Biomedical Engineering Reference
In-Depth Information
In general engineering mechanics, a number of methods are used to describe the system
that is being illustrated. The most common methods that are implemented in fluid
mechanics are the Lagrangian method and the Eulerian method. When it is easy to record
and solve for identifiable particles (with a real mass and volume) the Lagrangian method
is the most applicable. This method of solution tracks each particle and continually solves
the governing equations for each particle in the flow stream. This method is the point-by-
point differential approach that we described previously. By using this method, Newton's
second law of motion becomes a summation over all of the particles within the fluid. This
leads to why the solution may become difficult computationally in biofluid mechanics pro-
blems. For example, in the blood, cellular matter constitutes approximately 40% of the
total blood volume. To solve a Lagrangian biofluids problem, every cell within this 40%
must be solved individually. If your volume of interest is 1
μ
L of blood, your solution
10
6
10
3
white blood cells and
would consist of approximately 5
3
red blood cells, 6
3
10
5
platelets, as well as all of the plasma proteins, sugars, ions and other compounds
dissolved in blood. To monitor each particle properly, a lot of bookkeeping is required!
With the Lagrangian approach, the acceleration, velocity or position of interest is associ-
ated with the center of mass of the system of interest, and this may not make sense in
biological systems, in which the cell shape can and does change with time. Because of this
shape change, there would most likely be a redistribution of mass and, at each time point,
a new center of mass must be found for each particle. The Lagrangian approach, however,
is very useful in calculating particle trajectories in basic kinematic problems, if one can
assume that the center of mass stays the same throughout the problem.
When considering a real fluid, which is composed of a large number of finite particles,
the Eulerian method is more commonly used. This formulation is used because it is diffi-
cult (as described previously) to record all the information about each particle within the
fluid at every instant in time. The Eulerian method focuses on flow properties within a
certain volume over a certain time interval. Formulations that describe the flow are, there-
fore, spatially and temporally dependent. For instance, if one was interested in solving for
the velocity profile 5 cm downstream of the aortic valve for every 5 ms interval during
one cardiac cycle, the Eulerian approach is much more helpful then particle tracking with
the Lagrangian approach. This is because the solution for velocity, acceleration, among
others. is associated with this particular volume and time but not with individual particles
within the flow field. Using the Eulerian method leads to the assumption that a fluid can
be considered a continuous medium.
2
3
2.4 FLUID AS A CONTINUUM
In our previous discussions about fluid characteristics, we did not discuss the molecular
nature of a fluid. A fluid is composed of molecules in constant motion, within the fluid
itself (random motion of particles or diffusion) and possibly within the constraining vol-
ume (i.e., is the flow static or dynamic?). In most applications, the bulk properties of the
fluid are what we are interested in because this is what we can easily measure; however,
to fully characterize the fluid, a point-by-point analysis must be taken. The bulk properties
can also be considered the average of the particular property of all of the fluid particles
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