Biomedical Engineering Reference
In-Depth Information
be as applicable in purely liquid flow if there are not large changes in temperature or pres-
sure (which is typically the case in biological systems). Also, the ideal gas law assumes
that the particles in the fluid have no interactions but can exchange energy during any
of the infrequent particle collisions. Liquid molecules interact with each other constantly
and, therefore, this law would not be useful for many liquid flow problems. In subsequent
chapters, we will take the basic laws of mechanics and thermodynamics to derive constitu-
tive equations that are useful for solving a variety of fluid problems. As a caveat, not all
problems that we will encounter can be solved analytically. Chapter 13, In Silico Fluid
Mechanics, will discuss some of the numerical methods that can be used to solve computa-
tionally complex problems, along with some of the current research topics that use compu-
tational approaches to understand a particular system of interest.
2.3 ANALYSIS METHODS
The first step in solving a fluid mechanics problem is understanding the system that the
problem addresses. In engineering mechanics, the laws that govern a system, normally
represented within a free-body diagram, are used to analyze the problem. In thermody-
namics, we generally discuss the system as being either open or closed to other neighbor-
ing systems. Whatever approach is taken, governing equations are then written based on
our system convention choices. In fluid mechanics, we will discuss similar parameters that
should be familiar from engineering mechanics and thermodynamics. We define them in
this textbook as the system of interest and the volume of interest. By properly defining these
parameters and by choosing the proper laws that govern the entire system, the solutions
to fluid mechanics problems typically become significantly easier. By choosing these para-
meters in a different way, the fluid mechanics problems can still be solved, but this may
become laborious. This is not to say that by choosing the system of interest and the vol-
ume of interest in the most ideal way, the solution will be computationally easy (e.g., alge-
bra); the use of computational fluid dynamics may still be necessary to solve the problem
(see Chapter 13). Also, not choosing the best system of interest and volume of interest
does not mean that you cannot arrive at a solution by hand, but it will generally be more
difficult than the most ideal choice. A good analogy to reinforce this concept is from engi-
neering mechanics courses. When drawing a free-body diagram, an assumption must be
made for the Cartesian coordinates for the problem. If the problem consists of a mass on
an inclined plane, there are two common ways to set up the coordinate system
(
Figure 2.8
). In the first, the x axis of the coordinate system is aligned with the slope of the
inclined plane and, in the second, the x axis is horizontal. Using either of these coordinate
conventions, the problem could be solved; however, it is typically easier to solve this prob-
lem when the coordinate axis is aligned with the inclined plane because the frictional
forces, velocity and acceleration terms all align with that coordinate system. The solution
to fluid mechanics problems works in a similar way.
For fluid mechanics problems, a system of interest is defined here as a fixed quantity of
mass and this typically encompasses the entire system of interest. The boundaries of this
system separate the materials of interest and the surrounding material that is not of
interest. For instance, if our system were blood flowing through a blood vessel, the system
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