Basics of Fluid Mechanics, 2014a

CHAPTER 9
Dimensional Analysis
This chapter is dedicated to my adviser, Dr. E.R.G.
Eckert.
9.1 Introductory Remarks
Genick Bar-Meir
Dimensional analysis refers to techniques dealing with units or conversion to a unitless
system. The definition of dimensional analysis is not consistent in the literature which
span over various fields and times. Possible topics that dimensional analysis deals with
are consistency of the units, change order of magnitude, applying from the old and
known to unknown (see the Book of Ecclesiastes), and creation of group parameters
without any dimensions. In this chapter, the focus is on the applying the old to unknown
as different scales and the creation of dimensionless groups. These techniques gave birth
to dimensional parameters which have a great scientific importance. Since the 1940s 1 ,
the dimensional analysis is taught and written in all fluid mechanics textbooks. The
approach or the technique used in these books is referred to as Buckingham–π–theory.
The π–theory was coined by Buckingham. However, there is another technique which
is referred to in the literature as the Nusselt’s method. Both these methods attempt
to reduce the number of parameters which affect the problem and reduce the labor in
solving the problem. The key in these techniques lays in the fact of consistency of the
dimensions of any possible governing equation(s) and the fact that some dimensions
are reoccurring. The Buckingham–π goes further and no equations are solved and even
no knowledge about these equations is required. In Buckingham’s technique only the
1 The history of dimensional analysis is complex. Several scientists used this concept before Buckingham
and Nusselt (see below history section). Their work culminated at the point of publishing the
paper Buckingham’s paper and independently constructed by Nusselt. It is interesting to point out
that there are several dimensionless numbers that bear Nusselt and his students name, Nusselt number,
Schmidt number, Eckert number. There is no known dimensionless number which bears Buckingham
name. Buckingham’s technique is discussed and studied in Fluid Mechanics while almost completely
ignored by Heat and Mass Transfer researchers and their classes. Furthermore, in many advance fluid
mechanics classes Nusselt’s technique is used and Buckingham’s technique is abandoned. Perhaps this
fact can be attributed to tremendous influence Nusselt and his students had on the heat transfer field.
Even, this author can be accused for being bias as the Eckert’s last student. However, this author
observed that Nusselt’s technique is much more effective as it will demonstrated later.
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