Analysis of a mathematical model for fluid transport in poroelastic materials in 2D space

Roman Cherniha; Vasyl' Davydovych; Joanna Stachowska-Pietka; Jacek Waniewski; Roman Cherniha; Vasyl' Davydovych; Joanna Stachowska-Pietka; Jacek Waniewski

doi:10.3934/mbe.2026083

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 8: 2287-2306. doi: 10.3934/mbe.2026083

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Analysis of a mathematical model for fluid transport in poroelastic materials in 2D space

1.
Department of Mathematics, National University of Kyiv-Mohyla Academy, 2, Skovoroda Street, Kyiv 04070, Ukraine
2.
Institute of Mathematics, NAS of Ukraine, 3, Tereshchenkivs'ka Street, 01004 Kyiv, Ukraine
3.
Institute of Biocybernetics and Biomedical Engineering, PAS, Ks. Trojdena 4, 02 796 Warsaw, Poland

Received: 04 March 2026 Revised: 09 June 2026 Accepted: 22 June 2026 Published: 21 July 2026

Mathematical models that are based on the poroelastic theory are applied in biology, medicine, and biomedical engineering to describe a wide range of processes. In this work, a mathematical model for poroelastic materials (PEM) with variable volumes is developed in a multidimensional case. Governing equations of the model are constructed using continuity equations, which reflect the well-known physical laws. The deformation vector is specified using the Terzaghi effective stress tensor. In the two-dimensional space case, the model is studied by analytical methods. Using the classical Lie method, it is proven that the relevant nonlinear system of the $ (1+2) $-dimensional governing equations admits highly nontrivial Lie symmetries which lead to an infinite-dimensional Lie algebra. The radially-symmetric case is studied in details. It is shown how correct boundary conditions in the case of PEM in the form of a ring and an annulus are constructed. As a result, boundary-value problems with a moving boundary describing the ring (annulus) deformation are constructed. The relevant nonlinear boundary-value problems are analytically solved in the stationary case. In particular, the analytical formulae for unknown deformations and an unknown radius of the annulus are derived. Moreover, illustrative plots for parameters, which are typical for the tumor tissue deformation, are presented.
- poroelastic material,
- continuity equations,
- nonlinear PDE,
- Lie symmetry,
- exact solution,
- steady-state solution
Citation: Roman Cherniha, Vasyl' Davydovych, Joanna Stachowska-Pietka, Jacek Waniewski. Analysis of a mathematical model for fluid transport in poroelastic materials in 2D space[J]. Mathematical Biosciences and Engineering, 2026, 23(8): 2287-2306. doi: 10.3934/mbe.2026083

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Abstract

Mathematical models that are based on the poroelastic theory are applied in biology, medicine, and biomedical engineering to describe a wide range of processes. In this work, a mathematical model for poroelastic materials (PEM) with variable volumes is developed in a multidimensional case. Governing equations of the model are constructed using continuity equations, which reflect the well-known physical laws. The deformation vector is specified using the Terzaghi effective stress tensor. In the two-dimensional space case, the model is studied by analytical methods. Using the classical Lie method, it is proven that the relevant nonlinear system of the $ (1+2) $-dimensional governing equations admits highly nontrivial Lie symmetries which lead to an infinite-dimensional Lie algebra. The radially-symmetric case is studied in details. It is shown how correct boundary conditions in the case of PEM in the form of a ring and an annulus are constructed. As a result, boundary-value problems with a moving boundary describing the ring (annulus) deformation are constructed. The relevant nonlinear boundary-value problems are analytically solved in the stationary case. In particular, the analytical formulae for unknown deformations and an unknown radius of the annulus are derived. Moreover, illustrative plots for parameters, which are typical for the tumor tissue deformation, are presented.

