Biomedical Engineering Reference
In-Depth Information
This problem simplified to a similar form as many of the Navier-Stokes examples that
we have solved previously. It is important to note that we made the assumption that the
entire red blood cell will travel with the same velocity and that the flow is axisymmetric.
The validity of the red blood cells as a rigid body would need to be quantified.
In Stokes flow situations, the viscous forces dominate the flow profile. As a conse-
quence, there is a significant amount of drag on a particle that is within the fluid. To quan-
tify this, we will introduce the drag coefficient (C
D
), which relates the viscous stresses on
the surface of a particle to the fluid properties. In a general form, the drag coefficient is
defined as
F
C
D
5
ð
6
:
4
Þ
1
=
2
ρ
v
2
A
ρ
where F is the applied force on the particle due to the viscous fluid stresses,
is the fluids
density, V is the mean fluid velocity, and A is the cross-sectional area of the particle. For a
sphere with radius of r, the viscous forces are equal to
F
6
πμ
rv
ð
6
:
5
Þ
5
Substituting
Equation 6.5
into
Equation 6.4
, the drag coefficient for a sphere is equal to
6
πμ
rv
12
μ
C
D
ð
6
:
6
Þ
5
r
2
5
v
2
1
=
2
ρ
π
ρ
vr
We will see in a later chapter that this is strongly dependent on the Reynolds number
(Re). Incorporating the Re into
Equation 6.6
6
Re
C
D
ð
6
:
7
Þ
5
Example
Using the formulation from the previous example, calculate the viscous forces on a red blood
cell in the microcirculation as well as the drag coefficient. Assume that the velocity of the red
blood cell is 15 mm/s, the viscosity of blood is 5 cP, and the density of the blood is 1100 kg/m
3
.
Solution
The average blood velocity is approximately equal to half of the maximum blood velocity of
15 mm/s:
F
6
πð
5cP
Þð
4
μ
m
Þð
7
:
5mm
=
s
Þ
5
2
:
82 nN
5
12
ð
5cP
Þ
C
D
5
Þ
D
1820
ð
1100 kg
=
m
3
Þð
7
:
5mm
=
s
Þð
4
μ
m
In the microcirculation, it is common for the Reynolds number to range between 0.001
and 0.01. This is associated with a very large drag coefficient for any particle within the
fluid. As illustrated in the example above, the actual forces on each particle are not
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