Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 19

within the fluid element. If we can assume that each fluid particle has the same properties
as the bulk fluid properties, our analysis will be simplified. Making this assumption
allows us to consider a fluid as an infinitely divisible substance, which is the definition of
a continuum. Therefore, the effects of individual molecules and the relative changes from
the bulk fluids state properties induced by dividing the medium into very minute parts
are ignored. If you have taken a course in mechanics of materials, the continuum principle
should be familiar. In most mechanics examples when a deformation, stress or strain is
calculated within a certain material, the changes in the elastic modulus or deformities/
inhomogeneities throughout the material are ignored. In biofluid mechanics, the assump-
tion that a fluid is a continuum is legitimate when we consider flows within the macrocir-
culation (see Part 2). However, when discussing the microcirculation (see Part 3),
molecular effects must be considered and the fluid should not be classified as a contin-
uum. As a consequence of using the assumption that a fluid is a continuum, properties
such as density and temperature are continuous throughout the fluid. This is not necessar-
ily a bad assumption on a large scale (i.e., bulk or average properties within the macrocir-
culation). Remember though, these properties can be a function of time and/or spatial
location, but within that instant in time or one particular location, the medium is continu-
ously divisible and the value of interest will not change based on molecular interactions.
Once again, this is very useful for general fluid mechanics problems, but this assumption
breaks down in some biological systems.
A good illustration of this phenomenon is the viscosity of a fluid. If we look at a partic-
ular volume of fluid ( Figure 2.11 , light grey), which will be used to calculate the viscosity
of the fluid, we obtain the average bulk viscosity for that volume. In general, this value is
similar to the individual viscosities of smaller sections within that same volume (i.e., dark
grey section P ). Imagine if we choose to divide the light grey section into 10 equal

FIGURE 2.11

The bulk

Y

viscosity ( ) of a fluid ele-

ment can be described by the
average of the individual vis-
cosities

µ

Volume of interest, V
with viscosity, μ

µ P ) of differential
elements (

(

Volume ∂ V with
viscosity, μ P

V). The viscosity
within each differential ele-
ment can be greater or lower
than the bulk viscosity, but if
we ignore these changes, we
would assume that the fluid is
a continuum.

@

P(X 0 , Y 0 , Z 0 )

X

Z