References

[1]	L. A. Taber, Nonlinear Theory of Elasticity: Applications in Biomechanics, World Scientific, New Jersey, 2004.
[2]	B. Loret, F. M. F. Simoes, Biomechanical Aspects of Soft Tissue, CRC Press, Boca Raton, 2017.
[3]	E. Detournay, A. H. D. Cheng, Fundamentals of Poroelasticity. In: Analysis and Design Methods, Pergamon Press, Oxford, 1993.
[4]	O. Coussy, Mechanics of Porous Continua, Wiley & Sons Ltd., New York, 1995.
[5]	O. Coussy, Mechanics and Physics of Porous Solids, Wiley & Sons Ltd., New York, 2010.
[6]	U. Lacis, G. A. Zampogna, S. Bagheri, A computational continuum model of poroelastic beds, Proc. R. Soc. A: Math. Phys. Eng. Sci., 473 (2017), 20160932. https://doi:10.1098/rspa.2016.0932 doi: 10.1098/rspa.2016.0932
[7]	J. I. Siddique, A. Ahmed, A. Aziz, C. M. Khalique, A review of mixture theory for deformable porous media and applications, Appl. Sci., 7 (2017), 917. https://doi.org/10.3390/app7090917 doi: 10.3390/app7090917
[8]	V. Guerriero, S. Mazzoli, Theory of effective stress in soil and rock and implications for fracturing processes: A review, Geosciences, 11 (2021), 119. https://doi.org/10.3390/geosciences11030119 doi: 10.3390/geosciences11030119
[9]	S. Speziale, S. Sivaloganathan, Poroelastic theory of transcapillary flow: Effects of endothelial glycocalyx deterioration, Microvasc. Res., 78 (2009), 432–441. https://doi.org/10.1016/j.mvr.2009.07.007 doi: 10.1016/j.mvr.2009.07.007
[10]	S. Speziale, G. Tenti, S. Sivaloganathan, A poroelastic model of transcapillary flow in normal tissue, Microvasc. Res., 75 (2008), 285–295. https://doi.org/10.1016/j.mvr.2007.07.001 doi: 10.1016/j.mvr.2007.07.001
[11]	F. Travascio, R. Serpieri, S. Asfour, Articular cartilage biomechanics modeled via an intrinsically compressible biphasic model: Implications and deviations from an incompressible biphasic approach, in Proceedings of the ASME 2013 Summer Bioengineering Conference, 2013. https://doi.org/10.1115/SBC2013-14082
[12]	P. Mascheroni, M. Carfagna, A. Grillo, D. P. Boso, B. A. Schrefler, An avascular tumor growth model based on porous media mechanics and evolving natural states, Math. Mech. Solids., 23 (2018), 686–712. https://doi.org/10.1177/1081286517711217 doi: 10.1177/1081286517711217
[13]	D. R. Sowinski, M. D. McGarry, E. E. Van Houten, S. Gordon-Wylie, J. B. Weaver, K. D. Paulsen, Poroelasticity as a model of soft tissue structure: Hydraulic permeability reconstruction for magnetic resonance elastography in silico, Front Phys., 8 (2021), 617582. https://doi.org/10.3389/fphy.2020.617582 doi: 10.3389/fphy.2020.617582
[14]	R. Cherniha, J. Stachowska-Pietka, J. Waniewski, A mathematical model for transport in poroelastic materials with variable volume: Derivation, lie symmetry analysis, and examples, Symmetry, 12 (2020), 396. https://doi.org/10.3390/sym12030396 doi: 10.3390/sym12030396
[15]	R. Cherniha, V. Davydovych, J. Stachowska-Pietka, J. Waniewski, A mathematical model for transport in poroelastic materials with variable volume: Derivation, Lie symmetry analysis and examples–Part 2, Symmetry, 14 (2022), 109. https://doi.org/10.3390/sym14010109 doi: 10.3390/sym14010109
[16]	R. Cherniha, J. Stachowska-Pietka, J. Waniewski, A mathematical model for two solutes transport in a poroelastic material and its applications, Commun. Nonlinear Sci. Numer. Simulat., 132 (2024), 107905. https://doi.org/10.1016/j.cnsns.2024.107905 doi: 10.1016/j.cnsns.2024.107905
[17]	E. Samaniego, C. Anitescu, S. Goswami, V. M. Nguyen-Thanh, H. Guo, K. Hamdia, et al., An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications, Comput. Methods Appl. Mech. Eng., 362 (2020), 112790. https://doi.org/10.1016/j.cma.2019.112790 doi: 10.1016/j.cma.2019.112790
[18]	K. P. Soldatos, On the bending problem of a polar linearly elastic cantilever, Mech. Res. Commun., 147 (2025), 104448. https://doi.org/10.1016/j.mechrescom.2025.104448 doi: 10.1016/j.mechrescom.2025.104448
[19]	R. Cherniha, V. Davydovych, A. Vorobyova, New Lie symmetries and exact solutions of a mathematical model describing solute transport in poroelastic materials, Math. Comp. Appl., 29 (2024), 43. https://doi.org/10.3390/mca29030043 doi: 10.3390/mca29030043
