Biomedical Engineering Reference
In-Depth Information
Example
Find an expression for the velocity profile and the shear stress distribution for blood flowing
in an arteriole with a diameter of 500
m. Use the Navier-Stokes equations for cylindrical coordi-
nates to solve this problem. The pressure driving this flow is given in
Figure 3.21
.
μ
Solution
To solve this problem, assume that
v
r
5
0 and
v
z
is a function of
r
only. Assume that the
viscosity is constant and that the flow is incompressible and steady.
v
θ
5
1
2
v
z
@θ
2
v
z
@
@
v
z
@
v
r
@
v
z
@
v
r
@
v
z
@θ
1
v
z
@
v
z
@
g
z
2
@
p
1
r
@
r
@
v
z
@
r
2
@
1
1
@
ρ
5
ρ
z
1
μ
t
1
r
1
z
@
@
r
r
2
z
2
52
@
p
z
1
r
@
r
@
v
z
@
0
@
@
r
r
r
μ
@
p
d
dr
r
dv
z
dr
z
5
@
ð
r
μ
ð
dr
dv
z
dr
@
p
z
dr
5
@
r
2
2
@
p
r
dv
z
r
2
@
p
c
1
r
5
dv
z
dr
c
1
5
z
1
dr
-
z
1
μ
@
μ
@
dr
ð
ð
dv
z
r
2
@
p
c
1
r
z
1
5
μ
@
r
2
4
@
p
v
z
ð
r
Þ
5
c
1
ln
ð
r
Þ
1
c
2
z
1
μ
@
To find the exact solution, we need boundary conditions for the particular flow scenario. Due
to the no-slip boundary condition, we know that the velocity at the wall is zero and that the
shear stress at the centerline is zero. Therefore, our boundary conditions are
v
z
ð
R
Þ
5
0
dv
z
ð
0
Þ
0
5
dr
0, because in cylindrical coordi-
nates there is no negative radial direction. This location would be associated with 180
Note that we do not have a boundary condition of
v
z
(
2
R
)
5
in the
FIGURE 3.21
r
Pressure-driven flow in an arteriole with cylindrical coordi-
nates for the in text example. This is the same image as
Figure 3.20
, but choos-
ing a different coordinate system to illustrate the usage of Cartesian
coordinates versus cylindrical coordinates.
R
z
g
r
5 cm
50 mmHg
40 mmHg
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