Chapter
3
Physical Concepts and Basic Fluid Mechanics
3.1
Basic Mechanics and Dimensional Analysis
3.1.1
Mass,
Length
and
Time System
and
the Pi-Theorem
of
Buckingham
Description of physical quantities requires the use of dimensions. The
mass
(M),
length
(L)
and time (T) representation of a physical variable or
parameter, or the so-called the MLT system
is
the most common.
Dimensional analysis has its well-founded place in the physical sciences
and engineering.
We
must differentiate between physical quantities and physical
constants. The former always possess units, while the latter are not
always dimensionless (e.g. Plancks constant). The use of Buckingham's
Pi-theorem for dimensional analysis requires all physical quantities be
expressed in
M
(mass),
L
(length) and T (time). The theorem has wide
applications, as will be shown later.
Dimensional homogeneity, another requirement in order to use the Pi-
theorem, was first proposed by Fourier in
1882,
who stated that any
equation
applied to physical
phenomena or involving
physical
measurements must be dimensionally homogeneous.
Its usefulness can
be found in the Navier-Stokes equations describing incompressible fluid
flow in the longitudinal
direction, for instance,
(in
cylindrical
coordinates). Every term in the equation has the dimension
of
a pressure
gradient,
or
M
'L-%-2.
Many dimensionless numbers have found their way through the use
of the dimensional matrix. The matrix comprises columns representing
physical quantities, while rows are filled with basic units
(M,
L,
T). To
