Theory of Dimensionless Products: Reduction to Dimensionless Products

Reducing Non-Dimensionless Product to Dimensionless Form

K-transformation

Let x₁, x₂, …, x_n denote independent variables that represent magnitudes of physical quantities.

Geometrically, we may consider x₁, x₂, …, x_n as coordinates in a space S.

Also, let A, B and C be any positive constants and a, b and c be any integer such that the variables K₁, K₂, …, K_n are given by

K_i = A^a_i B^b_i C^c_i ∀ i = 1, 2, …, n Then, any independent variable x_i may be considered to be x_i = K_i x_i' ∀ i = 1, 2, …, n That is, a point transformation that carries the point x_i' to the point x_i in space S. This is called K-transformation.

K-space is the set of all points that can be derived from a given point x_i' by K-transformation.

The resultant of two K-transformations is also a K-transformation. And, the set of all K-transformations is algebraically a group.

If the magnitudes of a physical quantity x_i are restricted to have positive values, then the space S will consist entirely of points whose coordinates are positive. ❶

Four Lemmas

A K-space is generated by any one of its points.

Let the magnitudes of physical quantities x_i' i = 1, …, n represent points that by K-transformation generate the K-space.

Let x_i'' be another point in the K-space such that

x_i'' = K_i' x_i' where K_i' = (A')aⁱ (B')bⁱ (C')cⁱ

If the point generated by x_i' is

x_i = K_i x_i' where K_i = Aaⁱ Bbⁱ Ccⁱ and the point x_i also lies in the K-space, then x_i = K_i (x_i''/K_i')
x_i = K̄_i x_i'' where K̄_i ≜ (K_i/K_i'), thus $Kbar_i = (A/A')^a_i (B/B')^b_i (C/C')^c_i$ x_i = K̄_i x_i'' tells us that the point x_i in K-space is derived from the point x_i'' by a K-transformation.

Therefore, any point x_i derived from point x_i' by a K-transformation may also be derived from the point x_i'' by a K-transformation provided that the point x_i'' lies in the same K-space as the point x_i'.∎

The space S is completely partitioned by nonoverlapping K-spaces.

Since by definition any point in the space S generates a K-space and from the preceeding theorem that a K-space is generated by any one of its points therefore, the space S is partitioned into nonoverlapping K-spaces.∎

A dimensionally homogeneous dimensionless function, R = f(x₁, x₂, …, x_n) is constant in any K-space.

Since π is dimensionless K = A^a B^b C^c = A⁰ B⁰ C⁰ = 1 where A, B and C are any positive constants with respective dimensional exponents a, b and c.

Also, π is dimensionally homogeneous

Kπ = f(k₁x₁, k₁x₂, …, k_nx_n) substituting K = 1 we get π = f(k₁x₁, k₁x₂, …, k_nx_n) Thus, the point (x₁, x₂, …, x_n) generates π in K-space and π is a constant.∎

Since dimensionless products of x's are constants througout a K-space, if π₁, π₂, …, π_p is a complete set of dimensionless products of the x's, then to each K-space of the space S there is a corresponding single set of values of the π's. This is restated in the next theorem.

If π₁, π₂, …, π_p is a complete set of dimensionless products of the x's, there corresponds to each set of values of the π's a single K-space of the space S.

