Page 108 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

PHYSICAL AND FLOW PROPERTIES OF BLOOD 85

Since we are analyzing capillaries in which the RBCs are considered solid bodies traveling in a tube
and surrounded by a waterlike fluid (plasma), a good representation of the viscous shear forces
acting in the fluid phase is the newtonian flow,

2
μ
F shear =∇ u (3.43)
We now examine the four terms in the momentum equation from the vantage point of an observer
sitting on the erythrocytes. It is an observable fact that most frequently the fluid in the capillary
moves at least 10 to 20 vessel diameters before flow ceases, so that a characteristic time for the
unsteady term (A) is, say, 10 D/U. The distance over which the velocity varies by U is, typically, D.
(In the gap between the RBC and the wall, this distance is, of course, smaller, but the sense of our
argument is not changed.)
Dividing both sides of Eq. (3.42) by r, we have the following order-of-magnitude comparisons
between the terms:
() U /(10 D / ) UD
A
U
≈ =
D
() (vU /D 2 ) 10 v
(3.44)
2
() UD UD
B
/
≈ =
D
/
() (vU/D 2 ) v
The term UD/v is the well-known Reynolds number. Typical values for human capillaries are
2
−2
−3
U ª 500 mm/s, Dª 7 mm, vª 1.5 × 10 cm /s, so that the Reynolds number is about 2 × 10 . Clearly,
the unsteady (A) and convective acceleration (B) terms are negligible compared to the viscous forces
(Le-Cong and Zweifach, 1979; Einav and Berman, 1988).
This result is most welcome, because it allows us to neglect the acceleration of the fluid as it
passes around and between the RBCs, and to establish a continuous balance between the local net
pressure force acting on an element of fluid and the viscous stresses acting on the same fluid
element. The equation to be solved is therefore
2
μ
∇= ∇ u (3.45)
P
subject to the condition that the fluid velocity is zero at the RBC surface, which is our fixed frame
of reference, and U at the capillary wall. We must also place boundary conditions on both the pres-
sure and velocity at the tube ends, and specify the actual shape and distribution of the RBCs. This
requires some drastic simplifications if we wish to obtain quantitative results, so we assume that all
the RBCs have a uniform shape (sphere, disk, ellipse, pancake, etc.) and are spaced at regular inter-
vals. Then the flow, and hence the pressure, will also be subject to the requirement of periodicity,
and we can idealize the ends of the capillary as being substantially removed from the region being
analyzed. If we specify the relative velocity U between the capillary and the RBC, the total pressure
drop across the capillary can be computed.
3.5.3 Motion of a Single Cell
For isolated and modestly spaced RBC, the fluid velocities in the vicinity of a red cell is schemati-
cally shown in Fig. 3.12. In the gap, the velocity varies from U to zero in a distance h, whereas in
the “bolus” region between the RBC, the same variation is achieved over a distance of D/4. If h < D/4,
as is often observed in vivo, then the viscous shear force is greatest in the gap region and tends to
“pull” the RBC along in the direction of relative motion of the wall.
Counteracting this viscous force must be a net pressure, P − P , acting in a direction opposite to
d
u
the sense of the shear force. This balance of forces is the origin of the parachutelike shape shown in
Fig. 3.3 and frequently observed under a microscope.