# Buckingham π theorem

Buckingham π theorem This article needs additional citations for verification. Ajude a melhorar este artigo adicionando citações a fontes confiáveis. O material sem fonte pode ser contestado e removido. Encontrar fontes: "Buckingham π theorem" – notícias · jornais · livros · acadêmico · JSTOR (abril 2022) (Saiba como e quando remover esta mensagem de modelo) Edgar Buckingham circa 1886 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, p2, ..., 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 (por exemplo, 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.

Conteúdo 1 História 2 Declaração 3 Significado 4 Prova 4.1 Outline 4.2 Formal proof 5 Exemplos 5.1 Speed 5.2 The simple pendulum 5.3 Other examples 6 Veja também 7 Referências 7.1 Notas 7.2 Citações 7.3 Bibliografia 7.4 Original sources 8 External links History Although named for Edgar Buckingham, the π theorem was first proved by French mathematician Joseph Bertrand[1] dentro 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[Nota 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, para 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] e D. Riabouchinsky,[6] and again in 1914 by Buckingham.[7] It was Buckingham's article that introduced the use of the symbol " {estilo de exibição pi _{eu}} " for the dimensionless variables (or parameters), and this is the source of the theorem's name.

Statement More formally, o número {estilo de exibição p} of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, e {estilo de exibição k} is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.

Em termos matemáticos, if we have a physically meaningful equation such as {estilo de exibição f(q_{1},q_{2},ldots ,q_{n})=0,} Onde {estilo de exibição q_{1},ldots ,q_{n}} are the {estilo de exibição m} independent physical variables, and they are expressed in terms of {estilo de exibição k} independent physical units, then the above equation can be restated as {estilo de exibição F(pi_{1},pi_{2},ldots ,pi_{p})=0,} Onde {estilo de exibição pi _{1},ldots ,pi_{p}} are dimensionless parameters constructed from the {estilo de exibição q_{eu}} por {displaystyle p=n-k} dimensionless equations — the so-called Pi groups — of the form {estilo de exibição pi _{eu}=q_{1}^{uma_{1}},q_{2}^{uma_{2}}cdots q_{n}^{uma_{n}},} where the exponents {estilo de exibição a_{eu}} are rational numbers. (They can always be taken to be integers by redefining {estilo de exibição pi _{eu}} as being raised to a power that clears all denominators.) Significance 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. No entanto, 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.

Proof Outline 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" Operação, and raising to powers as the "scalar multiplication" Operação: 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). Por exemplo, the standard gravity {estilo de exibição g} has units of {displaystyle D/T^{2}=D^{1}T^{-2}} (distance over time squared), so it is represented as the vector {estilo de exibição (1,-2)} 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 proof Given a system of {estilo de exibição m} dimensional variables {estilo de exibição q_{1},ldots ,q_{n}} (with physical dimensions) dentro {estilo de exibição k} fundamental (basis) dimensões, the dimensional matrix is the {displaystyle ktimes n} matriz {estilo de exibição M} whose {estilo de exibição k} rows are the fundamental dimensions and whose {estilo de exibição m} columns are the dimensions of the variables: a {estilo de exibição (eu,j)} th entry (Onde {estilo de exibição 1leq ileq k} e {displaystyle 1leq jleq n} ) is the power of the {estilo de exibição eu} th fundamental dimension in the {estilo de exibição j} 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 {displaystyle ktimes 1} (coluna) vector that results from the multiplication {estilo de exibição M{começar{bmatriz}uma_{1}\vdots a_{n}fim{bmatriz}}} consists of the units of {estilo de exibição q_{1}^{uma_{1}},q_{2}^{uma_{2}}cdots q_{n}^{uma_{n}}} in terms of the {estilo de exibição k} fundamental independent (basis) units.[Nota 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 {displaystyle ker M} of this matrix.[Nota 2] By the rank–nullity theorem, a system of {estilo de exibição m} vectors (matrix columns) dentro {estilo de exibição k} linearly independent dimensions (the rank of the matrix is the number of fundamental dimensions) leaves a nullity {displaystyle p=dim(ker M)} satisfatório {displaystyle p=n-k,} 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 {displaystyle k=7} base units, which are the ampere, kelvin, segundo, 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]) Examples Speed 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 {estilo de exibição n=3} dimensioned variables: distance {estilo de exibição d,} time {estilo de exibição t,} e velocidade {estilo de exibição v,} and we are seeking some law of the form {displaystyle t=operatorname {Duration} (v,d).} These variables admit a basis of {displaystyle k=2} dimensões: time dimension {estilo de exibição T} and distance dimension {displaystyle D.} Thus there is {displaystyle p=n-k=3-2=1} dimensionless quantity.

