. Cath, A. E. Childress and M. Elimelech, Forward osmosis: Principles, applications, and recent developments,, Journal of Membrane Science, 281 (2006), 70. doi: 10.1016/j.memsci.2006.05.048. Google Scholar

[11]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization, vol. 234 of Translations of Mathematical Monographs,, American Mathematical Society, (2007). Google Scholar

[12]

D. Coelho, M. Shapiro, J. Thovert and P. Adler, Electroosmotic phenomena in porous media,, Journal of Colloid and Interface Science, 181 (1996), 169. doi: 10.1006/jcis.1996.0369. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.bpj.2008.12.3929. Google Scholar

[16]

D. Guell and H. Brenner, Physical mechanism of membrane osmotic phenomena,, Industrial and Engineering Chemistry Research, 35 (1996), 3004. doi: 10.1021/ie950787f. Google Scholar

[17]

D. Guell, The Physical Mechanism of Osmosis and Osmotic Pressure-a Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient,, Massachusetts Institute of Technology, (1991). Google Scholar

[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. doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. Google Scholar

[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. doi: 10.1017/S002211200900799X. Google Scholar

[20]

O. Kedem and A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes,, Biochimica et Biophysica Acta, 27 (1958), 229. doi: 10.1016/0006-3002(58)90330-5. Google Scholar

[21]

O. Kedem and A. Katchalsky, Thermodynamics of flow processes in biological systems,, Biophysical Journal, 2 (1962), 53. Google Scholar

[22]

A. Kufner, Weighted Sobolev Spaces, vol. 31 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics],, BSB B. G. Teubner Verlagsgesellschaft, (1980). Google Scholar

[23]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Revised English edition. Translated from the Russian by Richard A. Silverman, (1963). Google Scholar

[24]

B. E. Logan and M. Elimelech, Membrane-based processes for sustainable power generation using water,, Nature, 488 (2012), 313. doi: 10.1038/nature11477. Google Scholar

[25]

J. R. Looker and S. L. Carnie, Homogenization of the ionic transport equations in periodic porous media,, Transp. Porous Media, 65 (2006), 107. doi: 10.1007/s11242-005-6080-9. Google Scholar

[26]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043. Google Scholar

[27]

B. Opic and A. Kufner, Hardy-type Inequalities, vol. 219 of Pitman Research Notes in Mathematics Series,, Longman Scientific & Technical, (1990). Google Scholar

[28]

F. Reuss, Charge-induced flow,, Proceedings of the Imperial Society of Naturalists of Moscow, 3 (1809), 327. Google Scholar

[29]

N. Scales and N. Tait, Modelling electroosmotic flow in porous media for microfluidic applications,, in MEMS, (2004), 571. Google Scholar

[30]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system,, Math. Models Methods Appl. Sci., 19 (2009), 993. doi: 10.1142/S0218202509003693. Google Scholar

[31]

M. Schmuck, Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach,, Commun. Math. Sci., 9 (2011), 685. doi: 10.4310/CMS.2011.v9.n3.a3. Google Scholar

[32]

L. Tartar, Incompressible fluid flow through porous media. convergence of the homogenization process,, in Nonhomogeneous media and vibration theory), (1980). Google Scholar

[33]

J. van't Hoff, The role of osmotic pressure in the analogy between solutions and gases,, Zeitschrift fur physikalische Chemie, 1 (1887), 481. Google Scholar

[34]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,, Annalen der Physik, 326 (1906), 756. doi: 10.1002/andp.19063261405. Google Scholar

[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. doi: 10.1063/1.1680484. Google Scholar

[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. Google Scholar

[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). doi: 10.1152/ajpheart.00695.2005. Google Scholar

[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. doi: 10.1016/j.memsci.2011.12.023. Google Scholar

[39]

V. V. Zhikov, On an extension and an application of the two-scale convergence method,, Mat. Sb., 191 (2000), 31. doi: 10.1070/SM2000v191n07ABEH000491. Google Scholar

show all references

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. doi: 10.1016/0362-546X(92)90015-7. Google Scholar

[2]

G. Allaire, Homogenization of the Stokes flow in a connected porous medium,, Asymptotic Anal., 2 (1989), 203. Google Scholar

[3]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[4]

G. Allaire, Homogenization in porous media,, CEA-EDF-INRIA School on Homogenization, 2 (2010), 1. Google Scholar

[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. doi: 10.1016/j.physd.2014.05.007. Google Scholar

[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). doi: 10.1063/1.3521555. Google Scholar

[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. doi: 10.1017/S0022112082001542. Google Scholar

[8]

J. L. Anderson and D. M. Malone, Mechanism of osmotic flow in porous membranes,, Biophysical Journal, 14 (1974), 957. doi: 10.1016/S0006-3495(74)85962-X. Google Scholar

[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). doi: 10.1103/PhysRevE.87.033006. Google Scholar

[10]

T. Y. Cath, A. E. Childress and M. Elimelech, Forward osmosis: Principles, applications, and recent developments,, Journal of Membrane Science, 281 (2006), 70. doi: 10.1016/j.memsci.2006.05.048. Google Scholar

[11]

G. A. Chechkin, A. L. Piatnitski and A. S. Shamaev, Homogenization, vol. 234 of Translations of Mathematical Monographs,, American Mathematical Society, (2007). Google Scholar

[12]

D. Coelho, M. Shapiro, J. Thovert and P. Adler, Electroosmotic phenomena in porous media,, Journal of Colloid and Interface Science, 181 (1996), 169. doi: 10.1006/jcis.1996.0369. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.bpj.2008.12.3929. Google Scholar

[16]

