Mechanics of Fluids

Note: a suffix has been attached to the symbols φ1( )and φ2( )to
clarify the fact that they are different mathematical functions.
Step 7 Check calculation. Since n = 5, j = 3andk = 2, then (n−k) =
3 which is equal to j. QED.
The reader is encouraged to go through Steps 4 to 7 again, using as
repeating variables:
(a) d, u and µ resulting in the relation F/µud = φ3(Re),
(b) d, ϱ and µ resulting in the relation Fϱ/µ 2 = φ4(Re) and
(c) u, ϱ and µ resulting in the relation Fϱ/µ 2 = φ4(Re) ✷
Dimensional analysis 175
5.6.3 Some comments on dimensional analysis
One of the more difficult tasks for the inexperienced analyst is to decide Drawing up the list of
which independent variables to include in the dimensional analysis and independent variables
which to exclude. It is important to include in the list of variables all quantities
that influence the fundamental physics of the problem. On the other
hand, there is no virtue in bringing in quantities that have no bearing on the
situation. Here the discussion of the different kinds of dynamic similarity in
Section 5.3 provides guidance on the variables that might require consideration
in the field of fluid mechanics. In the context of dimensional analysis,
the term independent variable sometimes needs to be interpreted broadly.
This is because, in some experimental environments, it is not possible to vary
particular quantities, despite the fact that they have a fundamental influence
on an experiment. This is particularly so for problems in which gravity plays
a fundamental role. Some students new to dimensional analysis omit g from
the list of variables in a study where it in fact plays a fundamental role,
simply because it is a physical quantity that cannot be varied in the laboratory.
In deciding whether or not g should be included amongst the list of
independent variables, it is often helpful to consider whether the dependent
variable would be affected if the experiment were performed on the surface
of the moon, where g differs from the value on the surface of the earth.
Temperature is another quantity that causes difficulties, because physical
properties, such as dynamic viscosity, vary significantly with temperature.
Again, in fluid mechanics, it is sufficient to consider the question of dynamic
similarity, to appreciate that the physical property, but not temperature,
should be included in the list of variables. In problems of heat transmission,
which are outside the scope of this book, temperature levels are important
and must be included in the list of variables.
There are circumstances where, in specifying the list of independent variables,
a choice has to be made between a number of equally valid options.
For example, when considering problems of internal flow, the flow rate can
be represented by one of three variables, namely a representative mean velocity
u[LT −1 ], volumetric flow rate Q[L3T−1 ] or mass flow rate m[MT −1 ].
The case for any one of these can be argued but, by convention, velocity is
generally used. However, there are circumstances when the use of either m or