Buckingham π theorem

In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = nk dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)

Edgar Buckingham circa 1886

The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.

The Buckingham π theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (for example, pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and the theorem would not hold.

HistoryEdit

Although named for Edgar Buckingham, the π theorem was first proved by French mathematician Joseph Bertrand[1] in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem ("the method of dimensions") became widely known due to the works of Rayleigh. The first application of the π theorem in the general case[note 1] to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892,[2] a heuristic proof with the use of series expansions, to 1894.[3]

Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892,[4] then in 1911—apparently independently—by both A. Federman[5] and D. Riabouchinsky,[6] and again in 1914 by Buckingham.[7] It was Buckingham's article that introduced the use of the symbol " " for the dimensionless variables (or parameters), and this is the source of the theorem's name.

StatementEdit

More formally, the number   of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and   is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.

In mathematical terms, if we have a physically meaningful equation such as

 
where   are the   independent physical variables, and they are expressed in terms of   independent physical units, then the above equation can be restated as
 
where   are dimensionless parameters constructed from the   by   dimensionless equations — the so-called Pi groups — of the form
 
where the exponents   are rational numbers. (They can always be taken to be integers by redefining   as being raised to a power that clears all denominators.)

SignificanceEdit

The Buckingham π theorem provides a method for computing sets of dimensionless parameters from given variables, even if the form of the equation remains unknown. However, the choice of dimensionless parameters is not unique; Buckingham's theorem only provides a way of generating sets of dimensionless parameters and does not indicate the most "physically meaningful".

Two systems for which these parameters coincide are called similar (as with similar triangles, they differ only in scale); they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one. Most importantly, Buckingham's theorem describes the relation between the number of variables and fundamental dimensions.

ProofEdit

OutlineEdit

It will be assumed that the space of fundamental and derived physical units forms a vector space over the rational numbers, with the fundamental units as basis vectors, and with multiplication of physical units as the "vector addition" operation, and raising to powers as the "scalar multiplication" operation: represent a dimensional variable as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). For instance, the standard gravity   has units of   (distance over time squared), so it is represented as the vector   with respect to the basis of fundamental units (distance, time).

Making the physical units match across sets of physical equations can then be regarded as imposing linear constraints in the physical-units vector space.

Formal proofEdit

Given a system of   dimensional variables   (with physical dimensions) in   fundamental (basis) dimensions, the dimensional matrix is the   matrix   whose   rows are the fundamental dimensions and whose   columns are the dimensions of the variables: the  th entry (where   and  ) is the power of the  th fundamental dimension in the  th variable. The matrix can be interpreted as taking in a combination of the dimensions of the variable quantities and giving out the dimensions of this product in fundamental dimensions. So the   (column) vector that results from the multiplication

 
consists of the units of
 
in terms of the   fundamental independent (basis) units.[note 2]

A dimensionless variable is a quantity that has all of its fundamental dimensions raised to the zeroth power (the zero vector of the vector space over the fundamental dimensions). The dimensionless variables are exactly the vectors that belong to the kernel   of this matrix.[note 2]

By the rank–nullity theorem, a system of   vectors (matrix columns) in   linearly independent dimensions (the rank of the matrix is the number of fundamental dimensions) leaves a nullity   satisfying   where the nullity is the number of extraneous dimensions which may be chosen to be dimensionless.

The dimensionless variables can always be taken to be integer combinations of the dimensional variables (by clearing denominators). There is mathematically no natural choice of dimensionless variables; some choices of dimensionless variables are more physically meaningful, and these are what are ideally used.

The International System of Units defines   base units, which are the ampere, kelvin, second, metre, kilogram, candela and mole. It is sometimes advantageous to introduce additional base units and techniques to refine the technique of dimensional analysis. (See orientational analysis and reference.[8])

ExamplesEdit

SpeedEdit

This example is elementary but serves to demonstrate the procedure.

Suppose a car is driving at 100 km/h; how long does it take to go 200 km?

This question considers   dimensioned variables: distance   time   and speed   and we are seeking some law of the form   These variables admit a basis of   dimensions: time dimension   and distance dimension   Thus there is   dimensionless quantity.

The dimensional matrix is

 
in which the rows correspond to the basis dimensions   and   and the columns to the considered dimensions   where the latter stands for the speed dimension. The elements of the matrix correspond to the powers to which the respective dimensions are to be raised. For instance, the third column   states that   represented by the column vector   is expressible in terms of the basis dimensions as   since  

For a dimensionless constant   we are looking for vectors   such that the matrix-vector product   equals the zero vector   In linear algebra, the set of vectors with this property is known as the kernel (or nullspace) of (the linear map represented by) the dimensional matrix. In this particular case its kernel is one-dimensional. The dimensional matrix as written above is in reduced row echelon form, so one can read off a non-zero kernel vector to within a multiplicative constant:

 

If the dimensional matrix were not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant, replacing the dimensions by the corresponding dimensioned variables, may be written:

 

Since the kernel is only defined to within a multiplicative constant, the above dimensionless constant raised to any arbitrary power yields another (equivalent) dimensionless constant.

Dimensional analysis has thus provided a general equation relating the three physical variables:

 
or, letting   denote a zero of function  
 
which can be written in the desired form (which recall was  ) as
 

The actual relationship between the three variables is simply   In other words, in this case   has one physically relevant root, and it is unity. The fact that only a single value of   will do and that it is equal to 1 is not revealed by the technique of dimensional analysis.

