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

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.