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1

The Physical Basis of

DIMENSIONAL ANALYSIS
Ain A
...
Sonin

Department of Mechanical Engineering
MIT
Cambridge, MA 02139

First Edition published 1997
...
25
Advanced Fluid Mechanics and other courses at MIT since 1992
...
Introduction

1

2
...
1 Physical properties
2
...
3 Unit and numerical value
2
...
5 Physical equations, dimensional homogeneity, and
physical constants
2
...
7 Systems of units
2
...
Dimensional Analysis

4
5
10
12
15
19
22
27
29

3
...
2 Example: Deformation of an elastic sphere striking a wall 33
Step 1: The independent variables
Step 2: Dimensional considerations
Step 3: Dimensionless similarity parameters
Step 4: The end game
3
...


33
35
36
37

37
37
38
38

4

An incomplete set of independent quantities may
destroy the analysis
Superfluous independent quantities complicate the result
unnecessarily
On the importance of simplifying assumptions
On choosing a complete set of independent variables
The result is independent of how one chooses a dimensionally
independent subset
The result is independent of the type of system of units

40
40
41
42
43
43

4
...
This work was begun with
support from the Gordon Fund
...
”
“Think things, not words
...

Propositions arrived at by purely logical means are
completely empty as regards reality
...
Bridgman (1882-1961)3:
“
...
”

1

Catherine Drinker Bowen, 1963
2 Einstein, 1933
3 Bridgman, 1950

6

1
...
Bridgman (1969) explains it thus: "The principal
use of dimensional analysis is to deduce from a study of the dimensions of
the variables in any physical system certain limitations on the form of any
possible relationship between those variables
...

At the heart of dimensional analysis is the concept of similarity
...
For example, under some very
particular conditions there is a direct relationship between the forces
acting on a full-size aircraft and those on a small-scale model of it
...
Here the question is, what kind of
transformation works? Dimensional analysis addresses both these
questions
...
A problem
that at first looks formidable may sometimes be solved with little effort
after dimensional analysis
...
This is an inspectional form of
similarity analysis
...
Dimensional analysis is, however, the only
option in problems where the equations and boundary conditions are not
completely articulated, and always useful because it is simple to apply and
quick to give insight
...
By the 1920's
the principles were essentially in place: Buckingham's now ubiquitous
π−theorem had appeared (Buckingham, 1914), and Bridgman had
published the monograph which still remains the classic in the field
(Bridgman, 1922, 1931)
...

Applications now include aerodynamics, hydraulics, ship design,
propulsion, heat and mass transfer, combustion, mechanics of elastic and
plastic structures, fluid-structure interactions, electromagnetic theory,
radiation, astrophysics, underwater and underground explosions, nuclear
blasts, impact dynamics, and chemical reactions and processing (see for
example Sedov, 1959, Baker et al, 1973, Kurth, 1972, Lokarnik, 1991),
and also biology (McMahon & Bonner, 1983) and even economics (de
Jong, 1967)
...
The debate
over the method's theoretical-philosophical underpinnings, on the other
hand, has never quite stopped festering (e
...
Palacios, 1964)
...
g
...
The
problem is that dimensional analysis is based on ideas that originate at
such a substratal point in science that most scientists and engineers have
lost touch with them
...

Dimensional analysis is rooted in the nature of the artifices we
construct in order to describe the physical world and explain its
functioning in quantitative terms
...
Propositions arrived at by
purely logical means are completely empty as regards reality
...
We will clarify the terms used in
dimensional analysis, explain why and how it works, remark on its utility,
and discuss some of the difficulties and questions that typically arise in its
application
...
The procedure is the same
in all applications, a great variety of which may be found in the references
and in the scientific literature at large
...
Physical Quantities and Equations
2
...
It is at this very first step that we face the fact upon which
dimensional analysis rests: Description in absolute terms is impossible
...
When we say that something "is" a tree, we mean
simply that it has a set of attributes that are in some way shared by certain
familiar objects we have agreed to call trees
...
Physics starts by breaking the descriptive process down
into simpler terms
...
None of these
properties can be defined in absolute terms, but only by reference to
something else: an object has the length of a meter stick, we say, the color
of an orange, the weight of a certain familiar lump of material, or the
shape of a sphere
...
We can do no more than
compare one thing with another
...
(We shall use bold
symbols when we are referring not to numerical values, but to actual
physical attributes
...
Properties of the same kind (or simply, the
same properties) are compared by means of the same comparison
operation
...
Asking whether a particular mass
is physically equal to a particular length is meaningless: no procedure
exists for making the comparison
...
Shape and color are examples
...
But
asking whether a square shape is smaller or larger than a circular shape, or
whether green is smaller or larger than white, makes no sense
...

2
...
That the language of mathematics is ideally suited for expressing
those laws is not accidental, but follows from the constraints we put on the
types of physical properties that are allowed to appear in quantitative
analysis
...

Physical quantities are of two types: base quantities and derived
quantities
...
The
base and derived quantities together provide a rational basis for describing
and analyzing the physical world in quantitative terms
...

Base quantities with the same comparison and addition operations are of
the same kind (that is, different examples of the same quantity)
...
Quantities with different comparison and
addition operations cannot be compared or added; no procedures exist for
executing such operations
...
They are not themselves physical things or
events
...

