Transport in Porous Media

, Volume 53, Issue 1, pp 1–24 | Cite as

Thermomechanics of Swelling Biopolymeric Systems

  • Pawan P. Singh
  • John N. Cushman
  • Lynn S. Bennethum
  • Dirk E. Maier


A two-scale theory for the swelling biopolymeric media is developed. At the microscale, the solid polymeric matrix interacts with the solvent through surface contact. The relaxation processes within the polymeric matrix are incorporated by modeling the solid phase as viscoelastic and the solvent phase as viscous at the mesoscale. We obtain novel equations for the total stress tensor, chemical potential of the solid phase, heat flux and the generalized Darcy's law all at the mesoscale. The constitutive relations are more general than those previously developed for the swelling colloids. The generalized Darcy's law could be used for modeling non-Fickian fluid transport over a wide range of liquid contents. The form of the generalized Fick's law is similar to that obtained in earlier works involving colloids. Using two-variable expansions, thermal gradients are coupled with the strain rate tensor for the solid phase and the deformation rate tensor for the liquid phase. This makes the experimental determination of the material coefficients easier and less ambiguous.

biopolymeric swelling porous viscoelastic hybrid mixture theory microscale mesoscale 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Achanta, S. and Cushman, J. H.: 1994, Nonequilibrium swelling-pressure and capillary-pressure relations for colloidal systems, J. Colloid Interf. Sci. 168(1), 266–268.Google Scholar
  2. Achanta, S., Cushman, J. H. and Okos, M. R.: 1994, On multicomponent, multiphase thermomechanics with interfaces, Int. J. Eng. Sci. 32(11), 1717–1738.Google Scholar
  3. Achanta, S., Okos, M. R., Cushman, J. H. and Kessler, D. P.: 1997, Moisture transport in shrinking gels during saturated drying, AIChE J. 43(8), 2112–2122.Google Scholar
  4. Bennethum, L. S. and Cushman, J. H.: 1996a, Multiscale, hybrid mixture theory for swelling systems. 1. Balance laws, Int. J. Eng. Sci. 34(2), 125–145.Google Scholar
  5. Bennethum, L. S. and Cushman, J. H.: 1996b, Multiscale, hybrid mixture theory for swelling systems. 2. Constitutive theory, Int. J. Eng. Sci. 34(2), 147–169.Google Scholar
  6. Bennethum, L. S. and Cushman, J. H.: 1999, Coupled solvent and heat transport of a mixture of swelling porous particles and fluids: single time-scale problem, Transp. Porous Media 36(2), 211–244.Google Scholar
  7. Bennethum, L. S. and Giorgi, T.: 1997, Generalized Forchheimer equation for two-phase flow based on hybrid mixture theory, Transp. Porous Media 26(3), 261–275.Google Scholar
  8. Bennethum, L. S. and Weinstein, T.: Three pressures in swelling porous media, Transp. Porous Media (in press).Google Scholar
  9. Bennethum, L. S., Murad, M. A. and Cushman, J. H.: 1997, Modified Darcy's law, Terzaghi's effective stress principle and Fick's law for swelling clay soils, Comput. Geotech. 20(3–4), 245–266.Google Scholar
  10. Bennethum, L. S., Murad, M. A. and Cushman, J. H.: 2000, Macroscale thermodynamics and the chemical potential for swelling porous media, Transp. Porous Media 39(2), 187–225.Google Scholar
  11. Bowen, R. M.: Theory of mixtures, in: 1971, A.C. Eringen (ed.), Continuum Physics, Vol. 3, Academic Press, New York, pp. 2–127.Google Scholar
  12. Bowen, R. M.: 1982, Compressible porous-media models by use of the theory of mixtures, Int. J. Eng. Sci. 20(6), 697–735.Google Scholar
  13. Coleman, B. D. and Noll, W.: 1963, The thermodynamics of elastic materials with heat conduction and viscosity, Archiv. Rational Mech. Anal. 13, 167–178.Google Scholar
  14. Cushman, J. H.: 1997, The physics of fluids in hierarchical porous media: Angstroms of miles, Theory and Applications of Transport in Porous Media, Vol. 10, Kluwer Academic Publishers, Dordrecht.Google Scholar
