A Mixture Theory Approach for Modeling Fluid Flow Through Deformable Porous Tissues
In this thesis, the problems of flow and ion-induced deformation of soft porous biological tissues have been examined by using continuum mixture theory approach. In particular, we focus on the tissue deformation due to non-Newtonian fluid and externally applied magnetic field. In this regard, we first analyze the problem of non-Newtonian flow-induced deformation from pressurized cavities in absorbing porous tissues. Specifically, a model with a spherical cavity embedded in an infinite porous medium is used to find fluid pressure and solid displacement in the tissue as a function of non-dimensional radial distance and time. The governing nonlinear equations have been solved numerically to highlight effects of various emerging parameters. Furthermore, based on the geometry of the previous problem, the effect of the externally applied magnetic field on flow-induced deformation of absorbing porous tissues is investigated. A biphasic mixture theory approach has been used to develop a mathematical model. The governing dimensionless equations for fluid pressure and solid displacement have been solved numerically using the method of lines approach and the trapezoidal rule, respectively. The effect of magnetic parameter on fluid pressure and solid displacement is illustrated graphically. Finally, the problem of ion-induced deformation of articular cartilage with strain-dependent nonlinear permeability and magnetohydrodynamic effects is presented. The governing set of coupled partial differential equations are nondimensionalized using suitable dimensionless variables. Analytical solutions are provided for the constant permeability case whereas for the nonlinear permeability case the displacement equation is solved numerically using the method of lines technique. The influence of magnetic and permeability parameter on solid displacement and fluid pressure is illustrated graphically. In some cases, a graphical comparison to the previously reported literature is also presented.