The dimensional matrix is {estilo de exibição M={começar{bmatriz}1&0&;;;1\0&1&-1end{bmatriz}}} in which the rows correspond to the basis dimensions {estilo de exibição D} e {estilo de exibição T,} and the columns to the considered dimensions {estilo de exibição D,T,{texto{ e }}V,} 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. Por exemplo, the third column {estilo de exibição (1,-1),} afirma que {displaystyle V=D^{0}T^{0}V^{1},} represented by the column vector {estilo de exibição mathbf {v} =[0,0,1],} is expressible in terms of the basis dimensions as {displaystyle V=D^{1}T^{-1}=D/T,} desde {displaystyle Mmathbf {v} =[1,-1].} For a dimensionless constant {displaystyle pi =D^{uma_{1}}T^{uma_{2}}V^{uma_{3}},} we are looking for vectors {estilo de exibição mathbf {uma} =[uma_{1},uma_{2},uma_{3}]} such that the matrix-vector product {displaystyle Mmathbf {uma} } equals the zero vector {estilo de exibição [0,0].} Em álgebra linear, the set of vectors with this property is known as the kernel (or nullspace) do (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: {estilo de exibição mathbf {uma} ={começar{bmatriz}-1\;;;1\;;;1\fim{bmatriz}}.} 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: {displaystyle pi =d^{-1}t^{1}v^{1}=tv/d.} 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: {estilo de exibição F(pi )=0,} ou, de locação {estilo de exibição C} denote a zero of function {estilo de exibição F,} {displaystyle pi =C,} which can be written in the desired form (which recall was {displaystyle t=operatorname {Duration} (v,d)} ) Como {displaystyle t=C{fratura {d}{v}}.} The actual relationship between the three variables is simply {displaystyle d=vt.} Em outras palavras, nesse caso {estilo de exibição F} has one physically relevant root, and it is unity. The fact that only a single value of {estilo de exibição C} 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 {estilo de exibição T} of small oscillations in a simple pendulum. It will be assumed that it is a function of the length {estilo de exibição L,} the mass {estilo de exibição M,} and the acceleration due to gravity on the surface of the Earth {estilo de exibição g,} which has dimensions of length divided by time squared. The model is of the form {estilo de exibição f(T,M,eu,g)=0.} (Note that it is written as a relation, not as a function: {estilo de exibição T} is not written here as a function of {estilo de exibição M,eu,{texto{ e }}g.} ) Há {displaystyle k=3} fundamental physical dimensions in this equation: time {estilo de exibição t,} massa {estilo de exibição m,} and length {ell de estilo de exibição ,} e {displaystyle n=4} dimensional variables, {estilo de exibição T,M,eu,{texto{ e }}g.} Thus we need only {displaystyle p=n-k=4-3=1} dimensionless parameter, denotado por {estilo de exibição pi ,} and the model can be re-expressed as {estilo de exibição F(pi )=0,} Onde {estilo de exibição pi } É dado por {displaystyle pi =T^{uma_{1}}M^{uma_{2}}L^{uma_{3}}g^{uma_{4}}} for some values of {estilo de exibição a_{1},uma_{2},uma_{3},uma_{4}.} The dimensions of the dimensional quantities are: {displaystyle T=t,M=m,L=ell ,g=ell /t^{2}.} The dimensional matrix is: {estilo de exibição mathbf {M} ={começar{bmatriz}1&0&0&-2\0&1&0&0\0&0&1&1end{bmatriz}}.} (The rows correspond to the dimensions {estilo de exibição t,m,} e {ell de estilo de exibição ,} and the columns to the dimensional variables {estilo de exibição T,M,eu,{texto{ e }}g.} Por exemplo, the 4th column, {estilo de exibição (-2,0,1),} states that the {estilo de exibição g} variable has dimensions of {displaystyle t^{-2}m^{0}ell ^{1}.} ) We are looking for a kernel vector {displaystyle a=left[uma_{1},uma_{2},uma_{3},uma_{4}certo]} such that the matrix product of {estilo de exibição mathbf {M} } sobre {estilo de exibição a} yields the zero vector {estilo de exibição [0,0,0].} The dimensional matrix as written above is in reduced row echelon form, so one can read off a kernel vector within a multiplicative constant: {displaystyle a={começar{bmatriz}2\0\-1\1fim{bmatriz}}.} 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: {estilo de exibição {começar{alinhado}pi &=T^{2}M^{0}L^{-1}g^{1}\&=gT^{2}/Lend{alinhado}}.} In fundamental terms: {displaystyle pi =(t)^{2}(m)^{0}(bem )^{-1}deixei(ell /t^{2}certo)^{1}=1,} 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.

Neste exemplo, three of the four dimensional quantities are fundamental units, so the last (qual é {estilo de exibição g} ) must be a combination of the previous. Observe que se {estilo de exibição a_{2}} (the coefficient of {estilo de exibição M} ) had been non-zero then there would be no way to cancel the {estilo de exibição M} valor; Portanto {estilo de exibição a_{2}} deve ser zero. Dimensional analysis has allowed us to conclude that the period of the pendulum is not a function of its mass {displaystyle M.} (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, {estilo de exibição {vec {g}}+2{vec {T}}-{vec {eu}}} is the only nontrivial way to construct a vector of a dimensionless parameter.) The model can now be expressed as: {displaystyle Fleft(gT^{2}/Lright)=0.} Assuming the zeroes of {estilo de exibição F} are discrete and that they are labelled {estilo de exibição C_{1},C_{2},ldots ,} then this implies that {displaystyle gT^{2}/L=C_{eu}} for some zero {estilo de exibição C_{eu}} of the function {displaystyle F.} If there is only one zero, chame-o {estilo de exibição C,} então {displaystyle gT^{2}/L=C.} 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 {displaystyle C=4pi ^{2}.} 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.