Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Unidirectional infiltration in deformable porous media: mathematical modeling and self-similar solution

Authors: L. Billi and A. Farina
Journal: Quart. Appl. Math. 58 (2000), 85-101
MSC: Primary 76S05; Secondary 35Q35, 74F10
DOI: https://doi.org/10.1090/qam/1738559
MathSciNet review: MR1738559
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Abstract: This study is motivated by the necessity to develop models and to obtain mathematical results which can help in improving certain industrial processes used for manufacturing composite materials. We focus on unidirectional infiltration processes in deformable porous material and on the analytical study of the related mathematical problem. A key point in developing the model is the use of a set of Lagrangian coordinates fixed on the solid. This technique allows us to simplify the mathematical problem. In the Eulerian formalism, in fact, such a problem is a free boundary problem characterized by the presence of two time-dependent interfaces. The use of material coordinates fixed on the solid allows, vice-versa, to fix one boundary and so to obtain a nonlinear one-phase Stefan problem. The latter, then, has been studied from the analytical viewpoint. In particular, we have proved the existence of a solution exhibiting a self-similar form.

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  • [1] R. K. Everett and J. R. Arsenault, Treatise on Material Science and Technology, Academic Press, 1991
  • [2] A. I. Isayev, Injection and Compression Molding Fundamentals, Marcel Dekker, 1987
  • [3] D. Purslow and R. Child, Autoclave molding of carbon-fiber reinforcing epoxies, Composites 17, 127-136 (1986)
  • [4] R. K. Upadhyay and E. W. Liang, Consolidation of advanced composites having volatile generation, Polymer Compos. 12, 96-108 (1995)
  • [5] A. E. Scheidegger, Hydrodynamics in porous media, Handbuch der Physik, Bd. VIII/2, Springer, Berlin, 1963, pp. 625–662. MR 0161518
  • [6] Arianna Passerini, Steady flows of a viscous incompressible fluid in a porous half-space, Math. Models Methods Appl. Sci. 4 (1994), no. 5, 705–732. MR 1300813, https://doi.org/10.1142/S021820259400039X
  • [7] A. Lyaghfouri, The inhomogeneous dam problem with linear Darcy’s law and Dirichlet boundary conditions, Math. Models Methods Appl. Sci. 6 (1996), no. 8, 1051–1077. MR 1428145, https://doi.org/10.1142/S0218202596000432
  • [8] J. L. Sommer and A. Mortensen, Forced unidirectional infiltration of deformable porous media, J. Fluid. Mech. 311, 193-217 (1996)
  • [9] R. M. Bowen, The theory of mixtures, in Continuum Physics, A. C. Eringen, ed., Vol. 3, Academic Press, 1976
  • [10] K. R. Rajagopal and L. Tao, Mechanics of mixtures, Series on Advances in Mathematics for Applied Sciences, vol. 35, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370661
  • [11] Luigi Preziosi, The theory of deformable porous media and its application to composite material manufacturing, Surveys Math. Indust. 6 (1996), no. 3, 167–214. MR 1422885
  • [12] C. A. Truesdell and R. A. Toupin, The classical field theory, in Handbuck der Phisik, S. Flügge, C. Truesdell, eds., Vol. III/1, Springer-Verlag, 1960
  • [13] R. M. Bowen, Incompressible porous media models by use of the theory of mixtures, Internat. J. Engrg. Sci. 18, 1129-1148 (1980)
  • [14] D. Munaf, A. S. Wineman, K. R. Rajagopal, and D. W. Lee, A boundary value problem in groundwater motion analysis—comparison of predictions based on Darcy’s law and the continuum theory of mixtures, Math. Models Methods Appl. Sci. 3 (1993), no. 2, 231–248. MR 1212941, https://doi.org/10.1142/S0218202593000138
  • [15] K. R. Rajagopal, A. S. Wineman, and M. Gandhi, On boundary conditions for a certain class of problems in mixture theory, Internat. J. Engrg. Sci. 24 (1986), no. 8, 1453–1463. MR 858456, https://doi.org/10.1016/0020-7225(86)90074-1
  • [16] Douglas E. Kenyon, The theory of an incompressible solid-fluid mixture, Arch. Rational Mech. Anal. 62 (1976), no. 2, 131–147. MR 0413710, https://doi.org/10.1007/BF00248468
  • [17] L. Tao, K. R. Rajagopal, and A. S. Wineman, Unsteady diffusion of fluids through solids undergoing large deformations, Math. Models Methods Appl. Sci. 1 (1991), no. 3, 311–346. MR 1127571, https://doi.org/10.1142/S0218202591000162
  • [18] I. S. Liu, On chemical potential and incompressible porous media, J. Mech. 19, 327-342 (1980)
  • [19] G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech. 30, 197-207 (1967)
  • [20] J. S. Hou, M. H. Holmes, W. M. Lai, and V. C. Mow, Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications, J. Biomech. Eng. 111, 78-87 (1989)
  • [21] Y. Lanir, S. Sauob, and P. Maretsky, Nonlinear finite deformation response of open cell polyurethane sponge to fluid filtration, Trans. ASME E: J. Appl. Mech. 57, 449-454 (1990)
  • [22] K. H. Parker, R. V. Mehta, and C. G. Caro, Steady flow in porous, elastically deformable materials, Trans. ASME E: J. Appl. Mech. 54, 794-800 (1987)
  • [23] Y. R. Kim, S. P. McCarthy, and J. P. Fanucci, Compressibility and relaxation of fiber reinforcements during composites processing, Polymer Compos. 12, 13-19 (1991)
  • [24] Antonio Fasano and Mario Primicerio, Free boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J. Math. Anal. Appl. 72 (1979), no. 1, 247–273. MR 552335, https://doi.org/10.1016/0022-247X(79)90287-7
  • [25] Anvarbek M. Meirmanov, The Stefan problem, De Gruyter Expositions in Mathematics, vol. 3, Walter de Gruyter & Co., Berlin, 1992. Translated from the Russian by Marek Niezgódka and Anna Crowley; With an appendix by the author and I. G. Götz. MR 1154310

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DOI: https://doi.org/10.1090/qam/1738559
Article copyright: © Copyright 2000 American Mathematical Society

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