D. Guell and H. Brenner, Physical mechanism of membrane osmotic phenomena,, Industrial and Engineering Chemistry Research, 35 (1996), 3004. doi: 10.1021/ie950787f. Google Scholar

[17]

D. Guell, The Physical Mechanism of Osmosis and Osmotic Pressure-a Hydrodynamic Theory for Calculating the Osmotic Reflection Coefficient,, Massachusetts Institute of Technology, (1991). Google Scholar

[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. doi: 10.1002/(SICI)1097-0363(19960315)22:5<325::AID-FLD307>3.0.CO;2-Y. Google Scholar

[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. doi: 10.1017/S002211200900799X. Google Scholar

[20]

O. Kedem and A. Katchalsky, Thermodynamic analysis of the permeability of biological membranes to non-electrolytes,, Biochimica et Biophysica Acta, 27 (1958), 229. doi: 10.1016/0006-3002(58)90330-5. Google Scholar

[21]

O. Kedem and A. Katchalsky, Thermodynamics of flow processes in biological systems,, Biophysical Journal, 2 (1962), 53. Google Scholar

[22]

A. Kufner, Weighted Sobolev Spaces, vol. 31 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics],, BSB B. G. Teubner Verlagsgesellschaft, (1980). Google Scholar

[23]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Revised English edition. Translated from the Russian by Richard A. Silverman, (1963). Google Scholar

[24]

B. E. Logan and M. Elimelech, Membrane-based processes for sustainable power generation using water,, Nature, 488 (2012), 313. doi: 10.1038/nature11477. Google Scholar

[25]

J. R. Looker and S. L. Carnie, Homogenization of the ionic transport equations in periodic porous media,, Transp. Porous Media, 65 (2006), 107. doi: 10.1007/s11242-005-6080-9. Google Scholar

[26]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043. Google Scholar

[27]

B. Opic and A. Kufner, Hardy-type Inequalities, vol. 219 of Pitman Research Notes in Mathematics Series,, Longman Scientific & Technical, (1990). Google Scholar

[28]

F. Reuss, Charge-induced flow,, Proceedings of the Imperial Society of Naturalists of Moscow, 3 (1809), 327. Google Scholar

[29]

N. Scales and N. Tait, Modelling electroosmotic flow in porous media for microfluidic applications,, in MEMS, (2004), 571. Google Scholar

[30]

M. Schmuck, Analysis of the Navier-Stokes-Nernst-Planck-Poisson system,, Math. Models Methods Appl. Sci., 19 (2009), 993. doi: 10.1142/S0218202509003693. Google Scholar

[31]

M. Schmuck, Modeling and deriving porous media Stokes-Poisson-Nernst-Planck equations by a multi-scale approach,, Commun. Math. Sci., 9 (2011), 685. doi: 10.4310/CMS.2011.v9.n3.a3. Google Scholar

[32]

L. Tartar, Incompressible fluid flow through porous media. convergence of the homogenization process,, in Nonhomogeneous media and vibration theory), (1980). Google Scholar

[33]

J. van't Hoff, The role of osmotic pressure in the analogy between solutions and gases,, Zeitschrift fur physikalische Chemie, 1 (1887), 481. Google Scholar

[34]

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,, Annalen der Physik, 326 (1906), 756. doi: 10.1002/andp.19063261405. Google Scholar

[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. doi: 10.1063/1.1680484. Google Scholar

[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. Google Scholar

[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). doi: 10.1152/ajpheart.00695.2005. Google Scholar

[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. doi: 10.1016/j.memsci.2011.12.023. Google Scholar

[39]

V. V. Zhikov, On an extension and an application of the two-scale convergence method,, Mat. Sb., 191 (2000), 31. doi: 10.1070/SM2000v191n07ABEH000491. Google Scholar

[1]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006

[2]

Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29

[3]

Mario Ohlberger, Ben Schweizer. Modelling of interfaces in unsaturated porous media. Conference Publications, 2007, 2007 (Special) : 794-803. doi: 10.3934/proc.2007.2007.794

[4]

Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644

[5]

Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63

[6]

Cedric Galusinski, Mazen Saad. Water-gas flow in porous media. Conference Publications, 2005, 2005 (Special) : 307-316. doi: 10.3934/proc.2005.2005.307

[7]

Ioana Ciotir. Stochastic porous media equations with divergence Itô noise. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020010

[8]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[9]

Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91

[10]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305

[11]

Youcef Amirat, Laurent Chupin, Rachid Touzani. Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2445-2464. doi: 10.3934/cpaa.2014.13.2445

[12]

Cédric Galusinski, Mazen Saad. A nonlinear degenerate system modelling water-gas flows in porous media. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 281-308. doi: 10.3934/dcdsb.2008.9.281

[13]

Ting Zhang. The modeling error of well treatment for unsteady flow in porous media. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2171-2185. doi: 10.3934/dcdsb.2015.20.2171

[14]

T. L. van Noorden, I. S. Pop, M. Röger. Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness. Conference Publications, 2007, 2007 (Special) : 1013-1020. doi: 10.3934/proc.2007.2007.1013

[15]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265

[16]

D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685

[17]

Maurizio Verri, Giovanna Guidoboni, Lorena Bociu, Riccardo Sacco. The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics. Mathematical Biosciences & Engineering, 2018, 15 (4) : 933-959. doi: 10.3934/mbe.2018042

[18]

Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042

[19]

Fioralba Cakoni, Houssem Haddar, Isaac Harris. Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem. Inverse Problems & Imaging, 2015, 9 (4) : 1025-1049. doi: 10.3934/ipi.2015.9.1025

[20]

Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]