A Review of Mixture Theory for Deformable Porous Media and Applications

Abstract

Abstract:Mixture theory provides a continuum framework to model a multi-phase system. The basic assumption is, at any instant of time all phases are present at every material point and momentum and mass balance equations are postulated. This paper reviews the recent developments in mixture theory and focuses on the applications of the theory in particular areas of biomechanics, composite manufacturing and infiltration into deformable porous materials. The complexity based upon different permeability and stress functions is also addressed. The review covers the literature presented in the past fifty years and summarizes applications of mixture theory in specific areas of interest, for the sake of brevity, only necessary details are provided rather than complete modeling and simulation.

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

“…and the function e = ∂u ∂x is called dilatation,j V the volumetric flux across the PEM, X the initial position within PEM, and x (t) the position at time t, i.e., after displacement. For the derivation of this equation and subsequent to those, we used the theory described in [17][18][19]. It comes from continuity law for the mass ρdV of the infinitesimal element dV and the mass density ρ that:…”

“…All the notations arising in the above formulae are explained in Table 1. In order to write the governing equations in the explicit form, one needs to substitute the expressions for flows from (14)- (18) into Equations (6)-(9), and (11) and the Terzaghi effective stress tensor into (12). Taking into account the equalities (10) and θ M = 1 − θ F , we prefer to exclude the unknown functions ρ M and θ M for the solid phase (matrix) in order to have a simpler equation corresponding to (1), instead of (7).…”

“…The first attempt to investigate transport processes through such a porous, elastic medium that might be deformed is to consider one-dimensional transport, i.e., along one axis or as in the case of spherical symmetry. An approach that is typically applied for the description of (bio)mechanical characteristics of the poroelastic material combined with its transport parameters is the theory of poroelasticity [5,[11][12][13][14][15][16][17][18][19]. The theory in general involves higher order tensors, includes many parameters, and is difficult for mathematical, numerical, and experimental investigations [5,[17][18][19].…”

Modeling Multiphase Flow Within and Around Deformable Porous Materials: A Darcy‐Brinkman‐Biot Approach

“…A similar situation pertains with regard to the coupling between fluid flow and solid mechanics. Theoretical and numerical approaches based on Biot's Theory of poroelasticity (Biot, 1941), Terzaghi's effective stress principle (Terzaghi, 1943), and Mixture Theory (Siddique et al., 2017) have been successful at modeling systems with flow in deformable porous media including arteries, biofilms, boreholes, hydrocarbon reservoirs, seismic systems, membranes, soils, swelling clays, and fractures (Auton & MacMinn, 2017; Barry et al., 1997; Jha & Juanes, 2014; Lo et al., 2002, 2005; MacMinn et al., 2016; Mathias et al., 2017; Santillán et al., 2017). However, as mentioned above, we still have very little understanding of how flow‐induced deformation of these solid materials affects the macroscopic flow around them (and thus their boundary conditions) or how fluid‐fluid interfaces behave when pushed against a soft porous medium and vice versa.…”

Investigating the physical effects in bacterial therapies for avascular tumors

“…Here we describe a mathematical model for bacteria-based cancer therapy within tumor spheroids. The model is formulated in the context of mixture theory, a continuum theory with a long history of applications to biological problems - see for example Ambrosi and Preziosi (2002); Breward et al (2001, 2002, 2003); Byrne and Preziosi (2003); Chaplain et al (2006); Preziosi and Tosin (2009) and the recent reviews of Siddique et al (2017); Pesavento et al (2017). Our aim is to evaluate the impact of bacterial chemotaxis and anti-tumor activity on spheroid size and composition.…”