The simple pendulumEdit

We wish to determine the period   of small oscillations in a simple pendulum. It will be assumed that it is a function of the length   the mass   and the acceleration due to gravity on the surface of the Earth   which has dimensions of length divided by time squared. The model is of the form

 

(Note that it is written as a relation, not as a function:   is not written here as a function of  )

There are   fundamental physical dimensions in this equation: time   mass   and length   and   dimensional variables,   Thus we need only   dimensionless parameter, denoted by   and the model can be re-expressed as

 
where   is given by
 
for some values of  

The dimensions of the dimensional quantities are:

 

The dimensional matrix is:

 

(The rows correspond to the dimensions   and   and the columns to the dimensional variables   For instance, the 4th column,   states that the   variable has dimensions of  )

We are looking for a kernel vector   such that the matrix product of   on   yields the zero vector   The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant:

 

Were it not already reduced, one could perform Gauss–Jordan elimination on the dimensional matrix to more easily determine the kernel. It follows that the dimensionless constant may be written:

 
In fundamental terms:
 
which is dimensionless. Since the kernel is only defined to within a multiplicative constant, if the above dimensionless constant is raised to any arbitrary power, it will yield another equivalent dimensionless constant.

In this example, three of the four dimensional quantities are fundamental units, so the last (which is  ) must be a combination of the previous. Note that if   (the coefficient of  ) had been non-zero then there would be no way to cancel the   value; therefore   must be zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass   (In the 3D space of powers of mass, time, and distance, we can say that the vector for mass is linearly independent from the vectors for the three other variables. Up to a scaling factor,   is the only nontrivial way to construct a vector of a dimensionless parameter.)

The model can now be expressed as:

 

Assuming the zeroes of   are discrete and that they are labelled   then this implies that   for some zero   of the function   If there is only one zero, call it   then   It requires more physical insight or an experiment to show that there is indeed only one zero and that the constant is in fact given by  

For large oscillations of a pendulum, the analysis is complicated by an additional dimensionless parameter, the maximum swing angle. The above analysis is a good approximation as the angle approaches zero.

Other examplesEdit

An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method.[9]

The theorem has also been used in fields other than physics, for instance in sport sciences.[10]

See alsoEdit

ReferencesEdit

NotesEdit

  1. ^ When in applying the π–theorem there arises an arbitrary function of dimensionless numbers.
  2. ^ a b If these basis units are   and if   for every   then   so that for instance, the units of   in terms of these basis units are   For a concrete example, suppose that the   fundamental units are meters   and seconds   and that there are   dimensional variables:   By definition of vector addition and scalar multiplication in  
     
    so that   By definition, the dimensionless units are those of the form   which are exactly the vectors in   This can be verified by a direct computation:
     
    which is indeed dimensionless. Consequently, if some physical law states that   are necessarily related by a (presumably unknown) equation of the form   for some (unknown) function   with   (that is, the tuple   is necessarily a zero of  ), then there exists some (also unknown) function   that depends on only   variable, the dimensionless variable   (or any non-zero rational power   of   where  ), such that   holds (if   is used instead of   then   can be replaced with   and once again   holds). Thus in terms of the original variables,   must hold (alternatively, if using   for instance, then   must hold). In other words, the Buckingham π theorem implies that   so that if it happens to be the case that this   has exactly one zero, call it   then the equation   will necessarily hold (the theorem does not give information about what the exact value of the constant   will be, nor does it guarantee that   has exactly one zero).

CitationsEdit

  1. ^ Bertrand, J. (1878). "Sur l'homogénéité dans les formules de physique". Comptes Rendus. 86 (15): 916–920.
  2. ^ Rayleigh (1892). "On the question of the stability of the flow of liquids". Philosophical Magazine. 34 (206): 59–70. doi:10.1080/14786449208620167.
  3. ^ Strutt, John William (1896). The Theory of Sound. Vol. II (2nd ed.). Macmillan.
  4. ^ Quotes from Vaschy's article with his statement of the pi–theorem can be found in: Macagno, E. O. (1971). "Historico-critical review of dimensional analysis". Journal of the Franklin Institute. 292 (6): 391–402. doi:10.1016/0016-0032(71)90160-8.
  5. ^ Федерман, А. (1911). "О некоторых общих методах интегрирования уравнений с частными производными первого порядка". Известия Санкт-Петербургского политехнического института императора Петра Великого. Отдел техники, естествознания и математики. 16 (1): 97–155. (Federman A., On some general methods of integration of first-order partial differential equations, Proceedings of the Saint-Petersburg polytechnic institute. Section of technics, natural science, and mathematics)
  6. ^ Riabouchinsky, D. (1911). "Мéthode des variables de dimension zéro et son application en aérodynamique". L'Aérophile: 407–408.
  7. ^ Buckingham 1914.
  8. ^ Schlick, R.; Le Sergent, T. (2006). "Checking SCADE Models for Correct Usage of Physical Units". Computer Safety, Reliability, and Security. Lecture Notes in Computer Science. Berlin: Springer. 4166: 358–371. doi:10.1007/11875567_27. ISBN 978-3-540-45762-6.
  9. ^ Ramsay, Angus. "Dimensional Analysis and Numerical Experiments for a Rotating Disc". Ramsay Maunder Associates. Retrieved 15 April 2017.
  10. ^ Blondeau, J. (2020). "The influence of field size, goal size and number of players on the average number of goals scored per game in variants of football and hockey: the Pi-theorem applied to team sports". Journal of Quantitative Analysis in Sports. 17 (2): 145–154. doi:10.1515/jqas-2020-0009. S2CID 224929098.

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