Biomedical Engineering Reference
In-Depth Information
3.4 CONSERVATION OF MOMENTUM
Newton's second law of motion can be written in terms of linear momentum, which is
its most general form:
-
d
ð
Þ
-
ð
:
Þ
5
3
30
dt
For this analysis, we want to develop a relationship for the linear momentum within a
volume of interest. We will follow a similar technique as that used to develop a relationship
for the conservation of mass within a volume of interest. As we alluded to, prior to develop-
ing
Equation 3.22
, we know the system property (
-
) and its inherent property (
-
), but we
will take some space to define linear momentum and how it relates to fluid mechanics.
Linear momentum is defined as
-
ð
ð
-
dm
-
5
5
ρ
dV
ð
3
:
31
Þ
system mass
V
for a volume of interest. Also, recall from an earlier discussion that the summation of the
forces that act on a fluid element must include all body forces (denoted as
-
b
) and all sur-
face forces (denoted as
-
s
). Physically, linear momentum is a force of motion, which is
conserved unless other forces are applied to the system. By substituting the system prop-
erty and the inherent property into
Equation 3.22
, we can get the formulation for conserva-
tion of linear momentum:
ð
ð
dP
dt
5
@
d
-
-
-
-
ρ
dV
ρ
ð
3
:
32
Þ
1
U
@
t
V
area
Using Newton's relationship for momentum,
Equation 3.32
can be represented as
dP
dt
5
ð
ð
5
@
@
-
-
b
-
s
d
-
-
-
-
5
1
ρ
dV
1
ρ
U
ð
3
:
33
Þ
t
V
area
Equation 3.33
states that the summation of all forces acting on a volume of interest is
equal to the time rate of change of momentum within the control volume and the summa-
tion of momentum entering or leaving through the surface of interest. To solve conservation
of momentum problems, the first step will be to define the volume of interest and surfaces
of interest and label all of the forces that are acting on this system. This also applies when
you choose to define a coordinate system that is either aligned with or not aligned with the
majority of the forces; you will still need to define all forces and how they relate to the cho-
sen coordinate axis (remember the example of a block sliding down an incline from
Chapter 2). If the standard Cartesian coordinate system is chosen, then gravity aligns with
one of the axes, and typically gravity will be the only body force that acts on the system.
Surface forces are due to externally applied loads and are normally denoted through a pres-
sure acting on the system. The generalized surface force will be represented as
-
s
ð
area
2
pd
-
5
Search WWH ::
Custom Search