fluid_mechanics

7.7 Correlation of Experimental Data 351
the measurements?2. The simplest problems are obviously those involving the fewest pi terms,
and the following sections indicate how the complexity of the analysis increases with the increasing
number of pi terms.
If only one pi term
is involved in a
problem, it must be
equal to a constant.
7.7.1 Problems with One Pi Term
Application of the pi theorem indicates that if the number of variables minus the number of reference
dimensions is equal to unity, then only one pi term is required to describe the phenomenon.
The functional relationship that must exist for one pi term is
ß 1 C
where C is a constant. This is one situation in which a dimensional analysis reveals the specific
form of the relationship and, as is illustrated by the following example, shows how the individual
variables are related. The value of the constant, however, must still be determined by experiment.
E XAMPLE 7.3
Flow with Only One Pi Term
GIVEN As shown in Fig. E7.3, assume that the drag, d, acting
on a spherical particle that falls very slowly through a viscous
fluid, is a function of the particle diameter, D, the particle
velocity, V, and the fluid viscosity, .
FIND Determine, with the aid of dimensional analysis, how
the drag depends on the particle velocity.
D
= f(D,V, μ )
V7.7 Stokes flow
SOLUTION
From the information given, it follows that
d f(D, V, )
and the dimensions of the variables are
We see that there are four variables and three reference dimensions
(F, L, and T) required to describe the variables. Thus, according
to the pi theorem, one pi term is required. This pi term
can be easily formed by inspection and can be expressed as
Because there is only one pi term, it follows that
where C is a constant. Thus,
d F
D L
V LT 1
FL 2 T
ß 1
d
VD
d
VD C
d CVD
Thus, for a given particle and fluid, the drag varies directly with
the velocity so that
d r V
(Ans)
V
μ
F I G U R E E7.3
COMMENTS Actually, the dimensional analysis reveals that
the drag not only varies directly with the velocity, but it also
varies directly with the particle diameter and the fluid viscosity.
We could not, however, predict the value of the drag, since the
constant, C, is unknown. An experiment would have to be performed
in which the drag and the corresponding velocity are measured
for a given particle and fluid. Although in principle we
would only have to run a single test, we would certainly want to
repeat it several times to obtain a reliable value for C. It should be
emphasized that once the value of C is determined it is not necessary
to run similar tests by using different spherical particles and
fluids; that is, C is a universal constant so long as the drag is a
function only of particle diameter, velocity, and fluid viscosity.
An approximate solution to this problem can also be obtained
theoretically, from which it is found that C 3 so that
d 3VD
This equation is commonly called Stokes law and is used in the
study of the settling of particles. Our experiments would reveal that
this result is only valid for small Reynolds numbers (VD/ 1).
This follows, since in the original list of variables, we have