Pi theorem

Pi-theorem ( $\Pi$ ${\ displaystyle \ Pi}$ $\ Pi$ Theorem $\pi$ ${\ displaystyle \ pi}$ $\ pi$ Theorem) is a fundamental theorem of dimensional analysis . The theorem states that if there is a relationship between $n$ ${\ displaystyle n}$ $n$ physical quantities, which does not change its appearance when changing the scale of units in a certain class of systems of units, it is equivalent to the dependence between, generally speaking, a smaller number $p=n-k$ ${\ displaystyle p = nk}$ $p = n-k$ dimensionless quantities where $k$ ${\ displaystyle k}$ $k$ - the largest number of quantities with independent dimensions among the original $n$ ${\ displaystyle n}$ $n$ quantities. The Pi-theorem allows us to establish the general structure of the dependence, which follows only from the requirement of the invariance of the physical dependence when changing the scale of units, even if the specific form of the dependence between the initial quantities is unknown.

Content

1 Name Options
2 Historical background
3 Statement of the theorem
4 Proof
5 Special cases
- 5.1. Application to an equation resolved with respect to a single quantity
- 5.2 The case when the pi-theorem gives the form of a dependence accurate to a factor
6 Notes on the application of the pi-theorem
7 Application of the pi-theorem for physical modeling
8 Examples of application of the pi-theorem
9 See also
10 Links
11 Notes

Title Options

In Russian-language literature on dimensional theory and modeling, the name pi-theorem is usually used ( $\Pi$ ${\ displaystyle \ Pi}$ $\Pi$ Theorem $\pi$ ${\ displaystyle \ pi}$ $\pi$ Theorem ) ^[1] ^[2] ^[3] ^[4] , derived from the traditional designation of dimensionless combinations using the (capital or lowercase) Greek letter pi . In English-language literature, the theorem is usually associated with the name of Buckingham , and in French-language, with the name Yours .

Historical background

Apparently, the first pi-theorem was proved by J. Bertrand ^[5] in 1878. Bertrand considers particular examples of problems from electrodynamics and the theory of heat conduction, but his presentation clearly contains all the basic ideas of the modern proof of the pi-theorem, as well as a clear indication of the use of the pi-theorem for modeling physical phenomena. The method of applying the pi-theorem ( the method of dimensions ) was widely known thanks to the work of Rayleigh (the first application of the pi-theorem in general form ^[6] to the dependence of the pressure drop in the pipeline on the determining parameters probably dates back to 1892 ^[7] , heuristic proof using expansion in a power series - by 1894 ^[8] ).

A formal generalization of the pi-theorem to the case of an arbitrary number of quantities was first formulated by Yours in 1892 ^[9] , and later, apparently, independently by A. Federman ^[10] , D. Ryabushinsky ^[11] in 1911 and Buckingham ^{[ 12]} in 1914. Subsequently, the pi-theorem is generalized German Weil in 1926 .

Statement of the theorem

For simplicity, the wording for positives is given below. $q_{i}$ ${\ displaystyle q_ {i}}$ .

Suppose there is a relationship between $n$ ${\ displaystyle n}$ physical quantities $q_{1}$ ${\ displaystyle q_ {1}}$ , $q_{2}$ ${\ displaystyle q_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{n}$ ${\ displaystyle q_ {n}}$ :

f(q_{1},q_{2},\ldots ,q_{n})=0,

{\ displaystyle f (q_ {1}, q_ {2}, \ ldots, q_ {n}) = 0,}

the form of which does not change when changing the scale of units in the selected class of unit systems (for example, if the class of unit systems LMT is used, then the form of the function $f$ ${\ displaystyle f}$ does not change with any changes in the standards of length, time and mass, say, when switching from measurements in kilograms, meters and seconds to measurements in pounds, inches and hours).

Among the arguments of the function , we choose the largest set of quantities with independent dimensions (such a choice can, generally speaking, be made in various ways). Then if the number of quantities with independent dimensions is indicated $k$ ${\ displaystyle k}$ and they are numbered by indices $one$ ${\ displaystyle 1}$ , $2$ ${\ displaystyle 2}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $k$ ${\ displaystyle k}$ (otherwise they can be renumbered), then the initial dependence $f$ ${\ displaystyle f}$ equivalent to the relationship between $p=n-k$ ${\ displaystyle p = nk}$ dimensionless quantities $\pi _{1}$ ${\ displaystyle \ pi _ {1}}$ , $\pi _{2}$ ${\ displaystyle \ pi _ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $\pi _{p}$ ${\ displaystyle \ pi _ {p}}$ :

