Physical Concepts and Basic Fluid Mechanics
43
dz
v
=
-
=
[L]/[T]
dt
(3.1.5)
or with a physical unit of cm/sec;
z
is along the axis
of
the direction of
blood flow.
Heart rate in beats per minute
or
second has the dimension
of
fh
=[TI-'
(3.1.6)
3.1.2
Dimensional Matrix
When formulating a dimensional matrix, it is necessary to identify the
parameters that are considered pertinent to the problem at hand. These
parameters need to to be expressed in terms
of
[MI
[L]
and [TI. For
example, given arterial blood pressure (P), flow
(Q)
and heart rate (fh), a
dimensional matrix can be formed
P
Q
fh
M 100
L-130
T
-2
-1 -1
(3.1.7)
This is therefore, a 3 x 3 matrix, or a square matrix.
As
another example, suppose that one wishes to examine the
relationship between left ventricular wall tension
(T)
and left ventricular
diameter or radius (r) and left ventricular pressure (knowingly, this is the
Laplace's law), then a dimensional matrix can be formed in terms
of
the
three parameters, prior to the application of Buckingham's Pi-theorem.
This dimensional matrix is:
TPr
MllO
L 0-1 1
T
-2
-2
0
(3.1.8)
Again, this is a 3 x 3 square matrix.