The comparison and addition operations are physical, but they are
required to have certain properties that mimic those of the corresponding
mathematical operations for pure numbers:
(1) The comparison operation must obey the identity law (if A=B and
B=C, then A=C), and
(2) the addition operation must be commutative (A+B=B+A), associative
[A+(B+C)=(A+B)+C], and unique (if A+B=C, there exists no finite
D such that A+B+D=C)
...

A base quantity is thus a property for which the following mathematical
operations are defined in physical terms: comparison, addition,
subtraction, multiplication by a pure number, and division by a pure
number
...

This sets the stage for not only "catching the resemblance of things",
but also expressing that resemblance in the language of mathematics
...
No defining operation exists
for forming a tangible entity that represents the product of a mass and a
time, for example, or, for that matter, the product of one length and
another (more on this later)
...

Products, ratios, powers, and exponential and other functions such as

12

trigonometric functions and logarithms are defined for numbers, but have
no physical correspondence in operations involving actual physical
quantities
...
1 illustrates the comparison and addition operations of some
well-known physical quantities that can be chosen as base quantities
...
For our
present purpose we take these for granted, much like Dr
...


Figure 2
...


Figure 2
...


13

Figure 2
...


Figure 2
...


Figure 2
...


The comparison and addition operations for length and mass are
familiar
...

Time is omitted from the figure, largely because its defining
operations defy illustration in such simplistic terms
...
It is, after all, the warp of our
existence, the stuff, as some wit pointed out, that keeps everything from
happening at once
...
Aristotle
referred to time as a "dimension of motion", which pleases the poet in us,
but leaves the scientist unmoved
...
Whether those clocks are hourglasses or atomic clocks will affect
the precision of the operations, but not their intrinsic character
...

The concept of force arises in primitive terms from muscular effort,
and is formalized based on the observation that a net force on an object
(the vector sum of all the forces acting on the object) causes a rate of
change in its velocity
...

A reader accustomed to considering speed as "distance divided by
time" and area as "length squared" may be surprised to see them included
in figure 2
...
We include them to show that the set of base
quantities is very much a matter of choice
...
A self-propelled toy car
running across a tabletop has a speed, and we can define acceptable
procedures for establishing whether two speeds are equal or unequal and
for adding two speeds, as in figure 2
...
Speed can therefore be taken as a
base quantity, should we choose to do so
...
Note
that two areas may be equal without being congruent, provided one of
5

Sir Arthur Eddinton (1939): "It has come to be accepted practice in introducing new
physical quantities that they shall be regarded as defined by the series of measuring
operations and calculations of which they are the result
...


15

them can be "cut up" and reassembled (added back together) into a form
which is congruent with the other: the addition operation is invoked in
making a comparison, and the comparison operation in addition
...
That shape is disqualified is obvious: what is the sum of a
square and a circle? But why is color disqualified? We know that color in
the form of light can be added according to well-defined rules, as when
red light added to green produces yellow
...
Blue equals blue
...
The
addition operation is not unique: nA=A for any color A, where n is any
number
...
But, the
persistent reader may argue, the color of an object can be identified by the
wavelength of light reflected from it, and wavelength can be added
...
This is a rule
for adding lengths, not for adding the property we perceive as color
...
3 Unit and numerical value
The two operations that define a base quantity make it possible to
express any such quantity as a multiple of a standard sample of its own
kind, that is, to "measure it in terms of a unit"
...
The comparison operation allows the
replication of the unit, and the addition operation allows the identification
and replication of fractions of the unit
...
2)
...
If a is the unit chosen for quantities of type
A, the process of measurement yields a numerical value A (a number)
such that
A = Aa

(2
...
The only mathematics
involved is the counting of the number of whole and fractional units once

16

physical equality has been established between the quantity being
measured and a sum of replicas of the unit and fractions thereof
...
2: Measurement in terms of a unit and numerical value

It should be emphasized that numbers can be ascribed to properties in
many arbitrary ways, but such numbers will not represent numerical
values of physical quantities unless they are assigned by a procedure
consistent with the one defined above
...

A physical quantity exists, independent of the choice of unit
...
A quantity A can be measured in terms of a unit a or
in terms of another unit a', but the quantity itself remains physically the
same, that is,
A = Aa = A'a'
...
2)

If the unit a' is n times larger than a,
a' =na,

(2
...
2) that
A ′ = n −1 A
...
4)

If the size of a base quantity's unit is changed by a factor n, the quantity's
numerical value changes by a factor n-1
...
All base quantities of the same kind

17

thus change by the same factor when the size of that quantity's unit is
changed
...

Note also that when base quantities of the same kind are added
physically (A+B=C), the numerical values satisfy an equation of the same
form as the physical quantity equation (A+B=C), regardless of the size of
the chosen unit
...

2
...
We
determine the distance L that an object moves in time t, for example, and
calculate its speed V=Lt-1; or we measure a body's mass m and speed V and
calculate its kinetic energy K=mV2/2
...

Not all numbers obtained by inserting base quantities into formulas
can be considered physical quantities6
...
Bridgman (1931) postulated that this is in fact a defining
attribute of all physical quantities
...

Bridgman went on to show (Bridgman, 1931; see also the proof by
Barenblatt, 1996) that a monomial formula satisfies the principle of
6

From this point on, the term quantity will be used for the numerical value of a physical
quantity, unless otherwise noted
...