F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,

{\ displaystyle F (\ pi _ {1}, \ pi _ {2}, \ ldots, \ pi _ {p}) = 0,}

Where $\pi _{i}$ ${\ displaystyle \ pi _ {i}}$ - dimensionless combinations obtained from the remaining initial values $q_{k+1}$ ${\ displaystyle q_ {k + 1}}$ , $q_{k+2}$ ${\ displaystyle q_ {k + 2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{n}$ ${\ displaystyle q_ {n}}$ dividing by the selected values in the appropriate degrees:

\pi _{1}={\frac {q_{k+1}}{q_{1}^{a}\cdot q_{2}^{b}\cdot \ldots \cdot q_{k}^{z}}},

{\ displaystyle \ pi _ {1} = {\ frac {q_ {k + 1}} {q_ {1} ^ {a} \ cdot q_ {2} ^ {b} \ cdot \ ldots \ cdot q_ {k} ^ {z}}},}

\vdots

{\ displaystyle \ vdots}

\pi _{p}={\frac {q_{n}}{q_{1}^{A}\cdot q_{2}^{B}\cdot \ldots \cdot q_{k}^{Z}}}

{\ displaystyle \ pi _ {p} = {\ frac {q_ {n}} {q_ {1} ^ {A} \ cdot q_ {2} ^ {B} \ cdot \ ldots \ cdot q_ {k} ^ { Z}}}}

(dimensionless combinations always exist because $q_{1}$ ${\ displaystyle q_ {1}}$ , $q_{2}$ ${\ displaystyle q_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{k}$ ${\ displaystyle q_ {k}}$ - a set of dimensionally independent values of the largest size, and when one more quantity is added to them, a set with dependent dimensions is obtained).

Proof

The proof of the pi-theorem is very simple ^[13] . Initial dependency $f$ ${\ displaystyle f}$ between $q_{1}$ ${\ displaystyle q_ {1}}$ , $q_{2}$ ${\ displaystyle q_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{n}$ ${\ displaystyle q_ {n}}$ can be considered as some relationship between $q_{1}$ ${\ displaystyle q_ {1}}$ , $q_{2}$ ${\ displaystyle q_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{k}$ ${\ displaystyle q_ {k}}$ and $\pi _{1}$ ${\ displaystyle \ pi _ {1}}$ , $\pi _{2}$ ${\ displaystyle \ pi _ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $\pi _{p}$ ${\ displaystyle \ pi _ {p}}$ :

\Phi (q_{1},q_{2},\ldots ,q_{k},\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,

{\ displaystyle \ Phi (q_ {1}, q_ {2}, \ ldots, q_ {k}, \ pi _ {1}, \ pi _ {2}, \ ldots, \ pi _ {p}) = 0 ,}

Moreover, the type of function $\Phi$ ${\ displaystyle \ Phi}$ also does not change when changing the scale of units. It remains to note that due to the dimensional independence of the quantities $q_{1}$ ${\ displaystyle q_ {1}}$ , $q_{2}$ ${\ displaystyle q_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{k}$ ${\ displaystyle q_ {k}}$ you can always choose a scale of units such that these quantities become equal to unity, while $\pi _{1}$ ${\ displaystyle \ pi _ {1}}$ , $\pi _{2}$ ${\ displaystyle \ pi _ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $\pi _{p}$ ${\ displaystyle \ pi _ {p}}$ being dimensionless combinations, they will not change their values, therefore, for the scale of units so chosen, and therefore, due to invariance, in any system of units, the function $\Phi$ ${\ displaystyle \ Phi}$ actually only depends on $\pi _{i}$ ${\ displaystyle \ pi _ {i}}$ :

\Phi (1,1,\ldots ,1,\pi _{1},\pi _{2},\ldots ,\pi _{p})\equiv H(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0.

{\ displaystyle \ Phi (1,1, \ ldots, 1, \ pi _ {1}, \ pi _ {2}, \ ldots, \ pi _ {p}) \ equiv H (\ pi _ {1}, \ pi _ {2}, \ ldots, \ pi _ {p}) = 0.}

Special cases

Application to an equation resolved with respect to one quantity

A variant of the pi-theorem is often used for the functional dependence of one physical quantity $q$ ${\ displaystyle q}$ from several others $q_{1}$ ${\ displaystyle q_ {1}}$ , $q_{2}$ ${\ displaystyle q_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $q_{n}$ ${\ displaystyle q_ {n}}$ :

q=f(q_{1},q_{2},\ldots ,q_{n}).