Let {π₁', π₂', …, π_p'} be a complete set of constant values of π's and let x_i' and x_i'' be points (i = 1, …, n) of the space S that corresponds to the values of the elements in the set. Thus, π_h' = (x₁')^{k₁^h} (x₂')^{k₂^h} … (x_n')^{k_n^h}
   π_h' = (x₁'')^{k₁^h} (x₂'')^{k₂^h} … (x_n'')^{k_n^h}
or
(x₁')^{k₁^h} (x₂')^{k₂^h} … (x_n')^{k_n^h} = (x₁'')^{k₁^h} (x₂'')^{k₂^h} … (x_n'')^{k_n^h} Since the magnitudes of physical quantities x's are assumed to have only positive values, taking logarithm of both sides yields k₁^h log(x₁') + k₂^h log(x₂') + … + k_n^h log(x_n') = k₁^h log(x₁'') + k₂^h log(x₂'') + … + k_n^h log(x_n'')
or
k₁^h [log(x₁') − log(x₁'')] + k₂^h [log(x₂') − log(x₂'')] + … + k_n^h [log(x_n') − log(x_n'')] = 0
or
k₁^h log(x₁'/x₁'') + k₂^h log(x₂'/x₂'') + … + k_n^h log(x_n'/x_n'') = 0 Substituting the definition r_i ≜ log(x_i'/x_i'') we get r₁ k₁^h + r₂ k₂^h + … + r_n k_n^h = 0     for all h = 1, 2, …, p Recall that {π₁', π₂', …, π_p'} is a complete set of dimensionless products and from the theorem^{(ibid. 6.)} that exponents k₁, k₂, …, k_n
of a complete set of dimensionless products of the independent variables x₁, x₂, …, x_n
are a fundamental system of solutions of
a₁k₁ + a₂k₂ + … + a_nk_n = 0
b₁k₁ + b₂k₂ + … + b_nk_n = 0
c₁k₁ + c₂k₂ + … + c_nk_n = 0 Thus, the system r₁ k₁^h + r₂ k₂^h + … + r_n k_n^h = 0     for all h = 1, 2, …, p will have the solution k_i¹, k_i², …, k_i^p and the coefficients r₁, r₂, …, r_n are linearly related to the coefficients of the fundamental system, that is, there exists α, β and γ such that r_i = log(x_i'/x_i'') = αa_i + βb_i + γc_i     for all h = 1, 2, …, p For logarithm of base 10 log₁₀(x_i'/x_i'') = αa_i + βb_i + γc_i
x_i' = x_i'' 10^{(αa_i + βb_i + γc_i)} Take A = 10^α, B = 10^β, C = 10^γ, then x_i' = x_i'' A^a_i B^b_i C^c_i
x_i' = K_i x_i''                           where K_i ≜ A^a_i B^b_i C^c_i

Therefore, points x_i' and x_i'' belong in the same K-space.∎

❷

Buckingham's Theorem

If an equation is dimensionally homogeneous, it can be reduced to a relationship among a complete set of dimensionless products.

Let y = f(x₁, x₂, …, x_n) be a dimensionally homogeneous equation. Then, we have the theorem^{(ibid. 7.)} that tells us that there exists a product of powers of x's for such an equation. In other words, the dimensionally homogeneous equation y = f(x₁, x₂, …, x_n) can be expressed as π = F(x₁, x₂, …, x_n) where π is dimensionless.

Furthermore because of lemma

If π₁, π₂, …, π_p is a complete set of dimensionless products of the x's, there corresponds to each set of values of the π's a single K-space of the space S.

we know that each set of values of π₁, π₂, …, π_p corresponds to a single K-space.

And from lemma

A dimensionally homogeneous dimensionless function, R = f(x₁, x₂, …, x_n) is constant in any K-space.

each K-space will correspond to a single value of π

Thus, each set of values of π₁, π₂, …, π_p will correspond to a single value of π. That is, π is a single-valued function of π₁, π₂, …, π_p.

Therefore, an arbitrary dimensionally homogeneous equation y = f(x₁, x₂, …, x_n) is reduced to the form π = F(x₁, x₂, …, x_n). Alternatively, an equation that relates dimensionless products is dimensionally homogeneous.∎

Since^{(ibid. 6.)}

The number of products in a complete set of dimensionless products of the independent variables x₁, x₂, …, x_n is n − r, in which r is the rank of the dimensional matrix of the variables.

for the complete set of dimensionless products π₁, π₂, …, π_p p = n − r

❸

Systematic steps for deriving a complete set of dimensionless products

The illustration is made using the example by Langhaar (1951d). Imagine that the investigation involves a unknown function f which is dependent a collection of variables and/or parameters: P, Q, R, S, T, U, V and all of them can be derived from three base dimensions M, L, T. Thus, the problem f(P, Q, R, S, T, U, V) is a MLT-dimensional system.

The relationship of all of the independent variables/parameters of the f to the three base dimensions can be summarized as

	P	Q	R	S	T	U	V
M	2	−1	3	0	0	−2	1
L	1	0	−1	0	2	1	2
T	0	1	0	3	1	−1	2

This is the dimensional matrix. Hence, Step-1: Get the Dimensional Matrix is accomplished.

From the earlier discussions we know that the dimensionless products about to be derived will be of the form

π = P^k₁ ⋅ Q^k₂ ⋅ R^k₃ ⋅ S^k₄ ⋅ T^k₅ ⋅ U^k₆ ⋅ V^k₇ ⋅ The entries of each row of the dimensional matrix can be taken as the coefficients of the k's giving us the system of homogeneous equations 2k₁ − k₂ + 3k₃ + 0k₄ + 0k₅ − 2k₆ + k₇ = 0
   k₁ + 0k₂ − k₃ + 0k₄ + 2k₅ + k₆ + 2k₇ = 0
   0k₁ + k₂ + 0k₃ + 3k₄ + k₅ − k₆ + 2k₇ = 0 Thus, performing Step-2: Solve the Homogeneous Equation returns k₅ = −11k₁ + 9k₂ − 9k₃ + 15k₄
k₆ = 5k₁ − 4k₂ + 5k₃ − 6k₄
k₇ = 8k₁ − 7k₂ + 7k₃ − 12k₄      These are the expressions for k₅, k₆ and k₇. What about k₁, k₂, k₃ and k₄?

