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 = n − k 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.)
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.
Although named for Edgar Buckingham, the π theorem was first proved by French mathematician Joseph Bertrand 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, a heuristic proof with the use of series expansions, to 1894.
Formal generalization of the π theorem for the case of arbitrarily many quantities was given first by A. Vaschy in 1892, then in 1911—apparently independently—by both A. Federman and D. Riabouchinsky, and again in 1914 by Buckingham. 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.
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
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.
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.
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
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.)
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
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:
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 pendulum
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
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 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.
Cooling a drink with ice cubes
This section's factual accuracy is disputed. (November 2021)
Drinks cooled with small ice cubes cool faster than drinks cooled with the same mass of larger ice cubes. The common explanation for this phenomenon is that smaller cubes have greater surface area, and this greater area causes greater heat conduction and therefore faster cooling. For a given volume of ice, the total surface area of the ice is proportional to (the surface area of a single cube) times (the number of cubes), where is the length of the cube edges, and is the volume of ice. If the common explanation were correct, then it would imply that for a fixed volume of ice the rate of cooling should be proportional to and thus the time for the drink to cool should be proportional to In fact, dimensional analysis shows this common explanation to be incorrect, and gives the surprising result that the time to cool the drink is proportional to
The important dimensional quantities are the length scale of the cubes (dimension ), the time (dimension ), the temperature (dimension ), the thermal conductivity (dimensions ), and the volumetric heat capacity (dimensions ). The dimensional matrix is:
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.
The theorem has also been used in fields other than physics, for instance in sport sciences.
- When in applying the π–theorem there arises an arbitrary function of dimensionless numbers.
- 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).
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- 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.
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