{\ displaystyle q = f (q_ {1}, q_ {2}, \ ldots, q_ {n}).}

In this case, the pi-theorem states that the dependence is equivalent to the relation

\pi =F(\pi _{1},\pi _{2},\ldots ,\pi _{p}),

{\ displaystyle \ pi = F (\ pi _ {1}, \ pi _ {2}, \ ldots, \ pi _ {p}),}

Where

\pi ={\frac {q}{q_{1}^{\alpha }\cdot q_{2}^{\beta }\cdot \ldots \cdot q_{k}^{\omega }}},

{\ displaystyle \ pi = {\ frac {q} {q_ {1} ^ {\ alpha} \ cdot q_ {2} ^ {\ beta} \ cdot \ ldots \ cdot q_ {k} ^ {\ omega}}} ,}

but $\pi _{i}$ ${\ displaystyle \ pi _ {i}}$ are defined in the same way as above.

The case when the pi-theorem gives the form of a dependence accurate to a factor

In one important particular case, when depending

q=f(q_{1},q_{2},\ldots ,q_{n})

{\ displaystyle q = f (q_ {1}, q_ {2}, \ ldots, q_ {n})}

all arguments have independent dimensions; applying the pi-theorem gives

\pi ={\frac {q}{q_{1}^{\alpha }\cdot q_{2}^{\beta }\cdot \ldots \cdot q_{k}^{\omega }}}={\text{const}},

{\ displaystyle \ pi = {\ frac {q} {q_ {1} ^ {\ alpha} \ cdot q_ {2} ^ {\ beta} \ cdot \ ldots \ cdot q_ {k} ^ {\ omega}}} = {\ text {const}},}

that is, the type of functional dependence is determined accurate to a constant. The value of a constant is not determined by the methods of dimensional theory, and experimental or other theoretical methods must be used to find it.

Notes on applying the pi-theorem

The choice of arguments with independent dimensions, generally speaking, can be done in various ways, as a result of which, when applying the pi-theorem, formally different expressions can be obtained. However, in fact, the results are equivalent, and from one form of writing you can get another by switching to combinations of dimensionless parameters.
In the statement of the pi-theorem, the requirement of invariance of dependence is important. If, for example, when working in the International System of Units (SI) in an experiment, a path dependence was obtained $s$ ${\ displaystyle s}$ passed by a falling body, from time to time $t$ ${\ displaystyle t}$

s={\frac {9{,}81\cdot t^{2}}{2}},

{\ displaystyle s = {\ frac {9 {,} 81 \ cdot t ^ {2}} {2}},}

then in this form it does not satisfy the conditions of the pi-theorem.

Applying the Pi-Theorem to Physical Modeling

The Pi-theorem is used for the physical modeling of various phenomena in aerodynamics , hydrodynamics , elasticity theory , and oscillation theory . The simulation is based on the fact that if for two natural processes (“model” and “natural”, for example, for the air flow around an airplane model in a wind tunnel and the air flow around a real airplane), dimensionless arguments (they are called similarity criteria ) depending

\pi =F(\pi _{1},\pi _{2},\ldots ,\pi _{p})

{\ displaystyle \ pi = F (\ pi _ {1}, \ pi _ {2}, \ ldots, \ pi _ {p})}

coincide, which can be realized due to a special choice of parameters of the “model” object, and dimensionless values of the function $\pi$ ${\ displaystyle \ pi}$ also match. This allows you to "recount" the dimensional experimental values of the parameters from the "model" object to the "natural", even if the form of the function $F$ ${\ displaystyle F}$ unknown. If the coincidence of all similarity criteria for “model” and “full-scale” objects is impossible to achieve, then they often resort to approximate modeling, when similarity is achieved only by criteria that reflect the influence of the most significant factors, while the influence of secondary factors is taken into account approximately based on additional considerations (not following from dimensional theory).

Examples of application of the pi-theorem

Bell oscillation frequency

Radiation of sound by a bell occurs as a result of its own vibrations , which can be described in the framework of the linear theory of elasticity . Frequency $f$ ${\ displaystyle f}$ the sound produced depends on the density $\rho$ ${\ displaystyle \ rho}$ Young's modulus $E$ ${\ displaystyle E}$ and Poisson's ratio $\nu$ ${\ displaystyle \ nu}$ metal of which the bell is made, and of a finite number of geometric dimensions $l_{1}$ ${\ displaystyle l_ {1}}$ , $l_{2}$ ${\ displaystyle l_ {2}}$ , $\ldots$ ${\ displaystyle \ ldots}$ , $l_{N}$ ${\ displaystyle l_ {N}}$ bells:

f=F(\rho ,E,\nu ,l_{1},l_{2},\ldots ,l_{N}).

