# A Review of Mixture Theory for Deformable Porous Media and Applications
## 3 Applications of Mixture Theory
### Unidirectional Capillary Rise into Deformable Porous Material  
In this section a one-dimensional model of a sponge-like deformable porous material which is in contact with an infinite bath is discussed. The geometry of the model is shown in Figure 7. The porous material is initially dry and rigid having uniform solid volume fraction \(\phi_{0}\). The upper end of the deformable porous material is fixed and the contact position of porous material and liquid is represented by \(z=0\). The infiltration of liquid starts from infinite bath of liquid (i.e., \(z=0\)) and the material's upper end is open to atmospheric pressure \(p_{A}\). After time \(t>0\), the liquid starts to penetrate into initially dry porous material because of capillary forces associated with the pores of the porous material with the assumption that the capillary pressure \(p_{c}<0\), this causes the deformation in the porous material. The upper border of the wet porous material which we define as wet material-dry material interface \(z=h_{\ell}(t)\) and the bottom border which develops after the imbibition of liquid which we define as liquid-wet material interface \(z=h_{s}(t)\). The pressure of liquid bath is assumed to be hydrostatic, that is, \(p=p_{A}-\rho_{\ell}gh_{s}\) at the solid liquid interface. Figure 6: Solid volume fraction as a function of Lagrangian coordinate \(Y\) and normalized time \(T\) for pressure \(p(T)=1.25\sin(\frac{50\pi}{6}T)\). In the light of above assumptions, the set of Equations (7), (8), (15), and (17) for unidirectional capillary rise dynamics are obtained. The detailed equations are given in Siddique and Anderson [19]. It is important to note that for capillary rise dynamics Equation (15) in the presence of gravity effect takes the following form  
\[\
abla p=\
abla\cdot\sigma+(\rho_{s}+\rho_{\ell})\mathbf{g} \tag{26}\]  
where \(\rho_{s}\) and \(\rho_{\ell}\) are solid and liquid densities. After performing necessary mathematical manipulation steps Equations (7), (8), (17) along with (26) simplifies to the nonlinear partial differential equation (PDE) for solid volume fraction \(\phi\)[19]  
\[\frac{\partial\phi}{\partial t}+c(t)\frac{\partial\phi}{\partial z}=-\frac{ \partial}{\partial z}\left[\phi k(\phi)\left\{\sigma^{\prime}(\phi)\frac{ \partial\phi}{\partial z}-\rho\phi\right\}\right] \tag{27}\]  
The ordinary differential equations at the interface positions \(h_{s}(t)\) and \(h_{\ell}(t)\) are  
\[\frac{dh_{s}}{dt}=c(t)+k(\phi)\left.\left[\sigma^{\prime}(\phi)\frac{\partial \phi}{\partial z}-\rho\phi\right]\right|_{h_{\ell}^{+}} \tag{28}\]  
\[\frac{dh_{l}}{dt}=c(t)-\frac{\phi k(\phi)}{(1-\phi)}\left.\left[\sigma^{\prime }(\phi)\frac{\partial\phi}{\partial z}-\rho\phi\right]\right|_{h_{\ell}^{-}} \tag{29}\]  
The function \(c(t)\) that appears in Equations (27)-(29) has the following form [26]  
\[c(t)=-\frac{(1-\phi_{0})}{\phi_{0}}\left.\left[\frac{\phi k(\phi)}{(1-\phi)} \left(\sigma^{\prime}(\phi)\frac{\partial\phi}{\partial z}-\rho\phi\right) \right]\right|_{h_{\ell}^{-}} \tag{30}\]  
This completes the set of equations needed to capture the dynamics of capillary rise. Our attention here is to highlight two special cases with the help of above set of equations. In the absence of gravity effects, the above systems of Equations (27)-(29) admits similarity solution and interested readers can find the related details in [PERSON] and [PERSON] [19] and [PERSON] [29]. When similarity variables is introduced in the above set of equations, the outcome is an ordinary differential equation (ODE) for solid volume fraction and algebraic equations for interface positions. Finally these ODEs along with the interface position yields error function solution for special choices of permeability function \(k(\phi)\) and \(\sigma(\phi)\)[29]. In the presence of gravity effects, the PDE (27) for solid volume fraction \(\phi\) along with ODE Equations (28) and (29) is solved numerically. The details of numerical solution can be found in  
Figure 7: The dynamics of capillary rise after \(t>0\).