To get the solution we start from k₁ = 1 and set the rest to 0 and then set k₂ and rest to 0 and so on until k₄ = 1 as follows

Set: k₁ = 1, rest to 0     Then: k₅ = −11, k₆ = 5, k₇ = 8
Set: k₂ = 1, rest to 0     Then: k₅ = 9, k₆ = −4, k₇ = −7
Set: k₃ = 1, rest to 0     Then: k₅ = −9, k₆ = 5, k₇ = 7
Set: k₄ = 1, rest to 0     Then: k₅ = 15, k₆ = −6, k₇ = −12 Putting this in tabular form we get

k₁	k₂	k₃	k₄	k₅	k₆	k₇
1	0	0	0	−11	5	8
0	1	0	0	9	−4	−7
0	0	1	0	−9	5	7
0	0	0	1	15	−6	−12

This is the solution matrix. Hence, Step-3: Get the Solution Matrix is accomplished.

	k₁	k₂	k₃	k₄	k₅	k₆	k₇
	P	Q	R	S	T	U	V
π₁	1	0	0	0	−11	5	8
π₂	0	1	0	0	9	−4	−7
π₃	0	0	1	0	−9	5	7
π₄	0	0	0	1	15	−6	−12

Finally, for Step-4: Get the Dimensionless Products we have π₁ = PT⁻¹¹U⁵V⁸
π₂ = QT⁹U⁻⁴V⁻⁷
π₃ = RT⁻⁹U⁵V⁷
π₄ = ST¹⁵U⁻⁶V⁻¹²

❹

diman^© is my in-house Clojure based software for dimensional analysis. It is based on seven base dimensions: [M], [L], [T], [K], [A], [mol] and [cd] representing the quantities mass, length, time, thermodynamic temperature, electric current, amount of substance and luminous instensity respectively. Therefore, diman^© is based on this seven dimensional system.

diman^© is capable of performing dimensional consistency checks and derive dimensionless products. The program aims at simplifying the tedious computational steps particularly while deriving dimensionless products.

For a given problem before on can get the results of a complete set of dimensionless products the user must perform some minimum initialization steps.

Setting up the dimensional formulae of all the independent variables of the unknown function f
(def formula_of_manifold_eqn [{:quantity "term-p", :dimension "[M^(2)*L^(1)]"} {:quantity "term-q", :dimension "[M^(-1)*T^(1)]"} {:quantity "term-r", :dimension "[M^(3)*L^(-1)]"} {:quantity "term-s", :dimension "[T^(3)]"} {:quantity "term-t", :dimension "[L^(2)*T^(1)]"} {:quantity "term-u", :dimension "[M^(-2)*L^(1)*T^(-1)]"} {:quantity "term-v", :dimension "[M^(1)*L^(2)*T^(2)]"}])

Since these are derived dimensional formulae, it must be placed temporarily inside the standard_formula entity. This is done with
(update-sformula formula_of_manifold_eqn)

Finally, to call the dimensions for respective independent variable of f we define
(def varpars [{:symbol "P", :quantity "term-p"} {:symbol "Q", :quantity "term-q"} {:symbol "R", :quantity "term-r"} {:symbol "S", :quantity "term-s"} {:symbol "T", :quantity "term-t"} {:symbol "U", :quantity "term-u"} {:symbol "V", :quantity "term-v"}])

Steps-1, 2 and 3 in one code

The processes for generating the dimensional matrix, solving the homogeneous equation and determining the solution matrix can be achieved in one code block as shown
(def solution_matrix (get-solution-matrix (solve (get-augmented-matrix (generate-dimmat varpars)))))
The solution matrix is therefore
=> (view-matrix solution_matrix) [1 0 0 0 -11N 5N 8N] [0 1 0 0 9N -4N -7N] [0 0 1 0 -9N 5N 7N] [0 0 0 1 15N -6N -12N] Size -> 4 x 7

For the final Step-4: Get the Dimensionless Products we use get-dimensionless-products. Thus,
=> (pprint (get-dimensionless-products solution_matrix varpars)) [{:symbol "pi0", :expression "P^(1)*T^(-11)*U^(5)*V^(8)"} {:symbol "pi1", :expression "Q^(1)*T^(9)*U^(-4)*V^(-7)"} {:symbol "pi2", :expression "R^(1)*T^(-9)*U^(5)*V^(7)"} {:symbol "pi3", :expression "S^(1)*T^(15)*U^(-6)*V^(-12)"}]

Therefore, diman^© saves the analyst from labouring in computational tasks but at the same time provides the ability to follow each step of the derivation process.

