Biomedical Engineering Reference
In-Depth Information
FIGURE 3.12
Pressure force acting on a surface of interest.
Recall that the area vector for this surface would act in the negative
x-direction, whereas the pressure forces are acting in the positive
x-direction.
p
dA
Y
X
Z
The negative sign in this formulation is added to maintain the sign convention for the
forces acting on the system (see
Figure 3.12
). In
Figure 3.12
, the pressure is positive, but
because the pressure and the area vectors act in opposite directions, their vector product
would be a negative force, which does not correspond with the positive directions chosen
for the coordinate system.
Unlike the conservation of mass formula, the formulas derived for the conservation for
linear momentum are vector equations (compare with aortic arch example for conservation
of mass).
Equation 3.33
written in component form is:
ð
ð
F
sx
5
@
@
d
-
-
F
x
5
F
bx
1
u
ρ
dV
u
ρ
1
U
t
V
area
ð
ð
5
@
@
d
-
-
F
y
F
by
1
F
sy
v
ρ
dV
v
ρ
5
1
U
ð
:
Þ
3
34
t
area
V
ð
ð
5
@
@
d
-
-
F
z
F
bz
F
sz
w
ρ
dV
w
ρ
5
1
1
U
t
V
area
where
u
,
v
, and
w
are the velocity components in the x-, y- and z-directions, respectively.
As before, the product of
d
-
is a scalar whose sign depends on the directions of the nor-
mal area vector and the velocity vector. If these two vectors act in the same direction, the
product of the vectors is positive; if they act in opposite directions, then the product is nega-
tive. Remember that the velocity vector (
-
) in this product is not a component of velocity
but is the entire velocity vector. In scalar notation, the entire form of the product would be
represented as
-
ρ
U
2
Acos
is defined by the coordinate system of choice and the
positive or negative sign is defined through the velocity/normal area vectors relationship
and the direction of the velocity component. This angle appears for the
u
,
v
, and
w
direc-
tional velocities. However, remember that the product of
-
6
jρν
αj
, where
α
d
-
is a vector, and the sign
of this product depends on the coordinate system chosen (this defines the velocity vector
sign) and the sign of the scalar. The application of these sign conventions will become
apparent in some of the example problems. To determine the sign of the momentum flux
through a surface, first determine the sign associated with
-
ρ
U
jρν
Acos
αj
, and then determine
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