(2
...
are numerical values of base quantities and the
coefficient and exponents a, b, c, etc
...
All monomial
derived quantities have this power-law form; no other form represents a
physical quantity
...
A derived quantity does
not necessarily represent something tangible in the same sense as a base
quantity, although it may
...

To avoid talking of "units" for quantities that may have no physical
representation, but whose numerical values nevertheless depend on the
choice of base units, we introduce the concept of dimension
...
If A is the numerical
value of a length, we say it “has the dimension of length”, and write this as
[A]=L where the square brackets imply “the dimension of” and L
symbolizes the concept of length
...

The dimension of a derived quantity conveys the same information in
generalized form, for derived as well as base quantities
...
M1m1 M2 2
...


(2
...

If the length unit is changed by a factor nL , the mass unit by a factor nm ,
and the time unit by a factor nt , it follows from equations (2
...
6)
that the value of Q changes to
Q′ = n−1Q

(2
...


(2
...
By
analogy with the meaning of dimension for base quantities [see equation
(2
...


(2
...
A derived quantity's
dimension follows from its defining equation
...
8) for
the quantity defined in equation (2
...
6) the symbol for its dimension, omit the numerical
coefficient α, and obtain equation (2
...
The dimension of a
kinetic energy mV 2 2, for example, is M(L t) 2 = ML2 t −2
...
We shall see that a quantity’s dimension depends on
the choice of the system of units, and is therefore under the control of the
observer rather than an inherent attribute of that quantity
...
The
statement Q = 0
...
021 if mass is measured in kilograms and
time in seconds
...
In common parlance the
terms unit and dimension are often used synonymously, but such usage is
undesirable in a treatise where fundamental understanding is paramount
...
The dimension of any derived physical quantity is a product of
powers of the base quantity dimensions
...
Sums of derived quantities with the same dimension are derived
quantities of the same dimension
...

3
...

4
...
An example is Vt L , where
V = dx dt is a velocity, t is a time and L is a length
...

5
...
) of
dimensional derived quantities are in general not derived quantities
because their values do not in general transform like derived quantities
when base unit size changes
...
Special functions with
dimensionless arguments are therefore derived quantities with dimension
unity
...
5 Physical equations, dimensional homogeneity, and physical
constants
In quantitative analysis of physical events one seeks mathematical
relationships between the numerical values of the physical quantities that
describe the event
...
A
primitive soul may find it remarkable, or even miraculous, that his own
mass in kilograms is exactly equal to his height in inches
...

Science is concerned only with expressing a physical relationship between
one quantity and a set of others, that is, with “physical equations
...

We are interested, therefore, only in numerical relationships that remain
true independent of base unit size
...
Suppose that, in a specified physical event, the numerical value
Qo of a physical quantity is determined by the numerical values of a set
Q1
...
,Qn ),

(2
...
9) can be physically relevant only if Qo
and f change by the same factor when the magnitudes of any base units are
changed
...
Some reflection based on the points summarized at the end
of Section 2
...
9):
(1) both sides of the equation must have the same dimension;
(2) wherever a sum of quantities appears in f, all the terms in the sum
must have the same dimension;
(3) all arguments of any exponential, logarithmic, trigonometric or
other special functions that appear in f must be dimensionless
...
10)

C must be dimensionless, D1 and D2 must have the same dimension, and A,
B, D/E and F must have the same dimension
...

The following example may help to illustrate the reason for
dimensional homogeneity in physical equations and show how conceptual
errors that may arise if homogeneity isn’t recognized
...
We know of course that elementary
Newtonian mechanics gives the answer as

22

x=

1 2
gt ,
2

(2
...

This equation expresses the result of a general physical law, and is clearly
dimensionally homogeneous
...

Suppose we are ignorant of mechanics and conduct a large variety of
experiments in Cambridge, Massachusetts, on the time t it takes a body
with mass m to fall a distance x from rest in an evacuated chamber
...
91t 2
...
12)

This is a perfectly correct equation
...

However, it appears at first glance to be dimensionally non-homogeneous,
the two sides seemingly having different dimensions, and thus appears not
to be a true physical equation
...
91 remains invariant when units
are changed
...
91 represents not a dimensionless
number, but a particular numerical value of a dimensional physical
quantity which characterizes the relationship between x and t in the
Cambridge area
...
12) must transform when units are changed
...
12), which gives it in
meters, must be multiplied by 3
...

Thus, if x is measured in feet and t in seconds, the correct version of
equation (2
...
1t 2
...
13)

23

This same transformation could also have been obtained by arguing that
equation (2
...
91ms −2 )

(2
...
The units of c indicate its dimension and
show that a change of the length unit from meters to feet, with the time
unit remaining invariant, changes c by the factor 3
...
This gives c=16
...
13)
...
14) is the correct way of representing the data of equation
(2
...
It is dimensionally homogeneous, and makes the transformation to
different base units straightforward
...
A fitting formula
derived from correct empirical data may at first sight appear dimensionally
non-homogeneous because it is intended for particular base units
...

(2) Determine the dimensions of these constants by requiring that the new
equation be dimensionally homogeneous
...

This is of course how equation (2
...
12)
...
Suppose it is found that
the pressure distribution in the earth's atmosphere over much of the United
States can be represented (approximately) by the formula

24

p = 1
...
00012z

(2
...
This
expression applies only with the cited units
...
01× 105 Nm−2 , b = 0
...
16)

where a and b are physical quantities
...
The dimensions of a and b indicate
how these quantities change when units are changed
...
16) are physical constants in
the sense that their values are fixed once the system of units is chosen
...
Similarly, the
acceleration of gravity g in equation (2
...