  15. Eringen, A. C.: 1980, Mechanics of Continua, 2nd edn., R.E. Krieger Pub. Co., Huntington, New York.Google Scholar
  16. Etzler, F. M. and Conners, J. J.: 1991a, Structure transitions in vicinal water – pore-size and temperature-dependence of heat-capacity of water in small pores, Langmuir 7(10), 2293–2297.Google Scholar
  17. Etzler, F. M. and Conners, J. J.: 1991b, The structure and properties of vicinal water – recent advances, Abstracts of Papers of the American Chemical Society, Vol. 201, 246-COLL.Google Scholar
  18. Ferry, J.: 1980, Viscoelastic Properties of Polymers, Wiley, New York.Google Scholar
  19. Hassanizadeh, S.M. and Gray, W. G.: 1979a, General conservation equations for multiphase systems: 1. Averaging procedure, Adv. Water Resour. 2, 131–144.Google Scholar
  20. Hassanizadeh, S.M. and Gray, W. G.: 1979b, General conservation equations formultiphase systems: 2. mass, momenta, energy, and entropy equations, Adv. Water Resour. 2, 191–208.Google Scholar
  21. Hassanizadeh, S.M. and Gray, W. G.: 1980, General conservation equations for multiphase systems: 3. Constitutive theory for porous media flow, Adv. Water Resour. 3, 25–40.Google Scholar
  22. Hassanizadeh, S. M. and Gray, W.: 1987, High velocity flow in porous media, Transp. Porous Media 2, 521–531.Google Scholar
  23. Hornung, U.: 1997, Homogenization and porous media, Interdisciplinary Applied Mathematics, Vol. 6, Springer, New York.Google Scholar
  24. Low, P. F.: 1980, The swelling of clay. 2. Montmorillonites. Soil Sci. Soc. Am. J. 44(4), 667–676.Google Scholar
  25. Low, P. F.: 1981, The swelling of clay. 3. Dissociation of exchangeable cations, Soil Sci. Soc. Am. J. 45(6), 1074–1078.Google Scholar
  26. Low, P. F.: 1987,Structural component of the swelling pressure of clays, Langmuir 3(1), 18–25.Google Scholar
  27. Low, P. F. and Margheim, J. F.: 1979, Swelling of clay. 1. Basic concepts and empirical equations, Soil Sci. Soc. Am. J. 43(3), 473–481.Google Scholar
  28. Murad, M. A. and Cushman, J. H.: 1996, Multiscale flow and deformation in hydrophilic swelling porous media, Int. J. Eng. Sci. 34(3), 313–338.Google Scholar
  29. Murad, M. A. and Cushman, J. H.: 1997, Multiscale theory of swelling porous media: II. Dual porosity models for consolidation of clays incorporating physicochemical effects, Transp. Porous Media 28(1), 69–108.Google Scholar
  30. Murad, M. A. and Cushman, J. H.: 2000 Thermomechanical theories for swelling porous media with microstructrue, Int. J. Eng. Sci. 38(5), 517–564.Google Scholar
  31. Murad, M. A., Bennethum, L. S. and Cushman, J. H.: 1995, A multiscale theory of swelling porousmedia. 1. Application to one-dimensional consolidation, Transp. Porous Media 19(2), 93–122.Google Scholar
  32. Plumb, O. A. and Whitaker, S.: 2000, Diffusion, adsorption and dispersion in porous media: smallscale averaging and local volume averaging, in: J. H. Cushman (ed.), Dynamics of Fluid in Hierarchical Porous Media, Academic Press, New York, pp. 97–176.Google Scholar
  33. Schoen, M., Rhykerd, C. L., Diestler, D. J. and Cushman, J. H.: 1989, Shear forces in molecularly thin films, Science 245, 1223–1225.Google Scholar
  34. Segel, L. A. and Handelman, G. H.: 1977, Mathematics Applied to Continuum Mechanics, Macmillan, New York.Google Scholar
  35. Singh, P. P., Cushman, J. H. and Maier, D. E.: Multiscale fluid transport theory for swelling biopolymers, Chem. Eng. Sci. (submitted).Google Scholar
  36. Thomas, N. L. and Windle, A. H.: 1982, A theory of case-II diffusion, Polymer 23(4), 529–542.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Pawan P. Singh
    • 1
  • John N. Cushman
    • 2
  • Lynn S. Bennethum
    • 3
  • Dirk E. Maier
    • 1
  1. 1.Department of Agricultural and Biological EngineeringPurdue UniversityWest LafayetteU.S.A
  2. 2.Center for Applied Math, Math Science BuildingPurdue UniversityWest LafayetteU.S.A
  3. 3.Center for Computational MathematicsC.U. DenverDenverU.S.A

Personalised recommendations