Biomedical Engineering Reference
In-Depth Information
where the subscripts f and s stand for the fluid and solid, respectively. It
is assumed that the fluid (blood) is incompressible and Newtonian, and all
properties are constant.
1.3 Governing Equation for Blood Flow
Let us integrate the continuity equation (1.8) over a local control volume using
formula (1.6b) as
f
∂ε
u j
1
V
+
u j n j dA = 0
(1.12)
∂x j
A int
where A int is the local interface between the blood and solid matrix within
the control volume V , while n j is the unit vector pointing outward from the
fluid side to solid side. For sake of simplicity, the porosity ε is assumed to vary
moderately within a porous medium.
The second term describes the volume rate of the fluid bleeding off to the
solid matrix through the interfacial vascular wall, as illustrated in Figure 1.3.
In most microcirculatory systems of the body, there is a net filtration of fluid
from the intravascular to the extravascular compartment, such that capillary
fluid filtration exceeds reabsorption. However, this would not cause fluid to
accumulate within the interstitium since the lymphatic system removes excess
fluid from the interstitium and returns it back to the intravascular compart-
ment, as indicated in the figure. Thus, the second term describing the net
filtration is negligibly small, such that equation (1.12) reduces to
u j
∂x j
= 0
(1.13)
f
f
ω
p T
f
c
Filtration
s
f
ω
c
p T
f
Lymph flow
FIGURE 1.3
Capillary blood flow and extravascular flow.
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