Biomedical Engineering Reference
In-Depth Information
The cavitation number (Ca) relates the local pressure to the vapor pressure and the fluid
kinetic energy and provides a criterion for how likely it is for cavitation to occur. The cavi-
tation number is defined as
p2p v
1
Ca
5
ð
13
:
28
Þ
=
2
ρv 2
where p v is the vapor pressure of the fluid at a particular temperature. As the cavitation
number decreases, it is more likely for cavitation to arise in the fluid.
The Prandtl number (Pr) is the ratio of the momentum diffusion to the thermal diffu-
sion and provides criteria for whether the flow will deliver a significant amount of heat to
the surrounding container or if it retains its heat. The Prandtl number is defined as
5 μ C P
k
Pr
ð
13
:
29
Þ
where C P is the specific heat of the fluid and k is the thermal conduction coefficient. As
the Prandtl number reduces below 1, the fluid will not retain its heat very well. Flows
characterized by a larger Prandtl number can retain their heat better than lower Pr flows.
The Weber number (We) and the capillary number ( C ) are important for flows with a free
surface. The Weber number relates the inertial forces to the surface tension forces, and the
capillary number relates the viscous forces to the surface tension forces. They are defined as
5 ρ v 2 L
σ
We
ð
13
:
30
Þ
C5 μ u
σ
ð
13
:
31
Þ
where u is the average fluid velocity and
is the surface tension of the fluid. There are
many more dimensionless numbers that can be used to describe fluid properties, but we
will restrict ourselves to the ones listed here.
Depending on the exact conditions that describe the characteristics of a particular flow,
some of these dimensionless parameters may not be applicable. In general, in order to
model a particular flow condition accurately, as many of these dimensionless numbers
should be matched as possible. This leads us to a discussion of similarity within the exper-
imental or numerical model. For a model to be geometrically similar, the model has to be
the same shape as the real flow scenario and all lengths have to be related to the real sce-
nario by the same scaling factor. For a model to be kinetically similar, the velocity at each
location within the model has to be oriented in the same direction as the velocity under
the real flow conditions. The hardest criterion to meet is that all of the forces within the
model have to be identical (in magnitude by a constant scaling factor and direction). If
these three conditions are met, then the flows are dynamically similar. To maintain
dynamic similarity all of the viscous forces, the buoyancy forces, the inertial forces, the
pressure forces, and the surface tension forces, among others, must be matched. When
dynamic similarity is achieved, the data obtained in the model can be related back to the
real flow condition.
σ
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