Biomedical Engineering Reference
In-Depth Information
terms of concentrations of free cations c + , anions c , gas constant R and temper-
ature T and depends indirectly on the fixed charge density c fc
and the external salt
concentration c ex ,i.e.
= RT c fc 2
4 c ex 2
π = π π ex
2 RTc ex .
+
(23.1)
Not hydrostatic pressure, but the chemical potential of the fluid is the driving force
for fluid flow. Chemical potential is a measure for the free energy of the fluid. The
chemical potential of the fluid μ f
is defined per unit volume fluid
μ f
= p π,
(23.2)
where π is the osmotic pressure and p the hydrostatic pressure. The osmotic pres-
sure is determined by the empirical Van't Hoff equation, which defines the osmotic
pressure in terms of concentrations of free cations c + , anions c , gas constant R
and temperature T . Thus,
c fc 2
π = RT c + + c , + + c =
4 c ex 2 .
+
(23.3)
This osmotic pressure holds outside as well as inside the medium, but outside the
medium the fixed charge density c fc is zero and the osmotic coefficient may be
different. Electro-neutrality holds, therefore, the amount of negative charges are
equal to the amount of positive charges: c +
c fc
c + . Furthermore we introduce
=
π ex . The seepage flux q follows Darcy's law in the presence of concen-
tration gradients. The total equations are given as:
π
=
π
σ e −∇ μ f
+ π =
Momentum equation
∇·
σ
=∇·
0 ,
(23.4)
Stress-strain relation
σ e =
2 μ ε
+ λ tr ( ε ) I ,
(23.5)
u
∂t +∇·
Mass balance
∇·
q
=
0 ,
(23.6)
·∇ μ f ,
Darcy's law
q
=−
K
(23.7)
Swelling equation π = RT c fc 2
4 c ex 2
2 RTc ex , (23.8)
+
φ f i c f 0
tr ( ε )
Fixed charge c fc
=
.
(23.9)
φ i
+
=
+
=
The parameters μ
2 ν) are the Lamé constants,
and E , ν and μ are the Young's modulus, Poisson's ratio and shear modulus, re-
spectively. The tensor K
E/ 2 ( 1
ν) and λ
μ 2 ν/( 1
=
K I denotes the permeability tensor and is assumed to be
isotropic and constant in space and time.
The presence of ions fixed to the solid matrix results in prestress of the solid
matrix at the initial condition. Therefore, ε is the strain tensor which is separated
in an initial strain ε i and the deformation from an initial to the current state, i.e.
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