Biomedical Engineering Reference
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is equal to the average of the normal stress components (
Equation 2.14
). The negative sign
is added to the formulation to make sure that the forces balance each other. For any other
coordinate system (e.g., cylindrical) the hydrostatic pressure is equal to the average of the
stresses along the main diagonal of the Cauchy stress tensor. (Remember to include the
negative sign so that the forces can balance each other.)
5
2
1
p
hydrostatic
ðσ
1σ
1σ
Þ
ð
2
:
14
Þ
xx
yy
zz
3
2.6 KINEMATICS: VELOCITY, ACCELERATION, ROTATION
AND DEFORMATION
Kinematics is the generalized study of motion. Using kinematic relationships you can
describe the motion of any particle in space and time. In this section, we will develop the
formulations for the motion of an infinitesimal surface of fluid,
δ
x
δ
y, with a fixed identifi-
able mass,
m. (This derivation can be extended to three dimensions easily, by extending
the surface into a differential volume, denoted as
δ
δ
δ
δ
z.) As this mass moves throughout
the fluid, it may experience translation, rotation, extension, pure shear or any combination
of these four movement classifications (
Figure 2.15
). Translation and rotation are rigid
body motions and should be familiar from engineering mechanics courses. Translation is
defined when all of the particles within the mass move with the same displacement and
there is maintenance of the orientation of these particles. Rotation is defined when all the
particles rotate about some point, maintaining the orientation between all particles. As
depicted in
Figure 2.15
, not all of the particles experience the same displacement during
x
y
FIGURE 2.15
Different motions that a fluid
element can undergo.
These include translation,
rotation, extension/compression and pure shear. In
general, these motions can occur all at the same
time, which would be described as a general planar
rotation/deformation.
Y
Y
Translation
Rotation
X
X
Y
Y
Extension/Compression
Pure shear
β
α
X
X
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