The basic laws of physics also contain a number of universal physical
constants whose magnitudes are the same in all problems once the system
of units is chosen: the speed of light in vacuum c, the universal
gravitational constant G, Planck’s constant h, Boltzmann’s constant kB,
and many others
...
6 Derived quantities of the second kind
The classification of quantities as base or derived is not unique
...
Such transformations are useful because they reduce the
number of units that must be chosen arbitrarily, and simplify the forms of
physical laws
...
The floor area of a room may be measured by covering
the floor with copies of this postage stamp and parts thereof, and counting
the number of whole stamps required
...
17)

where the integral is taken over the area and c is a dimensional constant
the magnitude of which depends on the choice of base units of area and
length
...
18)

with dimension AL-2
...
17) can be thought of as a "physical law
...
Equation (2
...

Dimensional constants like c in Equation (2
...

The law (2
...
If we choose to measure area
in terms of a unit that is defined by the area of a square with sides equal to
the length unit, the physical coefficient c becomes unity, and equation
(2
...
19)

Area, in effect, has become a derived quantity that is defined in terms of
operations involving length
...
19) does not imply that area now
"is", in any physical sense, length squared
...
We have simply noted that, because our concepts of
"integral" and "area" are in fact similar, we may choose to measure area
via operations involving length, and have made a decision to do so
...

By transforming area in this way from a base to a derived quantity, we
have accomplished two simplifications: the length unit automatically
determines the area unit, and the dimensional physical constant c in
equation (2
...

There is nothing sacred about choosing c=1 in this kind of
transformation
...
19)
...
In other instances,
dimensionless coefficients other than unity are introduced deliberately to
make things more convenient
...
Similarly, the 4π terms that appear in the SI
system of units in the integral forms of Gauss's and Ampere's laws are
placed there in order to eliminate numerical coefficients in the differential
forms (Maxwell’s field equations)
...
We may define speed as a
base quantity with its own comparison and addition operations, and
choose for it a base unit—the speed of a certain very reliable wind-up toy
on a horizontal surface, say, to use again an absurd example
...
We can then choose to define speed as the derived quantity V=dx/dt,
which is equivalent to choosing a speed unit such that unit distance is
covered in unit time
...
Force may be taken as a
base quantity with an arbitrarily specified unit—the (equilibrium) force
required to extend a standard spring a given distance, say
...
20)

27

where the coefficient c is a universal constant with dimension Ft2M-1L-1 if
force, length, mass and time are all selected as base quantities
...
20) is a general physical law which expresses a relationship between the
numerical values of three different physical quantities that are involved in
the dynamics of a point mass—force, (gravitational) mass, and
acceleration
...
If we do this, we make c=1 in equation (2
...
21)

where force has a dimension MLt-2
...
21) does not imply that
force "is" in any physical sense a mass times an acceleration
...
In contrast with
the examples of area and speed, equation (2
...
Instead, we have imparted to force the
character of a derived quantity by making the force unit depend in a
particular way on the units of mass and length
...
Force is one example; heat and electric charge are also
treated in this way in the SI system
...
7 Systems of units
A system of units is defined by
(1) a complete set of base quantities with their defining comparison and
addition operations,
(2) the base units, and

28

(3) all relevant derived quantities, expressed in terms of their defining
equations (e
...
g
...

The set of derived quantities is open-ended; new ones may be introduced
at will in any analysis
...

In the SI system (Système International) there are six base quantities
(table 2
...
The units of length, time and mass are
the meter (m), the second (s) and the kilogram (kg), respectively
...

The temperature in any system of units must be a thermodynamic
temperature
...
(Is 2oC "twice as hot"
as 1 oC ? Is 0oC "zero" temperature in the sense of there being an absence
of temperature? For that matter, if temperature is defined in terms of a
thermometer with an arbitrarily marked temperature scale, is there any
reason why heat should flow from a "higher" to a "lower" temperature?)
The thermodynamic (or absolute) temperature is, however, defined in
terms of physical comparison and addition operations appropriate to a base
quantity7
...
e
...
The SI temperature unit is the
kelvin (K), which is defined as the fraction 1/273
...


7

The ratio of the numerical values of two physical thermodynamic temperatures T1 and
T2 can for example be defined in terms of the (physically measurable) efficiency η12 of a
Carnot engine which operates between heat reservoirs at the two temperatures
...
This provides a physical operation for determining the ratio of the numerical
values of T1 and T2, and implies a physical addition for the two quantities: if
T2=nT1,where n=T2 /T1 , then T1+T2≡T3=(1+n)T1
...


29

Table 2
...
The unit for current is the ampere (A), which is

30

defined as the current which, when passed through each of two infinite,
parallel conductors in vacuum one meter apart, will produce a force of
2x10-7 N per unit length on each conductor
...

The mole unit is retained in SI as an alternative way of specifying
number of things: instead of counting things one by one, one counts them
in lots of 6
...
One must, however, specify
what things one is counting, e
...
"one mole of atoms, of molecules, of
tennis balls", or whatever
...

