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.