# A Review of Mixture Theory for Deformable Porous Media and Applications
## 3 Applications of Mixture Theory
### Flow Induced Deformation of Porous Biological Tissues  
The modeling details of flow induced deformation from pressurized cavities in absorbing porous tissue (as presented by [PERSON] and [PERSON] [16]) are reproduced in this section. The schematic diagram of the mathematical model under consideration is presented in Figure 1.  
The mathematical model is constructed under following assumptions:  
* tissue to be deformable fully fluid saturated of infinite extent
* initially tissue is homogeneous and isotropic
* capillaries and lymphatics are distributed uniformly through the porous tissue
* tissue has ability to absorb the fluid at a rate proportional to local pressure. Using above assumptions the Equations (7), (8), (15) and (18), the equation for fluid pressure \(p\) is written  
\[\frac{\partial p}{\partial t}=\frac{\alpha}{r^{2}}\frac{\partial}{\partial r} \left\{r^{2}\kappa\left(p\delta\right)\frac{\partial p}{\partial r}\right\}-\beta p \tag{19}\]  
where \(\beta\) is proportionality constant and contribution of this term in the continuity equation comes from the assumption of fluid absorption in the tissue. The permeability of the porous tissue considered here is \(\kappa(p)=e^{mp}\), with \(m\) as a material constant. Initial and boundary conditions for fluid pressure \(p(r,t)\) are given by  
\[p(r,0)=f(r)=0,\quad p(a,t)=\Delta P(t)=1,\quad p(r,t)\to 0\quad\text{as} \quad r\to\infty \tag{20}\]  
Displacement of solid \(u(r,t)\) in terms of fluid pressure as derived from the relations \(\phi=\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}u)\), \(p(r,t)=H_{a}\phi(r,t)\) and boundary conditions for solid displacement are given by  
\[u(r,t)=\frac{\gamma}{r^{2}}\left\{\int_{a(t)}^{r}s^{2}p(s,t)ds+\frac{a^{3}(t) \Delta P(t)}{2(1-\lambda)}\right\} \tag{21}\]  
Here \(\alpha,\delta,\gamma,\lambda\) are dimensionless parameters. Equations (19) and (21) together with boundary conditions Equation (20) are now solved numerically for the fluid pressure \(p(r,t)\) and solid displacement \(u(r,t)\). The details of the solution procedure are not presented here. Figure 1: Radial flow after injection of fluid in an infinite porous media. The growth of cavity is occurring from radius \(1\) to \(a(t)\). In Figure 2, fluid pressure is plotted as a function of radial distance \(r\) at specific times \(t=2,4,6\). Graphs are generated for the constant, \(m=0\), as well as for the nonlinear permeability, \(m=4\). Generally the permeability parameter \(m\) is in the range 0-10 for biological tissues and the steady-state of the fluid pressure is achieved at time \(t=6\), this is shown by the solid curve in Figure 2. It is important to note that for the nonlinear permeability case pressure decays slowly with radial distance as compared to the constant permeability case. In addition, steady state for fluid pressure is achieved earlier for constant permeability than nonlinear permeability. Moreover the fluid pressure \(p\) and the dilatation \(\phi\) are related directly by the relation \(p(r,t)=H_{a}\phi(r,t)\), therefore Figure 2 also explains porosity as a function of radial distance. In general the fluid volume fraction in the tissue increases due to deformation of the porous material which results in an overall expansion of the porous material. Figure 3 presented the solid displacement \(u(r,t)\) as a function of radial distance \(r\) at specific times \(t=2,4,6\). Numerical computations are carried for both constant, \(m=0\) and nonlinear permeability \(m=4\). It is noted that solid displacement falls exponentially with radial distance for constant permeability case whereas a definite inflection occurs in the displacement due to nonlinearities in permeability parameter \(m\). Moreover, tissue solid displacement is increased for the nonlinear permeability as compared to the constant permeability. In addition in expanded pore region in Figure 3 is associated with the higher pressure and corresponding higher displacement. This is due to the coupled nature of solid permeability and displacement with fluid pressure decay.