Luminous intensity does not refer to radiant energy flow per unit solid
angle as such, which could be measured in watts per steradian, but only to
that portion of it to which "the human eye" (as defined by a standard
response curve) is sensitive
...
46x10-3 watt per steradian
...
We consider them derived
quantities because, though dimensionless, they are defined in terms of
operations involving length, much like area is defined in terms of length
operations
...
Indeed, quantities
with quite different physical meaning, like work and torque, can have the
same dimension
...
2, which shows the dimensions of some mechanical
quantities in three different types of systems of units
...

In the first type of system illustrated in the table, length, time and mass
are base quantities and the force unit is measured in terms of mass and
acceleration via Newton's law in the form F=ma
...
In the second
type, length, time and force are base, and mass is measured via F=ma
...
The pound-force is the
force exerted by standard gravity (32
...
Mass in this

Table 2
...
2 times as large as the pound-mass
...
The British Engineering System is an example
...
2 lbf s2 lbm-1 ft-1
...
2 illustrates the fact that, while an actual physical quantity like
force is the same regardless of the (arbitrary) choice of the system of units,
its dimension depends on that choice
...

An interesting point to note is that only a few of the available universal
laws are usually "used up" to make base quantities into derived ones of the
second kind
...
This leaves us with some interesting possibilities
...
In such systems all units of
measurement are related to some of the universal constants that describe
our universe
...
Unfortunately, the choice of such “natural”
systems of units turns out to be far from unique, which renders futile any
attempt to endow any one of them with unique significance
...
8 Recapitulation
1
...
The
comparison operation is a physical procedure for establishing whether two
samples of the quantity are equal or unequal; the addition operation
defines what is meant by the sum of two samples of that property
...
Base quantities are properties for which the following concepts are
defined in terms of physical operations: equality, addition, subtraction,
multiplication by a pure number, and division by a pure number
...


33

3
...

4
...
A derived quantity is defined in terms
of numerical value (which depends on base unit size) and does not
necessarily have a tangible physical representation
...
The dimension of any physical quantity, whether base or derived, is
a formula that defines how the numerical value of the quantity changes
when the base unit sizes are changed
...
The
same quantity (e
...
force) may have different dimensions in different
systems of units, and quantities that are clearly physically different (e
...

work and torque) may have the same dimension
...
Relationships between physical quantities may be represented by
mathematical relationships between their numerical values
...
5)
...

7
...
If a particular base quantity turns out to be
uniquely related to some other base quantities via some universal law, then
we can, if we so desire, use the law to redefine that quantity as a derived
quantity of the second kind whose magnitude depends on the units chosen
for the others
...
e
...

8
...
Both the type and the number of base quantities are
open to choice
...
Dimensional Analysis
This chapter introduces the procedure of dimensional analysis and
describes Buckingham’s π-theorem, which follows from it
...
1
lays down the procedure in general terms and defines the vocabulary
...
2 gives an example, which the reader may wish to read in
parallel with section 3
...
Section 3
...

3
...
Dimensional analysis derives the logical consequences of this
premise
...
By
this we mean that, once all the quantities that define the particular process
or event are specified, the value of Q0 follows uniquely
...
Qn that determine the value of
Q0,
Q0 = f(Q1, Q2,
...


(3
...
Qn is complete if, once the values of the members are specified,
no other quantity can affect the value of Q0, and independent if the value
of each member can be adjusted arbitrarily without affecting the value of
any other member
...
Qn is as important in dimensional
analysis as it is in mathematical physics to start with the correct
fundamental equations and boundary conditions
...
We defer to section 3
...

The relationship expressed symbolically in equation (3
...
It is our
premise that its form must be such that, once the values Q1
...
The steps that follow derive
the consequences of this premise
...
Qn
...
2), and we
must specify at least the type the system of units before we do this
...
2 and are dealing with a
purely mechanical problem, all quantities have dimensions of the form

[ Qi ] = Ll M m t
i

i

i

(3
...

We now pick from the complete set of physically independent
variables Q1
...
Qk
(k n), and express the dimension of each of the remaining independent
variables Qk+1
...
Qk
...
Alternatively, it is
possible to express the dimension of one quantity as a product of powers
of the dimensions of other quantities which are not necessarily base
quantities
...
Qk of the set Q1
...
And complete if the
dimensions of all the remaining quantities Qk+1
...
Qk
...
1) is dimensionally homogeneous, the dimension of
the dependent variable Q0 is also expressible in terms of the dimensions of
Q1
...


36

The dimensionally independent subset Q1
...
Its members may be picked in different ways (see section 3
...
Qn is unique to the set, and cannot exceed the number of base
dimensions which appear in the dimensions the quantities in that set
...
Qn involve only length, mass, and time,
then k≤3
...
Qk,
we express the dimensions of Q0 and the remaining quantities Qk+1
...
Qk
...
Qk Nik ]

(3
...
The exponents Nij are dimensionless real numbers and can in
most cases be found quickly by inspection (see section 3
...

The formal procedure can be illustrated with an example where length,
mass and time are the only base quantities, in which case all dimensions
have the form of equation (3
...
Let us take Q1, Q2, and Q3 as the complete
dimensionally independent subset
...
2) with that of equation (3
...


(3
...
Qk
which has the same dimension,
Πi =

Q

Q1

N (k +i )1

Q2

k +i
N (k +i ) 2


...
5)

where i=1, 2,
...


