Biomedical Engineering Reference
In-Depth Information
FIGURE 2.10
Surface of interest
This figure illustrates examples
of a volume of interest and a surface of interest
within a blood vessel.
The two highlighted sur-
faces of interest are imaginary. The boundary of the
dashed cylinder and the tube wall is a real surface.
By choosing the surfaces in this manner the solution
of fluid mechanics problem would be easier, com-
pared with other choices for the surfaces/volumes
of interest.
Direction
of flow
Volume of interest
are interested in because mass is exchanged with time. To circumvent this type of prob-
lem, a volume of interest is defined. This volume of interest is a volume in space with which
a fluid flows through. The boundaries of this volume are the surfaces of interest
(
Figure 2.10
). These surfaces can change with time; hence be at rest or in motion.
Figure 2.10
depicts a blood vessel with a fluid flowing through it. The dashed cylinder
within the blood vessel is our choice of a volume of interest. This choice depicts an exam-
ple in which the volume of interest has real and imaginary surfaces of interest. The real
surface is the inside of the blood vessel wall, which surrounds our volume of interest. Real
surfaces can be thought as the fluid boundaries, which prevent possible mass transfer.
However, the vertical grey circles, with which fluid will enter or exit our volume, are
imaginary surfaces. These surfaces are solely chosen for convenience, so that we can solve
the fluid mechanics problem more easily. Using these imaginary surfaces, we can impose
boundary conditions or initial conditions with ease. Other choices for this surface of inter-
est/volume of interest would make the boundary value and initial condition definitions
more difficult (remember problems depicted in
Figure 2.8
from engineering mechanics
courses). It is of the upmost importance to choose these surfaces wisely because the meth-
odology of fluid mechanics problems will change based on this surface and volume. Some
choices (as in
Figure 2.10
) can make the solution simple, whereas other choices can make
the problem difficult to solve.
Depending on the particular application of the fluid mechanics problem, there are dif-
ferent approaches to take to reach a solution to the problem. First, if the point-by-point
flow characteristics (an infinitesimal volume, i.e., one water molecule) are of interest
within the fluid, a differential approach to solve the equations would be used. This solu-
tion would be based on the differential equations of motion. This approach is useful in
biofluid mechanics if one is interested in two-phase flows and the location of individual
proteins or cells within the blood stream and how these individual particles interact with
other particles or the vessel wall. However, in some cases, this very detailed knowledge of
the flow profile is unnecessary and too labor intensive to calculate (by hand and/or com-
putationally). In this case, one is most likely interested in the bulk properties of the fluid.
Quantities of interest are most likely the velocity profile at a certain location and the pres-
sure gradient across a particular length, among others. The solution to these types of pro-
blems takes the form of the integral equations of motion, and the volumes of interest are
finite. An example of this approach is looking at one particular section of a blood vessel
(see the volume of interest in
Figure 2.10
). This method is generally easy to solve analyti-
cally and will be developed first in this textbook.
Search WWH ::
Custom Search