Mechanics of Fluids

moving in a straight line with constant acceleration a. If the body has an
initial velocity u, then after time t the distance s travelled is given by
s = ut + 1
2 at2
(5.6)
where we have taken s = 0att = 0. Using the terminology of mathematics,
in eqn 5.6 s is described as the dependent variable and t, u and a are the
independent variables. The fact that s depends upon t, u and a can be written
in the mathematical form
s = φ(t, u, a) (5.7)
where the symbol φ( ) simply stands for ‘some function of’.
Suppose that we know the general functional relationship (5.7) but do
not know the precise form of eqn 5.6. We could, in principle, investigate the
relation by means of experiment. We could systematically vary t, holding u
and a constant, then vary u, holding t and a constant, and finally vary a,
holding t and u constant. In this way we could build up a picture of how s
varied with t, u and a. Clearly this would be a time-consuming task, and if we
wished to plot the results they would occupy many charts. The tests would
reveal that s depended linearly on u and a, and that the relationship between
s and t was quadratic. It would still require some ingenuity to combine the
results to yield eqn 5.6.
The quantities s, t, u and a, respectively, have the dimensions [L], [T],
[LT−1 ] and [LT−2 ]. By inspection, it is easily verified that the quantities s, ut
and at2 which appear in eqn 5.6 all have the dimensions [L], demonstrating
that the equation satisfies the requirement of dimensional homogeneity (see
Section 1.2.7).
The equation can be written in a dimensionless form by dividing all terms
in eqn 5.6 by ut, yielding
s at
= 1 + (5.8)
ut 2u
Alternatively, eqn 5.8 can be written in functional form as
� �
s at
= φ
(5.9)
ut u
The quantities s/ut and at/u are said to be dimensionless or non-dimensional.
Equation 5.8 is equivalent to eqn 5.6 but now, instead of having the four
variables of eqn 5.6 – that is s, t, u and a – we have just two, s/ut and at/u.
Expressing the relation in dimensionless form, the number of variables has
effectively been reduced by two.
At this stage we must take stock. To derive the two dimensionless groups
s/ut and at/u, we have taken advantage of the fact that we knew the original
relationship, eqn 5.6, between s, t, u and a. When we first come to
investigate a new problem in fluid mechanics, such detailed information is
not generally available. But, in fluid mechanics, it is always possible to write
down functional relations similar to the functional relation (5.7), which can
be established from a careful assessment of the physical processes at play.
So, we need a technique which, starting from the functional relation (5.7),
Dimensional analysis 171