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Open AccessArticle

A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples

1
Institute of Mathematics, National Academy of Sciences of Ukraine, 3, Tereshchenkivs’ka Street, 01004 Kyiv, Ukraine
2
Institute of Biocybernetics and Biomedical Engineering, PAS, Ks. Trojdena 4, 02 796 Warsaw, Poland
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(3), 396; https://doi.org/10.3390/sym12030396
Received: 27 December 2019 / Revised: 1 February 2020 / Accepted: 5 February 2020 / Published: 4 March 2020
(This article belongs to the Special Issue Lie Symmetries at Work in Biology and Medicine)
Fluid and solute transport in poroelastic media is studied. Mathematical modeling of such transport is a complicated problem because of the volume change of the specimen due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the variables adds also substantial complication in both theoretical and experimental investigations. The one-dimensional version of the theory is less complex and may serve as an approximation in some problems, and therefore, a one-dimensional (in space) model of fluid and solute transport through a poroelastic medium with variable volume is developed and analyzed. In order to obtain analytical results, the Lie symmetry method is applied. It is shown that the governing equations of the model admit a non-trivial Lie symmetry, which is used for construction of exact solutions. Some examples of the solutions are discussed in detail. View Full-Text
Keywords: poroelastic material; continuity equations; nonlinear PDE; Lie symmetry; exact solution; steady-state solution poroelastic material; continuity equations; nonlinear PDE; Lie symmetry; exact solution; steady-state solution
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Cherniha, R.; Stachowska-Pietka, J.; Waniewski, J. A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples. Symmetry 2020, 12, 396.

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