{\ displaystyle f = F (\ rho, E, \ nu, l_ {1}, l_ {2}, \ ldots, l_ {N}).}

If the class system of units LMT is used, then as quantities with independent dimensions, for example, you can choose $\rho$ ${\ displaystyle \ rho}$ , $E$ ${\ displaystyle E}$ and $l_{1}$ ${\ displaystyle l_ {1}}$ (selected values included in the maximum dimensionally independent subsystem are underlined):

f=F({\underline {\rho }},{\underline {E_{\!}}},\nu ,{\underline {l_{1}}},l_{2},\ldots ,l_{N}),

{\ displaystyle f = F ({\ underline {\ rho}}, {\ underline {E _ {\!}}}, \ nu, {\ underline {l_ {1}}}, l_ {2}, \ ldots, l_ {N}),}

and applying the pi-theorem gives

{\frac {fl_{1}}{\sqrt {E/\rho }}}=G\left(\nu ,{\frac {l_{2}}{l_{1}}},{\frac {l_{3}}{l_{1}}},\ldots ,{\frac {l_{N}}{l_{1}}}\right).

{\ displaystyle {\ frac {fl_ {1}} {\ sqrt {E / \ rho}}} = G \ left (\ nu, {\ frac {l_ {2}} {l_ {1}}}, {\ frac {l_ {3}} {l_ {1}}}, \ ldots, {\ frac {l_ {N}} {l_ {1}}} right).}

If there are two geometrically similar bells made of the same material, then for them the function arguments $G$ ${\ displaystyle G}$ coincide, therefore, the ratio of their frequencies is inversely proportional to the ratio of their sizes (or inversely proportional to the cubic root of the ratio of their masses). This pattern is confirmed experimentally ^[14] .

Note that if other quantities were chosen as quantities with independent dimensions, for example $\rho$ ${\ displaystyle \ rho}$ , $E$ ${\ displaystyle E}$ and $l_{2}$ ${\ displaystyle l_ {2}}$ , then applying the pi-theorem would formally give a different result:

{\frac {fl_{2}}{\sqrt {E/\rho }}}=H\left(\nu ,{\frac {l_{1}}{l_{2}}},{\frac {l_{3}}{l_{2}}},\ldots ,{\frac {l_{N}}{l_{2}}}\right),

{\ displaystyle {\ frac {fl_ {2}} {\ sqrt {E / \ rho}}} = H \ left (\ nu, {\ frac {l_ {1}} {l_ {2}}}, {\ frac {l_ {3}} {l_ {2}}}, \ ldots, {\ frac {l_ {N}} {l_ {2}}} \ right),}

but the conclusions obtained would naturally remain the same.

Resistance during slow motion of a ball in a viscous fluid

With a slow (at low Reynolds numbers ) stationary motion of a sphere in a viscous fluid, the resistance force $F$ ${\ displaystyle F}$ depends on fluid viscosity $\mu$ ${\ displaystyle \ mu}$ as well as speed $V$ ${\ displaystyle V}$ and radius $R$ ${\ displaystyle R}$ spheres (fluid density is not among the determining parameters, since at low speeds the influence of fluid inertia is negligible). Applying to Addiction

F=f(\mu ,V,R)

{\ displaystyle F = f (\ mu, V, R)}

pi-theorem, we obtain

{\frac {F}{\mu VR}}={\text{const}},

{\ displaystyle {\ frac {F} {\ mu VR}} = {\ text {const}},}

i.e., in this problem, the resistance force is accurate to a constant. The value of the constant is not found for dimensional reasons (the solution of the corresponding hydrodynamic problem gives the value $6\pi$ ${\ displaystyle 6 \ pi}$ , which is confirmed experimentally).

