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

Abstract

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.

A Mathematical Model for Transport in Poroelastic Materials with Variable Volume: Derivation, Lie Symmetry Analysis and Examples—Part 2

“…where σ denotes sieving coefficient for tissue, λ * = λ + 2µ is the Lame coefficient [6,21] and the parameter κ describes the contribution of squared dilatation to the stress tensor. Obviously, setting κ = 0 one obtains the standard, linear expression for τ.…”

“…Having the formulae for fluxes q V , q C , and q S presented above, we have constructed the model consisting of five nonlinear PDEs in a quite similar way as it was done in [21] (we remind that internal sources/sinks were not taken into account in that paper). The basic equations of the model read as…”

“…All the notations arising in the above formulae and in those below are explained in Table 1. density of volumetric fluid absorption by lymphatics q 0 rate of lymphatic absorption for p = 0 q 1 sensitivity of lymphatic absorption to increase in p q C density of volumetric fluid flow across blood capillary wall q V net fluid flow into the tissue from internal sources/sinks q S net solute flow into the tissue from internal sources/sinks Obviously, the nonlinear PDE system (1)-( 5) is an essential generalization of that derived in paper [21] (see Equations ( 21)-(25) therein). Now we present a theorem presenting Lie's symmetry properties of (1)-( 5).…”

“…Now we present a theorem presenting Lie's symmetry properties of (1)-( 5). One contains Theorem 1 from [21] as a particular result (see item (v) below).…”

“…Our previous analysis considered nonperfused poroelastic material (PEM). The linear theory of poroelasticity was applied in order to model its volume/shape deformation due to external hydrostatic and/or osmotic pressure [21]. In this study, we extend previously developed model focusing on the biological tissue perfused by blood and drained by lymphatic system [10,17,22].…”