Biomedical Engineering Reference
In-Depth Information
Fig. 18.2
The foundation
from which the house was
removed represents the
Eulerian point considered in
the analysis of mixtures of
fluids
18.2 Mixture Theory
A mixture is a material with two or more ingredients, the particles of which are
separable, independent, and uncompounded with each other. If the distinct phases of
a mixture retain their identity, the mixture is said to be immiscible; if they lose their
identity, the mixture is said to be miscible. The constituents include a porous solid
of possibly a number of constituents, as well as solvents and solutes. The theory
of mixtures is based on diffusion models and stems from a fluid mechanics and
thermodynamics tradition and goes back to the century before last. Fick (
1855
) and
Stefan (
1871
) suggested (Truesdell and Toupin,
1960
, Sects. 158 and 295) that each
place in a fixed spatial frame of reference might be occupied by several different
particles, one for each constituent of the mixture (Fig.
18.2
). Truesdell and Toupin
(
1960
) assigned to each constituent of a mixture in motion a density, a body force
density, a partial stress, a partial internal energy density, a partial heat flux and a
partial heat supply density.
Truesdell and Toupin (
1960
) postulated equations of balance of mass, momentum
and energy for each constituent and derived the necessary and sufficient conditions
that the balance of mass, momentum and energy for the mixture be satisfied. Bowen
(
1967
) summarized the formative years of this subject. A readable history of the
subject and its applications in the period 1957-1975 is given by Atkin and Craine
(
1976a
,
b
). De Boer (
1996
,
2000
) has presented more up-to-date histories. Of key
importance in the development of the mixture theories is the application by Bowen
(
1967
,
1976
,
1980
,
1982
) of a thermodynamically-based analytical approach devel-
oped by Coleman and Noll (
1963
) to restrict the form of constitutive equations.
There have been many notable contributions of the mixture theory approach to
the modeling of tissue growth that are not cited here, as this is not a review of that
broad subject. Others have employed some of the modifications of mixture theory
employed here, but one would have to trace each modification through the literature
to determine which authors first employed it. Such a review is not the objective of
this contribution.
18.3 Poroelasticity
Poroelasticity is a theory that models the interaction of deformation and fluid flow
in a fluid-saturated porous, elastic medium. The deformation of the medium influ-
