44
Dynamics
of
the Vascular System
3.1.3
Dynamics Similitude in Vascular Biology
Dimensionless numbers provide useful scaling laws, particularly in
modeling and
similarity transformation.
Dimensional analysis is a
powerful tool, not limited to just mathematics, physics and modeling, but
has immense applicability to many biological phenomena.
Despite its many useful applications, dimensional analysis is not
without shortfalls. For a given set of physical quantities and basic units,
we can generate new dimensionless numbers, which are not necessarily
always invariant for a given system. They cannot therefore, be regarded
as similarity criteria.
The definition of dimensionless numbers as
similarity criteria (Stahl, 1963), is therefore inadequate.
Let us consider blood flow in vessels and see how similarity criteria
are obtained. A dimensional matrix is first formed by incorporating
parameters that are pertinent to the analysis. These are the fluid density
(p)
and viscosity
(q),
diameter (D) of the blood vessel, velocities of the
flowing blood (v) and of the pulse wave (c).
In terms
of
the
dimensioning mass
(M),
length
(L)
and time (T) system, we can write
down the following dimensional matrix,
P
cDqv
(g/cm3> (cm/s) (cm) (poise) (cm/s)
M
1
00
1
0
L
-3
1
1
-1
1
T
0
-1
0
-1
-1
(3.1.9)
where kn's are Rayleigh
indices referring to the exponents of the
parameters. According to Buckingham's Pi-theorem (Li, 1983
,
1986),
two
dimensionless pi-numbers (5-3
=
2) can be deduced.