Links

Some review works and primary sources on the history of pi-theorems and similarity theory

Notes

↑ G. Barenblatt. Similarity, self-similarity, intermediate asymptotic behavior. Theory and applications to geophysical hydrodynamics. - L .: Gidrometeoizdat, 1978.- S. 25. - 208 p.
↑ Sedov L.I. Methods of similarity and dimension in mechanics . - M .: Nauka, 1981 .-- S. 31 .-- 448 p.
↑ Bridgman P. Dimension Analysis . - Izhevsk: RHD, 2001 .-- S. 45. - 148 p.
↑ Huntley G. Dimension Analysis . - M .: Mir, 1970 .-- S. 6 .-- 176 p. (preface to the Russian edition)
↑ Bertrand J. Sur l'homogénété dans les formules de physique // Comptes rendus. - 1878. - T. 86 , No. 15 . - S. 916-920 .
↑ When, after applying the pi-theorem, an arbitrary function of dimensionless combinations arises.
↑ Rayleigh. On the question of the stability of the flow of liquids // Philosophical magazine. - 1892. - T. 34 . - S. 59-70 .
↑ Strett J.V. (Lord Rayleigh). Theory of sound . - M .: GITTL, 1955. - T. 2. - S. 348. - 476 p.
↑ Vaschy A. Sur les lois de similitude en physique // Annales Télégraphiques. - 1892. - T. 19 . - S. 25–28 . Quotations from your article with the statement of the pi-theorem are given in the article: Macagno E. O. Historico-critical review of dimensional analysis // Journal of the Franklin Institute. - 1971. - T. 292 , no. 6 . - S. 391-402 .
↑ Federman A. On some general methods for integrating first-order partial differential equations // Bulletin of the St. Petersburg Polytechnic Institute of Emperor Peter the Great. Department of Engineering, Natural Sciences and Mathematics. - 1911. - T. 16 , no. 1 . - S. 97-155 .
↑ Riabouchinsky D. Méthode des variables de dimension zéro et son application en aérodynamique // L'Aérophile. - 1911. - S. 407–408 .
↑ Buckingham E. On physically similar systems: illustrations of the use of dimensional equations // Physical Review. - 1914. - T. 4 , No. 4 . - S. 345-376 .
↑ Sena L. A. Units of physical quantities and their dimensions. - M .: Nauka , 1977 .-- S. 91-92.
↑ Pukhnachev Yu. Scattering, attenuation, refraction - three keys to unraveling the paradox // Science and Life. - 1983. - No. 2 . - S. 117-118 .

[1] G. Barenblatt. Similarity, self-similarity, intermediate asymptotic behavior. Theory and applications to geophysical hydrodynamics. - L .: Gidrometeoizdat, 1978.- S. 25. - 208 p.

[2] Sedov L.I. Methods of similarity and dimension in mechanics . - M .: Nauka, 1981 .-- S. 31 .-- 448 p.

[3] Bridgman P. Dimension Analysis . - Izhevsk: RHD, 2001 .-- S. 45. - 148 p.

[4] Huntley G. Dimension Analysis . - M .: Mir, 1970 .-- S. 6 .-- 176 p. (preface to the Russian edition)

[5] Bertrand J. Sur l'homogénété dans les formules de physique // Comptes rendus. - 1878. - T. 86 , No. 15 . - S. 916-920 .

[6] When, after applying the pi-theorem, an arbitrary function of dimensionless combinations arises.

[7] Rayleigh. On the question of the stability of the flow of liquids // Philosophical magazine. - 1892. - T. 34 . - S. 59-70 .

[8] Strett J.V. (Lord Rayleigh). Theory of sound . - M .: GITTL, 1955. - T. 2. - S. 348. - 476 p.

[9] Vaschy A. Sur les lois de similitude en physique // Annales Télégraphiques. - 1892. - T. 19 . - S. 25–28 . Quotations from your article with the statement of the pi-theorem are given in the article: Macagno E. O. Historico-critical review of dimensional analysis // Journal of the Franklin Institute. - 1971. - T. 292 , no. 6 . - S. 391-402 .

[10] Federman A. On some general methods for integrating first-order partial differential equations // Bulletin of the St. Petersburg Polytechnic Institute of Emperor Peter the Great. Department of Engineering, Natural Sciences and Mathematics. - 1911. - T. 16 , no. 1 . - S. 97-155 .

[11] Riabouchinsky D. Méthode des variables de dimension zéro et son application en aérodynamique // L'Aérophile. - 1911. - S. 407–408 .

[12] Buckingham E. On physically similar systems: illustrations of the use of dimensional equations // Physical Review. - 1914. - T. 4 , No. 4 . - S. 345-376 .

[13] Sena L. A. Units of physical quantities and their dimensions. - M .: Nauka , 1977 .-- S. 91-92.

[14] Pukhnachev Yu. Scattering, attenuation, refraction - three keys to unraveling the paradox // Science and Life. - 1983. - No. 2 . - S. 117-118 .