Theory of Dimensionless Products: Complete Set of Dimensionless Products

Complete Set of Dimensionless Products

Dimensionless products and complete set of dimensionless products

Recall that in dimensional analysis the product^{(ibid. 3.)} is given by the expression

y = x₁^k₁ ⋅ x₂^k₁ ⋅ … ⋅ x_n^k_n

If the product has no dimension it is called dimensionless product. In dimensional analysis the symbol π is used as a convention to denote dimensionless products. It does not equate to the number ≈ 3.14159.

To be more precise, π is used to symbolize the complete set of dimensionless products π_i of a function.

π₁ = x₁^k₁^' ⋅ x₂^k₁^' ⋅ … ⋅ x_n^k_n^'
π₂ = x₁^k₁^'' ⋅ x₂^k₁^'' ⋅ … ⋅ x_n^k_n^''
⋮
π_p = x₁^{k₁^p} ⋅ x₂^{k₁^p} ⋅ … ⋅ x_n^{k_n^p}
π = {π₁, π₂, …, π_p}

Recall that,

The (product) function y = ƒ (x₁, x₂, …, x_n)
= x₁^k₁ ⋅ x₂^k₁ ⋅ … ⋅ x_n^k_n is dimensionally homogeneous if, and only if, the exponents k₁, k₂, …, k_n are a solution of the linear equations a = a₁k₁ + a₂k₂ + … + a_nk_n
b = b₁k₁ + b₂k₂ + … + b_nk_n
⋮
g = g₁k₁ + g₂k₂ + … + g_nk_n
where a, b, …, g are the the dimensional exponents.

Therefore,

The product π_i = x₁^k₁ⁱ ⋅ x₂^k₁ⁱ ⋅ … ⋅ x_n^k_nⁱ is dimensionless if, and only if, the exponents k_jⁱ of the independent variables x_j are the solution of 0 = a₁k₁ⁱ + a₂k₂ⁱ + … + a_nk_nⁱ
0 = b₁k₁ⁱ + b₂k₂ⁱ + … + b_nk_nⁱ
⋮
0 = g₁k₁ⁱ + g₂k₂ⁱ + … + g_nk_nⁱ
where j = 1 to n the total number of independent variables, and the coefficients a_j, b_j, …, g_j are the dimensional exponents making up rows in the dimensional matrix of the x's.

The set of dimensionless products is said to be complete if

A set of dimensionless products of given variables is complete, if each product in the set is independent of the others, and evey other dimensionless product of the variables is a product of powers of dimensionless products in the set.

The exponents of x's can be displayed in a matrix

	x₁	x₂	…	x_n
π₁	k₁'	k₂'	…	k_n^'
π₂	k₁''	k₂''	…	k_n''
⋮	⋮	⋮	⋮	⋮
π_p	k₁^p	k₂^p	…	k_n^p

This is referred to as the dimensional matrix of exponents.

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Independent dimensionless products

The products π₁, π₂, …, π_p are said to be independent, if other than h₁ = h₂ = … = h_p = 0 there are no constants h₁, h₂, …, h_p such that π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p ≡ 1

This is a special case of the definition of linear dependence. This may be defined from the perspective of linear algebra. But before that we define the linear combination of rows in a matrix as

If there exist constants corresponding to several rows of a matrix such that the sum of the products of several rows with their respective constants is another row of the matrix, that row is said to be a linear combination of other rows.

Then,

The rows of a matrix are said to be linearly dependent if there exists at least one row that is a linear combination of other rows.

In the matrix of exponents, this means that there exists constants h₁, h₂, …, h_p where not all are zero such that, for a given column the sum of the products of a row element by its corresponding constant is zero, and this is so for all columns. Symbolically this means, h₁k_i' + h₂k_i'' + … + h_pk_i^p = 0, for i = 1, 2, …, n.

Notice that for a matrix of two rows linear dependency is equivalent to a proportion between two rows. Therefore,

Linear dependecy is a generalization of the concept of proportionality.

From the defintion of linear dependency it follows that

A necessary and sufficient condition that the products π₁, π₂, …, π_p be independent is that the rows in the matrix of exponents be linearly independent.

Hypothesis: Dimensionless products are independent. That is, π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p ≡ 1 iff, h_i = 0, for i = 1, 2, …, p. And, the rows of matrix of exponents are linearly dependent.

We know that the matrix of exponents is composed from

π₁ = x₁^k₁^' ⋅ x₂^k₁^' ⋅ … ⋅ x_n^k_n^'
π₂ = x₁^k₁^'' ⋅ x₂^k₁^'' ⋅ … ⋅ x_n^k_n^''
⋮
π_p = x₁^{k₁^p} ⋅ x₂^{k₁^p} ⋅ … ⋅ x_n^{k_n^p} . Multiplying them we get, π₁ ⋅ π₂ ⋅ … ⋅ π_p = x₁^(k₁^' + ^k₁^'' ^{+ … +} ^{k₁^p)} ⋅ x₂^(k₂^' + ^k₂^'' ^{+ … +} ^{k₂^p)} ⋅ … ⋅ x_n^(k_n^' + ^k_n^'' ^{+ … +} ^{k_n^p)}.

