fluid_mechanics

1Recall from Chapter 1 that the notation is used to indicate dimensional equality.2 The dimensions
of the variables in the pipe flow example are ¢p m FL 2 / FL 3 , D L, r FL 4 T 2 , T,
and V LT 1 . 3Note that the pressure drop per unit length has the dimensions of
A quick check of the dimensions of the two groups that appear in Eq. 7.2 shows that they are in
fact dimensionless products; that is,
and
7.2 Buckingham Pi Theorem
D ¢p /
rV 2
rVD
m
1FL 2 2L FL 3 .4
L1FL 3 2
1FL 4 T 2 21LT 1 2 F 0 L 0 T 0 2
1FL4 T 2 21LT 1 21L2
1FL 2 T2
7.2 Buckingham Pi Theorem 335
F 0 L 0 T 0
Not only have we reduced the number of variables from five to two, but the new groups are
dimensionless combinations of variables, which means that the results presented in the form of
Fig. 7.2 will be independent of the system of units we choose to use. This type of analysis is called
dimensional analysis, and the basis for its application to a wide variety of problems is found in
the Buckingham pi theorem described in the following section.
Dimensional analysis
is based on the
Buckingham pi
theorem.
A fundamental question we must answer is how many dimensionless products are required to replace
the original list of variables? The answer to this question is supplied by the basic theorem
of dimensional analysis that states the following:
If an equation involving k variables is dimensionally homogeneous, it can be reduced
to a relationship among k r independent dimensionless products, where r is the
minimum number of reference dimensions required to describe the variables.
The dimensionless products are frequently referred to as “pi terms,” and the theorem is called the
Buckingham pi theorem. 2 Edgar Buckingham used the symbol ß to represent a dimensionless
product, and this notation is commonly used. Although the pi theorem is a simple one, its proof is
not so simple and we will not include it here. Many entire books have been devoted to the subject
of similitude and dimensional analysis, and a number of these are listed at the end of this chapter
1Refs. 1–152. Students interested in pursuing the subject in more depth 1including the proof of the
pi theorem2 can refer to one of these books.
The pi theorem is based on the idea of dimensional homogeneity which was introduced in
Chapter 1. Essentially we assume that for any physically meaningful equation involving k variables,
such as
u 1 f 1u 2 , u 3 , . . . , u k 2
the dimensions of the variable on the left side of the equal sign must be equal to the dimensions
of any term that stands by itself on the right side of the equal sign. It then follows that we can
rearrange the equation into a set of dimensionless products 1pi terms2 so that
ß 1 f1ß 2 , ß 3 , . . . , ß kr 2
where f1ß 2 , ß 3 , . . . , ß kr 2 is a function of ß 2 through ß kr .
The required number of pi terms is fewer than the number of original variables by r, where
r is determined by the minimum number of reference dimensions required to describe the original
list of variables. Usually the reference dimensions required to describe the variables will be
the basic dimensions M, L, and T or F, L, and T. However, in some instances perhaps only two
dimensions, such as L and T, are required, or maybe just one, such as L. Also, in a few rare cases
2 Although several early investigators, including Lord Rayleigh 11842–19192 in the nineteenth century, contributed to the development of
dimensional analysis, Edgar Buckingham’s 11867–19402 name is usually associated with the basic theorem. He stimulated interest in the subject
in the United States through his publications during the early part of the twentieth century. See, for example, E. Buckingham, On Physically
Similar Systems: Illustrations of the Use of Dimensional Equations, Phys. Rev., 4 119142, 345–376.