N 01

Q1 Q2 N02
...
6)

Step 4: The end game and Buckingham’s -theorem
An alternative form of equation (3
...
, Qk;

1,

2

,
...
7)

in which all quantities are dimensionless except Q1
...
The values of the
dimensionless quantities are independent of the sizes of the base units
...
Qk, on the other hand, do depend on base unit size
...
From the principle that any
physically meaningful equation must be dimensionally homogeneous, that
is, valid independent of the sizes of the base units, it follows that Q1
...
7), that is,
= f( 1,

2

,
...


(3
...

The theorem derives its name from Buckingham's use of the symbol for
the dimensionless variables in his original 1914 paper
...
This can simplify the problem enormously, as will be
evident from the example that follows
...
It does not tell us the forms of the dimensionless variables
...
Nor
does the π-theorem, or for that matter dimensional analysis as such, say
anything about the form of the functional relationship expressed by
equation (3
...
That form has to be discovered by experimentation or by
solving the problem theoretically
...
2 An example: Deformation of an elastic ball striking a wall
Suppose we wish to investigate the deformation that occurs in elastic balls
when they impact on a wall
...
1)
...
We begin by specifying the problem more
clearly
...
We also agree to adopt a system of units of type 1 in table 2
...


Fig
...
1: A freshly dyed elastic ball leaving imprint after impact with rigid wall
...

Experience suggests that these should include at least the following: the
ball's diameter D and velocity V just prior to contact (the initial
conditions) and its mass m
...
Our theoretical understanding of solid mechanics tells us
that the quasi-static response of a perfectly elastic material is characterized
by two material properties, the modulus of elasticity E and Poisson's ratio
, and that the inertial effects which inevitably come into play during
collision and rebound will also depend on the material’s density
...
We know, however, by thinking of how the problem would have
to be set up as a theoretical one, that the answer for the numerical value d
will also depend on the values of all universal constants that appear in the
physical laws that control the ball's impact dynamics
...
Having chosen a system of units of type 1 in table 2
...
Nor are there any physical constants in the law of mass
conservation
...
This is a complete set, as required, but not an independent
set: once the ball's mass and diameter are specified, its density follows
...
(Other quantities
like V2, DE1/2, etcetera, all involving quantities that affect the value of d,
are excluded for the same reason: they are not independent of the
quantities already included
...


(3
...
One could just as well have chosen V2, , D, E, and , say, or V, m,
D, E, —see section 3
...
It should also be noted that further assumptions
have been taken for granted in equation (3
...
We have presumed, for
example, that the ball’s motion is unaffected by the properties of the fluid
through which it approaches the wall (which is certainly OK if the ball
moves through vacuum and a good approximation in air, but may not
apply to small balls in viscous liquids), and that gravitational effects play a
negligible role
...
3
...
9) are:
independent:

8

[V]=Lt-1
[ ]=ML-3
[D]=L
[E]=ML-1t-2
[ ]=1

(3
...
1, the value of
the dimensional constant c=F/ma for the chosen system of units would have to be
specified, and would affect the value of d
...
The dimension of any one of these three
cannot be made up of the dimensions of the other two
...
11)

dependent:

[d] = L = [D]

We have written down these results very simply by inspection
...
1
...

Step 3: Dimensionless similarity parameters
We non-dimensionalize the remaining independent variables E and and
the dependent variable d by dividing them by V 2 , D, and unity,
respectively, as suggested by equation (3
...
12)
d
D

42

Step 4: The end game
Using the logic that led to Buckingham's π−theorem, we now conclude
that
Π o = f (Π1 ,Π 2 ),
or
d
=
D

 E


...
13)

The number of independent variables has been reduced from the original
n=5 that define the problem to n-k =2
...
3 On the utility of dimensional analysis, and some difficulties and
questions that arise in its application
Similarity
Dimensional analysis provides a similarity law for the phenomenon under
consideration
...
The collisions
of two different elastic spheres 1 and 2 with a rigid wall, each with its own
values of V, , D, E, and , may appear to be quite different
...
14)
2

=

1,

equation (3
...

(3
...
14) apply, the two dynamic events
are similar in the sense of equation (3
...

Out-of-scale modeling
Scale modeling deals with the following question: If we want to learn
something about the performance of a full-scale system 1 by testing a
geometrically similar small-scale system model 2 (or vice-versa, if the
system of interest inaccessibly small), at what conditions should we test
the model, and how should we obtain the full-scale performance from
measurements at the small scale? Dimensional analysis provides the
answer
...
In that case we need only perform one small-scale
test with a model 2 of diameter D2, selecting its properties and test
conditions such that equations (3
...
The full-scale value d1 of the big ball’s imprint diameter at its
"design conditions" can then be obtained from equation (3
...

Dimensional analysis reduces the number of variables and minimizes
work
Dimensional analysis reduces the number of variables that must be
specified to describe an event
...
In our example of the impacting ball the answer depends on
five independent variables (equation 3
...
Suppose we set out to obtain the answer in a certain
region (a certain volume) of this variable-space, by either computation or
experimentation, and decide that 10 data points will be required in each
variable, with the other four being held constant
...
Dimensional analysis, however, shows us that in
dimensionless form the answer depends only on two similarity parameters
...
1% of the effort
...
1 shows some “experimental data” for impacts with balls of
three materials and various values of impact velocity
...
1: Computed “Experimental data”
Material

E

ρ

γ

V

E

Symbol

D

(Fig
...
2)

(MPa)

(kg m )

(m s )

