Buckingham's Π-theorem

The Buckinghamsche Π theorem (read: Pi theorem ) by Edgar Buckingham (1867-1940) is a fundamental theorem of similarity theory and dimensional analysis .

It describes how a physically meaningful equation with n dimensional quantities can be rewritten into an equation with nm dimensionless quantities , where m is the number of independent basic quantities used. Furthermore, Buckingham's Π theorem makes it possible to determine dimensionless key figures for a problem from the output variables, even if the exact relationship is not yet known in the form of an equation.

Apparently, while investigating problems in electrodynamics and the theory of heat conduction in 1878 , Joseph Bertrand first drew attention to the core content of the Π theorem and possible applications for modeling physical phenomena. The new methods of dimensional analysis became particularly well known in 1892 through Rayleigh's work on the pressure drop in a pipeline using the generalized The-theorem. The formalized generalization of the Π-theorem to the case of any number of physical quantities was first carried out in 1892 by Aimé Vaschy , then apparently independently by A. Federman and Dmitri Pawlowitsch Rjabuschinski in 1911 and finally in 1914 by Edgar Buckingham. In 1926 Hermann Weyl dealt with the Π-theorem.

Determination of the influencing factors

The determination of the influencing variables that describe a physical problem is the difficulty in applying Buckingham's Π-Theorem. Intuition and / or physical knowledge is required in this phase. With a consistent choice of influencing variables, however , a conversion into n - m dimensionless variables is always possible. Natural constants (e.g. the speed of light ) can also play a role here.

A conversion of the dimensional influencing variables into dimensionless parameters is only possible if each base dimension occurs in at least two dimensional influencing variables of the physical system. This requirement is necessary, but not sufficient. If it turns out that a conversion is not possible, this means that either too many, too few or the wrong influencing variables have been selected. Regardless of this, important influencing variables may have been forgotten and superfluous variables may have been used in the event of a successful conversion.

Examples

Friction-free pendulum

Pendulum (frictionless)

For small deflections, the pendulum length l , the acceleration due to gravity g , as well as the mass m can be assumed as the three essential descriptive quantities for the period of oscillation t of a pendulum ( n = 4 ). It should be the basic dimensions

• L length ( SI unit : m ),
• M mass (SI unit: kg ) and
• T time (SI unit: s )

can be used ( m = 3 ). The dimensions of the influencing variables can be expressed as the power product of the basic dimensions:

• l : dimension L ;
• g : dimension L / T ² ;
• m : dimension M ;
• t : Dimension T .

The product approach

(1) ${\displaystyle \Pi =l^{x_{1}}\cdot g^{x_{2}}\cdot m^{x_{3}}\cdot t^{x_{4}}}$

can only become dimensionless if

(2) ${\displaystyle L^{x_{1}}\cdot (L/{T^{2}})^{x_{2}}\cdot M^{x_{3}}\cdot T^{x_{4}}=1}$

and thus

(3) Length: ${\displaystyle x_{1}+x_{2}\;=\;0}$

(4) Mass: ${\displaystyle x_{3}\;=\;0}$

(5) Time: ${\displaystyle -2\cdot x_{2}+x_{4}=0}$

applies. Because of (1) and (4), the mass, contrary to the above assumption, has no significance for the period of oscillation . This is a first result of Buckingham's theorem. Since , and are only determined by the two equations (3) and (5), any one of the three unknowns can be selected freely (but not equal to 0) (for example ). Then: ${\displaystyle \Pi }$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle x_{4}}$ ${\displaystyle x_{4}=1}$

(6) , and . ${\displaystyle x_{4}=1}$ ${\displaystyle x_{2}=1/2}$ ${\displaystyle x_{1}=-1/2}$

Because nm = 1, a single dimensionless quantity is sufficient to describe the relationship sought. With (1) and (6) this becomes:

(7) . ${\displaystyle \Pi ={\sqrt {g/l}}\cdot m^{0}\cdot t={\sqrt {g/l}}\cdot t}$

Since no further dimensionless quantities are involved, Buckingham's theorem must be the result ${\displaystyle \Pi }$

(8th) ${\displaystyle {\sqrt {g/l}}\cdot t={\text{const.}}}$

be valid. The unknown constant of proportionality can be determined with a single experiment , and one obtains ${\displaystyle {\text{const.}}=6{,}283...}$

(9) ${\displaystyle t=6{,}283...\cdot {\sqrt {l/g}}}$

as the period of oscillation. Note that this relationship was determined without using the underlying differential equation of the motion of the pendulum. A solution to these differential equations provides the analog result

(10) . ${\displaystyle t=2\cdot \pi \cdot {\sqrt {l/g}}}$

The interpretation of the constant of proportionality as ( ) can neither provide Buckingham's theorem nor the experiment. ${\displaystyle {\text{const.}}=2\pi }$ ${\displaystyle \approx 6{,}283...}$ ${\displaystyle \Pi }$

Summary

Procedure	Results
Buckingham's theorem ${\displaystyle \Pi }$	${\displaystyle t={\text{const.}}\cdot {\sqrt {l/g}}}$
Buckingham's theorem and experiment ${\displaystyle \Pi }$	${\displaystyle t=6{,}283...\cdot {\sqrt {l/g}}}$
Solution of the differential equation	${\displaystyle t=2\cdot \pi \cdot {\sqrt {l/g}}}$

