Newtonian and non-Newtonian Flows into Deformable Porous Materials

Abstract

In this dissertation we examine fluid flow of Newtonian and non-Newtonian fluid into deformable porous materials. The one dimensional free boundary problems are modeled using mixture theory. The first problem we examine in this category of flows is a mathematical model for capillary rise of a fluid into an initially dry and deformable porous material. We use mixture theory to formulate the model. We obtain analytic results for steady state positions of the wet porous material-dry porous material interface as well as liquid-wet material interface. The time-dependent free-boundary problem is solved numerically and the results compared to the steady state predictions. In the absence of gravity, the liquid rises to an infinite height and the porous material deforms to an infinite depth, following square-root in time scaling. In contrast, in the presence of gravity, the liquid rises to a finite height and porous material deforms to a finite depth. Dependence on model parameters such as the solid liquid density ratio is also explored. We also examine the one-dimensional drainage of an incompressible liquid into an initially dry and deformable porous material. Here, we identify numerical solutions that quantify the e ffects of gravity, capillarity and solid to liquid density ratio on the time required for a finite volume of liquid to drain into a deformable porous material. We also study the capillary rise of a non-Newtonian liquid into a rigid and deformable porous materials in the presence and in the absence of gravity effects. In the case of rigid porous materials when gravity effects are present in the model, equilibrium heights are reached for both Newtonian and non-Newtonian cases. The evolution towards the equilibrium solution diff ers between Newtonian and non-Newtonian cases. In the case of deformable porous material where both fluid and solid phases move, we use mixture theory to formulate the problem. In contrast to the rigid porous materials where there is only one moving boundary, here both solid and liquid interface moves. In the absence of gravity effects, the model admits a similarity solution, which we compute numerically. If the effects of gravity are included, the free boundary problem is solved numerically where numerically computed zero gravity solution is used as an initial condition. In this case, the liquid rises to a finite height and the porous material deforms to a finite depth, following a scaling law that depends on the power law index n and power law consistency index μ*. In this case, steady state solutions exist and are the same for both Newtonian and non-Newtonian cases. We finally model a problem of fluid flow interactions within a deformable arterial wall. Again we use mixture theory to compute both the structural displacement of the solid and fluid motion. The coupled system of equations is solved numerically. We compare the mixture theory model to a hierarchy of models including simple spring models as well as elastic deformation models. The applications of the model are to understand the deformation of the wall as a function of its material properties and the relation of this deformation to the growth and rupture of aneurysms.