3960

43

0
...
150

3
...
22

26937

0
...
66E+05

3960

77

0
...
190

Aluminum 6
...
33

3973

0
...
90E+04

2705

126

0
...
300

6
...
33

215

0
...
93E+00

1060

5

0
...
500

3
...
47

79

0
...
93E+00

1060

12

0
...
700

Rubber

ρV

d

3
...
2: Experimental data of d/D vs E/ρV2 for γ=0
...
5
...
2: Plot of “experimental data” in dimensionless form
...
2: Plot of “experimental data” in dimensionless form
...
Fig
...
2 shows these data as a plot of d/D vs
...
The simulated “experimental scatter” in Fig
...
2 actually
results from the coarseness of the computational grid
...
The influence of the Poisson’s ratio turns out to be
virtually negligible, given the scatter, and all the data may be curve-fitted
with an equation of the form d / D = f (E / V 2 )
...
Qn is in fact properly identified in step 1
...
If, however, the analysis is based on
a set which omits even one independent quantity that affects the value of
Qo, dimensional analysis will give erroneous results
...
9)
...
13), we would then
have obtained the absurd result
d
= f ( ),
D

(3
...

Superfluous independent quantities complicate the result unnecessarily
Errors on the side of excess have a less traumatic effect
...
For every superfluous independent quantity included
in the set, there will be in the final dimensionless relationship a
superfluous dimensionless similarity parameter
...
This would change equation (3
...
17)

where gD V 2 is a dimensionless gravity
...
17) is “wrong” only in the sense that it suggests a
dependence on g that is not noticeably there, and thus unnecessarily
complicates our thinking
...

On the importance of simplifying assumptions
The previous example illustrates an important point about most problems
in dimensional analysis: Completeness in the set of independent variables
is not an absolute matter, but depends on how we choose to circumscribe
the problem
...
This
would be the case for balls with such low coefficient of elasticity, or large
diameter, or low impact velocity, that their deformation at rest on a wall in
the gravitational field would be significant compared with their
deformation upon impact
...

If dimensional analysis depended on a truly complete identification of
the independent variables that specify a given physical event, we would in
most cases be reduced to impotence
...


47

some measure have been affected by the wing-beat of a butterfly in Brazil
a month earlier
...

Assuming tentatively that I am correct, what does dimensional analysis
tell me?”
On choosing a complete set of independent variables
Given what has been said above, how does one go about choosing a
complete set of independent variables that define a particular problem?
If we know the mathematical forms of all the equations and boundary
(and initial) conditions that completely specify a particular type of process
or event, one can deduce from them a complete set of independent
parameters that define the event
...
The set may include
position and time, if the variable of interest depends on them, universal
physical constants (e
...
gravitational constant, universal gas constant R,
etc
...
g
...
) and all other quantities that
appear not only in the equations but also in the boundary and initial
conditions that determine the answer to the particular problem at hand
...

If the equations and boundary conditions are not well known, as for
example when one is trying to use dimensional analysis to help correlate
experimental data for a complex phenomenon that is not well understood,
one has to proceed by trial and error based on an (educated) guess about
the physics of the problem at hand
...
There should be sufficient
data to be able to show that one has neither missed an important quantity
nor included one that is irrelevant
...
Non-dimensionalizing d and D with combinations
of V, E and , we might have obtained the result

d
D


2
1 / 3,
1/3 = F
 (mV / E)

( V / E)
2

(3
...
19)

where F and f are different functions of their arguments
...
19) is
of course identical to equation (3
...

The result is independent of the type of system of units
The choice of system of units may affect the dimensions of physical
quantities as well as the values of the physical constants that appear in the
underlying physical laws
...

Consider our example of the dyed ball, but viewed in terms of a
system of units like the British Engineering System (type3 in table 2
...
In such a
system Newton’s law reads F= cma, where c is a physical constant with
dimension Ft2m-1 L -1
...
Since
the impact process is controlled by Newton’s law, which now contains the
constant c, the value of which must be specified, we now have
d = d(V ,D, E,m, ,c)
...
20)

49

The number of independent variable (n=6) has increased by one
...
21)

[d]=L

The quantities V, D, m and c comprise a convenient dimensionally
independent subset
...
The dimensions of the remaining quantities can be expressed in
terms of these four as
[d]=[D]
[E]=[cV2D-3]
[ ]=0

(3
...
23)

This differs from our previous result, equation (3
...
Equations
(3
...
23) are, however, functionally identical
...
It is just that in the second system of
units, E must be non-dimensionalized with cmV2/D3 instead of mV2/D3,
since the latter no longer has the same dimension as E
...
This is, of
course, as we expected; the choice of system of units is arbitrary, and
should not affect the physical “bottom line
...
Dimensional Analysis in Problems where Some
Independent Quantities Have Fixed Values
Engineering practice often involves problems where some of the quantities
that define the problem have the same fixed values in all the applications
being considered
...
Basic fluid
mechanics tells us that, barring surface roughness effects, the drag force
should be completely determined by the cable’s length L and diameter d,
the ship’s (or water’s) velocity V, and the water’s density and viscosity

...
One way of writing this relationship
is
D

d
2 2 = f  Re, 
V L
L

(4
...
2)

is a Reynolds number based on cable length and d/L is the cable’s
“fineness coefficient,” which defines its geometry
...
1) is a
general relationship for the cable-towing problem as stated
...
The number of quantities that actually
vary from case to case are thus actually three, not five
...
1), that is, to a reduction in the number of
independent similarity parameters? Simply omitting the quantities that
have fixed values and performing dimensional analysis on the rest cannot

51

answer this question 9
...
All the
quantities whose values determine the quantity of interest must be
included, regardless of whether some of them happen to be the same
values in the problems that are of interest
...

