Fundamentals of Fluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 29

Temporal driving forces Spatial driving forces

Constant driving force

120

Step-like driving force

60

0

Time

Arterial
side

Capillary
side

FIGURE 2.17 Examples of temporal and spatial driving forces that may arise in the vascular system. In

general, temporal and spatial driving forces can be coupled within the same problem.

FIGURE 2.18 Y Y Rotation of a rigid

body used to calculate, angular velocity,
vorticity and shear rate. The rate of
change of the displacement of point B is
defined from the change in velocity over
time across line segment AB. The same
analysis conducted on all other line seg-
ments in three dimensions would yield
the angular velocity formula. Remember
that

v x + Δ v x

C C

C 0 C 0

β

B B

α

A A

X X

B 0 B 0

v x

α and β do not have to be the same.

v y v y + Δ v y

Rotation components can be broken down into velocity components by realizing that the
rotation of a small fluid element ( Figure 2.18 ) can be described as the mathematical limit
(lim) as time approaches zero of the time rate of change of the angles (i.e.,

α , Figure 2.18 )

that the fluid element makes with the particular fixed coordinate system. From

Figure 2.18 , ω z , would be defined as

0 Δα

ω z 5 lim

Δ

ð 2 : 22 Þ

Δ t

t -

Using this approach, the angular velocity about the z axis can also be defined mathe-
matically as the average of the angular velocity of two perpendicular line segments in the
normal plane (x-y plane) of that axis system. This formulation will end up with fluid
quantities that are more easily quantified. The per pen dicula r lin e segments that will be
used to define the changes in angular velocity are AB 0 and AC 0 in Figure 2.18 . At some
later time, B 0 and C 0 will translate to B and C respectively ( Figure 2.18 ), forming the angle