In Silico Biofluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 348

a white blood cell. If we wanted to derive a relationship between the drag force and fluid
properties, we would first make a list of all of the properties that we felt were important
in determining the drag force. For instance, we would probably include the fluid's veloc-
ity, the viscosity, and the density as well as some geometric properties of the white blood
cell. In symbolic form, we would state that the drag force F is a function of our fluid prop-
erties of interest:

Þ
In Equation 13.22 , d is the characteristic diameter of the white blood cell. Now in the rela-
tionship that we derived, we may have unknowingly omitted parameters that are impor-
tant (e.g., the material properties of the white blood cell membrane), and we may have
included parameters which are not important. If we wanted to experimentally determine
the relationship between the drag force and the four parameters of interest that we have
chosen, let us consider the time it would take to complete this experiment. First, a
suitable facility would be needed to conduct these tests. Once this facility is built, we can
start our experiments. For these experiments, we would need to hold the fluid viscosity,
density, and white blood cell diameter at fixed values and vary the velocity (imagine the
difficulty of finding white blood cells with the exact same diameter). We would then vary
viscosity and hold the other three parameters fixed. This would then need to be repeated
for the remaining two variables. Even if we only conducted these experiments for five dif-
ferent fixed velocity, viscosity, density, and diameter values, we would need to complete
625 experiments. For 10 different values, we would conduct 10,000 experiments. For fun,
calculate how much time this would consume if each experiment took 1 hour to complete
and you worked 40 hours/week at 100% efficiency (for 10 different values, you would be
working for just shy of 5 years for this one relationship, without considering the time it
takes to build the facility). The Buckingham Pi Theorem is a relatively easy approach to
minimize this work and obtain a potentially meaningful relationship between all of the
fluid properties of interest.
Using the same example as above, we would assume that some physical property of
interest is dependent on n independent variables. This can be represented as a function, h,
which is dependent on any (or all) of the parameters of interest. For instance, we stated
that under our conditions drag force was a function of velocity, viscosity, density, and
diameter ( Equation 13.22 ). We would therefore build a function, h, such that some combi-
nation of all of the properties equate to zero:

F5 f ðv; μ; ρ; dÞ ð 13 : 22

hðF; v; μ; ρ; dÞ 5 0 ð 13 : 23 Þ

The Buckingham Pi Theorem states that for any grouping of n parameters, they can be
arranged into n - m independent dimensionless ratios (termed parameters). The number

m is normally equal to the minimum number of independent dimensions represented by
the quantities of interest. By knowing all of the possible

Π

Π parameters, it is then possible

to simplify the results by testing which of the groups match the data obtained. The

Buckingham Pi Theorem begins by listing all of the dimensional parameters involved in
the particular problem (this is equal to n ). A consistent fundamental dimensions must be
chosen (for SI units it is mass, length, time represented as MLT; for English units it is typi-
cally force, length, and time or FLT). We then list the fundamental dimensions for each

Π