✪

Bibliography

Anton, H. (1977a). Elementary Linear Algebra (2nd ed.). John Wiley & Sons, Inc.
As of 2021 this book is in its 11th edition but I consider the earlier editions particular those published in the 1970s to be an excellent pedagogical text for a learner interested in the concepts of Linear Algebra. Calculus is not a prerequisite for this text.
Anton, H. (1977b). Vector Spaces. In Elementary Linear Algebra (2nd ed., pp. 121–204). John Wiley & Sons, Inc.
The concepts of vector space discussed here^{(ibid. 4.)} were based on this chapter.
BIPM (2020). Base unit definitions. Retrieved from https://www.bipm.org/en/measurement-units/base-units.html
Bureau International des Poids et Mesures is the organization whose mission is to provide standards on matters related to measurement science. diman^© is a software for doing dimensional analysis; It can do consistency checks and derive dimensionless products. Diman^(c) is based on the Internation System of Units (SI), base units.
Buckingham, E. (1914). On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Phys. Rev., 4(4), 345–376. https://doi.org/10.1103/PhysRev.4.345
This is the paper where Buckingham illustrates the possibility of reducing a given dimensionally homogeneous equation into a relationship among the complete set of dimensionless products of the equation. This has come to be known as Buckingham's Theorem.
Jerrard, H. G., & McNeill, D. B. (1992). Dictionary of Scientific Units: Including dimensionless numbers and scales (6th ed.). Chapman & Hall.
This is a good resource for someone doing science let alone dimensional analysis.
Langhaar, H. L. (1951a). Algebraic Theory of Dimensional Analysis. In Dimensional Analysis and Theory of Models (pp. 47–59). John Wiley & Sons, Inc.
Most of the materials for this lecture "Theory of Dimensionless Products" were influenced by this chapter.
Langhaar, H. L. (1951b). General Remarks on Dimensional Analysis. In Dimensional Analysis and Theory of Models (pp. 14–16). John Wiley & Sons, Inc.
This section provides the definition of dimensional analysis.
Langhaar, H. L. (1951c). Principles and Illustrations of Dimensional Analysis. In Dimensional Analysis and Theory of Models (pp. 13–28). John Wiley & Sons, Inc.
This preliminary chapter provides an overall view of why one might want to consider using dimensionless products. The section "General Remarks on Dimensional Analysis" is part of this chapter.
Langhaar, H. L. (1951d). Systematic Calculation of Dimensionless Products. In Dimensional Analysis and Theory of Models (pp. 29-46). John Wiley & Sons, Inc.
This chapter provides the steps for deriving dimensionless products. It has two examples. This chapter and the chapter on algebraic theory (Langhaar, 1951a) were the foundation for how diman^© can derive dimensionless products.
McNish, A. G. (1957, April 1). Dimensions units and standards. Physics Today, 10(4), 19. https://doi.org/10.1063/1.3060330
This is from a talk at the National Bureau of Standards, 1956. The talk addresses various fundamental questions like What is a dimension? What is the purpose of a dimensional system? Why should there be at least seven base/elemental dimensions? How should we choose the minimum number of quantities and hence the "absolute" units to build a consistent system of units for the given experimentally derived equations?
Preussner, G. M. (2018, May 24). Dimensional Analysis in Programming Languages. Personal Homepage. https://gmpreussner.com/research/dimensional-analysis-in-programming-languages
This article provides a fairly exaustive landscape of the software implementation — in terms of available packages that deal with dimensions or dimensional analysis and in some cases at the level of some high-level programming language — with regards to dimensional analysis. From this paper one notices that the notion of "dimensional analysis" used for most softwares are often not defined or poorly defined and hence vague and not necessarily alligned to its mathematical definition. Consequently, out of more than fifty or so dimensional analysis related softwares mentioned in the article only a very few addressed dimensionless product.

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K-transformation

Four Lemmas

Buckingham's Theorem

Systematic steps for deriving a complete set of dimensionless products

Accomplishing the steps in diman©

Accomplishing the steps in diman^©