Physical Concepts and Basic Fluid Mechanics
Mathematically, we have
45
(3.1
.
1
0)
or in terms of M, L and T, then
Since pi-numbers are dimensionless, this means the exponent needs to be
zero. Equating the exponents of M, L and T to zero and solve, we obtain
two
pi numbers or similarity criteria (Li, 1983):
c1
and
n2
=-=-
7
v
Ma
PVD
n,
=-=Re
(3.1.12)
The first pi-number is clearly identified as the Reynolds number, Re.
The second is the inverse of the Mach number, Ma. The Mach number is
the ratio of flow speed to the local sonic speed, or in this case the ratio of
flow velocity
to
the pulse wave velocity. It is also termed the velocity
fluctuation ratio (VFR). Recalling that to assume linearity of the arterial
system, the flow velocity should be small as compared to the pulse wave
velocity, or that VFR should be small.
The requirements for dynamic similarity (Rosen,
1978)
are that
two
flows must possess both geometric and kinematic similarity. Thus the
effects of, for instance, viscous forces, pressure forces, surface tension,
(Li,
1996)
need to be considered. Here we have only the ratio
of
inertial
forces to viscous forces i.e. Reynolds number, and the ratio
of
inertial
forces to compressibility
forces i.e.
Mach’s number or velocity
fluctuation ratio. For a truly incompressible fluid, c>>v such that Ma
=
0.
For
the analysis of blood flow in arteries, both blood and arterial walls
are normally assumed to be incompressible. The Poisson ratio (op) for
the aorta is about
0.48
close to an incompressible material (op=0.5). The
assumptions of linearity
and linear system
analysis
applied to
hemodynamic studies often require the ratio v/c
<<
1,
or that the