The general question is the following: What reduction, if any, can be
obtained in the number of similarity parameters if a certain number of the
independent quantities that define the problem always have the same fixed
values? This can be answered by the following analysis
...
Let the quantities that
may vary be the first (n-nF) of Qi, and designate the nF quantities with
fixed values by Fi:
Q = f (Q1,Q2 ,
...
,FkF ,FkF +1,FkF +2 ,
...
3)

Choose a complete, dimensionally independent subset of the set Fi
...
3)
...
3) in the alternative form
∗
Q = f (Q1,Q2 ,
...
,FkF ,F k+1 ,Fk∗+1 ,
...
4)

where the asterisked indicate dimensionless quantities involving only
quantities with fixed values
...
For these cases, therefore,
we can write (4
...
,Qn− n F ;F1,F2 ,
...
5)

The value of Q is thus completely determined by a set of n-nF+kF
independent quantities consisting of those dependent quantities that are
9 Were we to simply omit the density and viscosity in the present problem, for example,
we would imply a relationship D=f(L,h,V) which is not dimensionally homogeneous and
therefore unacceptable
...


52

not fixed plus the dimensionally independent subset of the fixed
quantities
...
5)
...
5) a complete, dimensionally independent subset of k
quantities
...
Since equation
(4
...

According to equation (4
...
Dimensional analysis thus yields
the result
Q∗ = f (Π1,Π2 ,
...
6)

N = (n − k) − (nF − kF )
...
7)

where

We have arrived at the following theorem:
Theorem
If a quantity Q is completely determined by a set of n
independent quantities, of which k are dimensionally
independent, and if nF of these quantities have fixed values in
all the cases being considered, a number kF of these being
dimensionally independent, then a suitable dimensionless Q
will be completely determined by (n-k)-(nF-kF) dimensionless
similarity parameters
...
This theorem is a generalization of Buckingham’s -theorem, and
reduces to it when nF=0
...
The
similarity law equation (4
...

Simplification occurs only when some of the fixed quantities are
dimensionally dependent on the rest
...
I
...

Baker, W
...
, Westine, P
...
, and Dodge, F
...
, 1973, Similarity Methods
in Engineering Dynamics, Hayden, Rochelle Park, N
...

Bathe, Mark, 2001
...

Brand, L
...

Bridgman, P
...
, 1931, Dimensional Analysis, 2nd edition (the first edition
appeared in 1922), Yale University Press, New Haven
...
W
...

Bridgman
...
W
...
Haley, Editor-in-Chief), Vol
...
439-449:
Encyclopaedia Britannica, Chicago
...
, 1914, “On Physically Similar Systems; Illustrations of
the Use of Dimensional Analysis”, Physical Review, 4, 345-376
...
J
...

Drinker Bowen, Catherine, 1963, Francis Bacon: the Temper of a Man,
Little, Brown and Company, Boston
...
S
...

Einstein, A
...
, 1933, Essays in Science, translation published by
Philosophical Library, New York
...
, 1972, Dimensional Analysis and Group Theory in Astrophysics,
Pergamon Press, Oxford
...
, 1991, Dimensional Analysis and Scale-Up in Chemical
Engineering, Springer Verlag, Berlin
...
O
...

Maxwell, J
...
, 1891, A Treatise on Electricity and Magnetism, 3rd
Edition, Clarendon Press, Cambridge, republished by Dover, New York,
1954
...
A
...
T
...

Palacios, J
...

Sedov, L
...
, 1959, Similarity and Dimensional Analysis in Mechanics,
Academic Press, New York
...
C
...
2, 9, 237-253
...
I
...

Birge, R
...
, 1934, “On Electric and Magnetic Units and Dimensions”,
The American Physics Teacher, 2, 41-48
...
T
...

Birge, R
...
, 1935, “On the Establishment of Base and Derived Units, with
Special Reference to Electrical Units: Part II”, The American Physics
Teacher, 3, 171-179
...
, 1950, Hydrodynamics, Princeton University
...
W
...
, 1982, Experimental Modelling in
Engineering, Butterworths, London
...
, 1942, “On the Dimensions of Physical Magnitudes”,
Philosophical Magazine Ser
...

Duncan, W
...
, 1955, Physical Similarity and Dimensional Analysis,
Arnold, London
...
M
...

Huntley, H
...
, 1952, Dimensional Analysis, MacDonald & Co
...

Ipsen, D
...
, 1960, Units, Dimensions and Dimensional Numbers,
McGraw Hill, New York
...
de St
...
and Isaacson, M
...
Q
...

Kline, S
...
, 1965, Similitude and Approximation Theory, McGraw-Hill,
New York
...
L
...

Murphy, G
...


57

Pankhurst, R
...
, 1964, Dimensional Analysis and Scale Factors,
Reinhold, New York
...
J
...

Sena, L
...
, 1972, Units of Physical Quantities and Their Dimensions,
MIR, Moscow
...
, 1980, Similitude and Modeling, Elsevier, Amsterdam
...
J
...
Scranton
...
S
...

Zierep, J
...




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