[20]	D. A. Narmoneva, J. Y. Wang, L. A. Setton, A noncontacting method for material property determination for articular cartilage from osmotic loading, Biophy. J., 81 (2001), 3066–3076. https://doi.org/10.1016/S0006-3495(01)75945-0 doi: 10.1016/S0006-3495(01)75945-0
[21]	P. Li, M. Schanz, Wave propagation in a 1-D partially saturated poroelastic column, Geophys. J. Int., 184 (2011), 1341–1353. https://doi.org/10.1111/j.1365-246X.2010.04913.x doi: 10.1111/j.1365-246X.2010.04913.x
[22]	G. W. Bluman, A. F. Cheviakov, S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York, 2010.
[23]	D. J. Arrigo, Symmetry Analysis of Differential Equations, John Wiley & Sons, Hoboken, 2015.
[24]	R. Cherniha, M. Serov, O. Pliukhin, Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Wymmetry, Exact Solutions and Their Applications, CRC Press, Boca Raton, 2018.
[25]	S. Mandal, S. Ghosh, Modelling and nonclassical symmetry analysis of a complex porous media flow in a dilating channel, Physica D, 481 (2025), 134834. https://doi.org/10.1016/j.physd.2025.134834 doi: 10.1016/j.physd.2025.134834
[26]	R. Cherniha, Comments on the paper Modelling and nonclassical symmetry analysis of a complex porous media flow in a dilating channel, preprint, arXiv: 2511.08656.
[27]	R. Cherniha, V. Davydovych, A hunter-gatherer-farmer population model: Lie symmetries, exact solutions and their interpretation, Euro. J. Appl. Math., 30 (2019), 338–357. https://doi.org/10.1017/S0956792518000104 doi: 10.1017/S0956792518000104
[28]	E. Alimirzaluo, M. Nadjafikhah, J. Manafian, Some new exact solutions of $(3+1)$-dimensional Burgers system via Lie symmetry analysis, Adv. Differ. Equ., 2021 (2021), 60. https://doi.org/10.1186/s13662-021-03220-3 doi: 10.1186/s13662-021-03220-3
[29]	R. Cherniha, V. Davydovych, A hunter-gatherer-farmer population model: New conditional symmetries and exact solutions with biological interpretation, Acta Appl. Math., 182 (2022), 4. https://doi.org/10.1007/s10440-022-00537-3 doi: 10.1007/s10440-022-00537-3
[30]	A. Paliathanasis, Symmetry analysis for the 2D Aw-Rascle traffic-flow model of multi-lane motorways in the Euler and Lagrange variables, Symmetry, 15 (2023), 1525. https://doi.org/10.3390/sym15081525 doi: 10.3390/sym15081525
[31]	S. Shagolshem, B. Bira, S. Sil, Application of symmetry analysis to viscoelastic fluid model, Commun. Nonlinear. Sci. Numer. Simulat., 125 (2023), 107417. https://doi.org/10.1016/j.cnsns.2023.107417 doi: 10.1016/j.cnsns.2023.107417
[32]	M. Gorgone, F. Oliveri, E. Sgroi, Hierarchy of coupled Burgers-like equations induced by conditional symmetries, Z. Angew. Math. Phys., 76 (2025), 8. https://doi.org/10.1007/s00033-024-02405-6 doi: 10.1007/s00033-024-02405-6
[33]	K. Terzaghi, Relation between soil mechanics and foundation engineering, in Proceedings of the International Conference on Soil Mechanics and Foundation Engineering, (1936), 13–18.
[34]	R. Cherniha, J. Stachowska-Pietka, J. Waniewski, A mathematical model for fluid-glucose-albumin transport in peritoneal dialysis, Int. J. Appl. Math. Comput. Sci., 24 (2014), 837–851. https://doi.org/10.2478/amcs-2014-0062 doi: 10.2478/amcs-2014-0062
[35]	S. P. Lloyd, The infinitesimal group of the Navier–Stokes equations, Acta Mech., 38 (1981), 83–98. https://doi.org/10.1007/BF01351464 doi: 10.1007/BF01351464
[36]	R. Cherniha, V. Davydovych, Lie symmetries, reduction and exact solutions of the $(1+2)$-dimensional nonlinear problem modeling the solid tumour growth, Commun. Nonlinear. Sci. Numer. Simulat., 80 (2020), 104980. https://doi.org/10.1016/j.cnsns.2019.104980 doi: 10.1016/j.cnsns.2019.104980
[37]	P. A. Netti, L. T. Baxter, Y. Boucher, R. Skalak, R. K. Jain, Macro- and microscopic fluid transport in living tissues: Application to solid tumors, AIChE J., 3 (1997), 818–834. https://doi.org/10.1002/aic.690430327 doi: 10.1002/aic.690430327
[38]	R. Cherniha, J. R. King, Lie and conditional symmetries of a class of nonlinear (1 + 2)-dimensional boundary value problems, Symmetry, 7 (2015), 1410–1435. https://doi.org/10.3390/sym7031410 doi: 10.3390/sym7031410

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