Biomedical Engineering Reference
In-Depth Information
HOMEWORK PROBLEMS
*13.1
*Using the central difference method in two-dimensional space, where each point is the
average of its four nearest neighbors, calculate the velocity through the specified channel
with the known boundary conditions and the known inlet conditions (see
Figure 13.10
). The
velocity along the wall grid points (1, 2, 3, 19, 20, and 21) is 0 mm/sec. The velocity at the
inlet grid points (4, 9, and 14) is 40 mm/sec. The velocity at the outflow grid points 8 and 18
is 20 mm/sec, and the velocity at the grid point 13 is 35 mm/sec. It may be easiest to use lin-
ear algebra to solve the system of equations that will be generated. Assume that the grid is
uniformly spaced in the x- and y-directions.
FIGURE 13.10
Y
Grid for homework problem 13.1.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
X
13.2
Using an available computational fluid dynamics software package, make a simple geome-
try for a rapid constriction (similar to
Figure 13.3
). Calculate the flow through this constric-
tion. If no software is available, make a prediction for the flow through the constriction.
13.3
The spin on particulate matter within blood may play an important role in the activation or
recycling of the cell. Therefore, the spin dissipation rate may be an important parameter to
know. We believe that the torque on the particle,
T
, is dependent on the particle's speed (
v
),
the blood density (
ρ
), the blood viscosity (
μ
), the cell diameter (
d
), and the spin rate (angular
velocity,
ω
). Determine the dimensionless parameters that can be formed from this grouping.
13.4
When a mechanical heart valve closes rapidly, a hammer pressure wave is initiated in the
fluid. This high-pressure wave may damage the blood vessel wall distal to the mechanical
heart valve (potentially shearing endothelial cells off the inner lining). The maximum ham-
mer pressure generated (
p
max
) is a function of the blood density (
), the blood velocity at
time of closure (
v
), and the blood deformation modulus (
E
). Determine the relationship
between hammer pressure and the variables provided.
ρ
13.5
To match the Reynolds number between blood flow and a water flow model, using a two-
fold larger model, which flow will require the higher flow speed? How much higher will
it be?
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