Spring pendulum

Spring-mass pendulum

If one assumes the mass m and the spring constant c as the essential parameters for calculating the oscillation period t of a spring pendulum , the following approach can be used:

${\displaystyle \Pi =m^{x_{1}}\cdot c^{x_{2}}\cdot t^{x_{3}}}$

Since in this approach only the dimensions mass M and time T occur,

${\displaystyle M^{x_{1}}\cdot {\left({\frac {M}{T^{2}}}\right)}^{x_{2}}\cdot {T}^{x_{3}}=1}$

there are only two equations:

(1) Mass: ${\displaystyle x_{1}+x_{2}\;=0}$

(2) Time: ${\displaystyle -2\cdot x_{2}+x_{3}=0}$

for the three unknowns . Follow with assumption and . If you insert the results for in the approach, you get ${\displaystyle x_{1},x_{2},x_{3}}$ ${\displaystyle x_{3}\;=1}$ ${\displaystyle x_{2}=x_{3}/2=\;1/2}$ ${\displaystyle x_{1}=\;-x_{2}=\;-1/2}$ ${\displaystyle x_{1},x_{2},x_{3}}$

${\displaystyle \Pi =t\cdot {\sqrt {c/m}}={\text{const.}}}$

and thus

${\displaystyle t\sim {\sqrt {m/c}}}$.

Rotating ring

Rotating ring

For the stresses σ that arise in a rotating ring, a dependence on the rotational speed ω, the radius r and the density ρ is assumed (n = 4). The resulting approach

${\displaystyle \Pi =\sigma ^{x_{1}}\cdot \rho ^{x_{2}}\cdot \omega ^{x_{3}}\cdot r^{x_{4}}}$

can only become dimensionless if for the m = 3 basic dimensions used here ( L length, M mass, T time)

${\displaystyle {\left({\frac {M}{T^{2}\,L}}\right)}^{x_{1}}\cdot {\left({\frac {M}{L^{3}}}\right)}^{x_{2}}\cdot {\left({\frac {1}{T}}\right)}^{x_{3}}\cdot L^{x_{4}}=1}$

and thus

(1) Mass: ${\displaystyle x_{1}+x_{2}\;=\;0}$

(2) Time: ${\displaystyle -2x_{1}-x_{3}\;=\;0}$

(3) Length: ${\displaystyle -x_{1}-3x_{2}+x_{4}\;=\;0}$

applies. Only 3 equations are available for the four unknowns ( ). The system of equations becomes unambiguous with the arbitrary assumption . Follow from (1) and (2) and . With (3) can be determined. ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$ ${\displaystyle x_{1}=1}$ ${\displaystyle x_{2}=-1}$ ${\displaystyle x_{3}=-2}$ ${\displaystyle x_{4}=-2}$

The Π-theorem thus says:

${\displaystyle \sigma ^{1}\cdot \rho ^{-1}\cdot \omega ^{-2}\cdot r^{-2}={\text{const.}}\quad \Rightarrow \quad \sigma ={\text{const.}}\cdot \rho \cdot \omega ^{2}\cdot r^{2}}$.

The stress σ depends linearly on the density and quadratically on the angular velocity and the radius. The unknown constant of proportionality cannot be determined with the π-theorem.

literature

• E. Buckingham: The principle of similitude. In: Nature. 96, 1915, pp. 396-397.
• E. Buckingham: Model experiments and the forms of empirical equations. In: Trans. ASME 37, 1915, pp. 263-296.
• JH Spurk: Dimension analysis in fluid mechanics. Springer-Verlag, 1992, ISBN 3-540-54959-5 .

See also

• Similarity theory in the context of physical processes and allometry in biology
• Dimension (size system)
• Dimension analysis There you can find the formation via matrices.

Individual evidence

1. ^ Bertrand J .: Sur l'homogénété dans les formules de physique . In: Comptes rendus . tape 86 , no. 15 , 1878, p. 916-920 .
2. ^ Rayleigh: On the question of the stability of the flow of liquids . In: Philosophical Magazine . tape 34 , 1892, p. 59-70 .
3. ^ Vaschy A .: Sur les lois de similitude en physique . In: Annales Télégraphiques . tape 19 , 1892, p. 25-28 .
4. ^ Macagno EO: Historico-critical review of dimensional analysis . In: Journal of the Franklin Institute . tape 292 , no. 6 , 1971, p. 391-402 .
5. ↑ Федерман А .: О некоторых общих методах интегрирования уравнений с частными производными первордо первордо . In: Известия Санкт-Петербургского политехнического института императора Петра Великого. Отдел техники, естествознания и математики . tape 16 , no. 1 , 1911, p. 97-155 .
6. ↑ Riabouchinsky D .: М éthode des variables de dimension zéro et son application en aerodynamique . In: L'Aérophile . 1911, p. 407-408 .
7. ^ Buckingham E .: On physically similar systems: illustrations of the use of dimensional equations . In: Physical Review . tape 4 , no. 4 , 1914, pp. 345-376 .