Biomedical Engineering Reference
In-Depth Information
For the fluid constituents of a mixture, the driving forces are the gradients in the
constituents' mechanoelectrochemical potential,
μ α . This potential represents the
sum of mechanical, electrical and chemical contributions, thus
˜
z α
M α F c ψ
p
ρ T +
μ α
μ α .
˜
=
+
(17.8)
In this expression, ψ is the electric potential in the mixture and μ α is the chemical
potential of constituent α . The mechanical contribution is evidently proportional to
the fluid pressure p ; the electrical contribution is proportional to ψ , but reduces to
zero for a neutral constituent ( z α
0). The chemical potential represents the rate
of change of the mixture free energy density with changes in the relative mass con-
tent (apparent density) of constituent α . A constitutive relation is needed for μ α to
relate it to the state variables in an analysis, such as solute concentrations. In the mo-
mentum equation for fluids, the gradient in the mechanoelectrochemical potential is
balanced by the frictional interactions between constituents, inertia forces, external
body forces, and the momentum supply from active transport (Bowen, 1976 ;Lai
et al., 1991 ; Ateshian et al., 2010 ).
When the volume fraction of solutes is negligible, solute-solute frictional inter-
actions may be neglected relative to solute-solvent, solute-solid, and solvent-solid
interactions. Similarly, the mechanical potential of solutes in
=
μ α becomes negligi-
ble. Under quasi-static conditions, in the absence of external body forces and active
transport mechanisms, the resulting linear momentum balance equations may be
inverted to produces the following expressions for the fluxes,
˜
ρ T 2 grad
μ α ,
T ) 2
ρ w
d α
d 0 · ρ α grad
m w
=− k
μ w
·
+
(17.9)
α
=
s,w
ρ α M α
ρ α
ρ w
d α
d 0 ·
m α
d α
μ α
m w
=−
·
grad
˜
+
=
s,w,
(17.10)
where d α is the diffusivity tensor of solute α in the mixture (solid and fluid), d 0 is
the isotropic diffusivity of the solute in free solution (fluid), R is the universal gas
constant, θ is the absolute temperature, and k is the hydraulic permeability tensor
of the porous solid to the interstitial fluid (solvent and solutes), given by
k 1
I
1
ϕ w
α
c α
d 0
d α
d 0
k
=
+
,
(17.11)
=
s,w
where k is the hydraulic permeability tensor of the porous solid to the interstitial
solvent. Constitutive relations must be provided for k , d α , and d 0 , which relate
them to state variables such as solid matrix strain and suitable measures of solute
concentrations.
In a finite element modeling framework, it is necessary to define nodal variables
(degrees of freedom) that are continuous across the boundaries of elements. In a
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