Definition of linear dependency tell us that there must exist constants h₁, h₂, …, h_p where not all are zero such that,

h₁k_i' + h₂k_i'' + … + h_pk_i^p = 0, for i = 1, 2, …, n. Taking the power of each dimensionless product π₁, π₂, …, π_p by its corresponding constant h₁, h₂, …, h_p the above equation becomes, π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p = x₁^(h₁k₁^' + ^h₁k₁^'' ^{+ … +} ^{h₁k₁^p)} ⋅ x₂^(h₂k₂^' + ^h₂k₂^'' ^{+ … +} ^{h₂k₂^p)} ⋅ … ⋅ x_n^(h_pk_n^' + ^h_pk_n^'' ^{+ … +} ^{h_pk_n^p)} which gets reduced to π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p = x₁⁰ ⋅ x₂⁰ ⋅ … ⋅ x_n⁰. Suggesting that π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p = 1. But, this is contrary to the hypothesis.

Therefore, the matrix of exponents must be linearly independent for the dimensionless products to be linearly independent.∎

Hypothesis: Rows in matrix of exponents are linearly independent. That is, h₁k_i' + h₂k_i'' + … + h_pk_i^p = 0 iff, h_i = 0, for i = 1, 2, …, n. And, the dimensionless products are linearly dependent. That is, there exist constants h₁, h₂, …, h_p where not all are zero such that, π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p ≡ 1.

Substituting the expressions for π's shown belown

π₁ = x₁^k₁^' ⋅ x₂^k₁^' ⋅ … ⋅ x_n^k_n^'
π₂ = x₁^k₁^'' ⋅ x₂^k₁^'' ⋅ … ⋅ x_n^k_n^''
⋮
π_p = x₁^{k₁^p} ⋅ x₂^{k₁^p} ⋅ … ⋅ x_n^{k_n^p} . we get, π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p = x₁^(h₁k₁^' + ^h₁k₁^'' ^{+ … +} ^{h₁k₁^p)} ⋅ x₂^(h₂k₂^' + ^h₂k₂^'' ^{+ … +} ^{h₂k₂^p)} ⋅ … ⋅ x_n^(h_pk_n^' + ^h_pk_n^'' ^{+ … +} ^{h_pk_n^p)}. But, from π₁^h₁ ⋅ π₂^h₂ ⋅ … ⋅ π_p^h_p ≡ 1 we have, x₁^(h₁k₁^' + ^h₁k₁^'' ^{+ … +} ^{h₁k₁^p)} ⋅ x₂^(h₂k₂^' + ^h₂k₂^'' ^{+ … +} ^{h₂k₂^p)} ⋅ … ⋅ x_n^(h_pk_n^' + ^h_pk_n^'' ^{+ … +} ^{h_pk_n^p)} = 1 Suggesting that ∑_{i = 1 to p, j = 1 to n} (h_ik_j' + h_ik_j'' + … + h_ik_i^p) = 0. and that, h₁k_i' + h₂k_i'' + … + h_pk_i^p = 0, ∀i. But, we assumed that not all constants h₁, h₂, …, h_p are zero. Therefore, the observed result is contrary to the hypothesis.

Therefore, the condition that the matrix of exponents must be linearly independent is sufficient for the dimensionless products to be linearly independent.∎

❷

Computation of dimensionless products

Any fundamental system of solutions of a₁k₁ + a₂k₂ + … + a_nk_n = 0
b₁k₁ + b₂k₂ + … + b_nk_n = 0
⋮
g₁k₁ + g₂k₂ + … + g_nk_n = 0
furnishes exponents k₁, k₂, …, k_n of a complete set of dimensionless products of the independent variables x₁, x₂, …, x_n.

Conversely, the 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
⋮
g₁k₁ + g₂k₂ + … + g_nk_n = 0

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.

We know from definition of a dimensionless product that the product has exponents k₁, k₂, …, k_n that are a solution of

a₁k₁ + a₂k₂ + … + a_nk_n = 0
b₁k₁ + b₂k₂ + … + b_nk_n = 0
⋮
g₁k₁ + g₂k₂ + … + g_nk_n = 0 Any such fundamental system of solutions will furnish linearly independent sets of exponents k₁^', k_i^'', …, k_i^{(n − r)}, for i = 1, 2, …, n where r is the rank of the dimensional matrix of the coefficients $dimensional matrix of coefficients$ . Theorem

A necessary and sufficient condition that the products π₁, π₂, …, π_p be independent it that the rows in the matrix of exponents be linearly independent.

tells us that because the exponents k₁^', k_i^'', …, k_i^{(n − r)} are linearly independent, they should yield independent dimensionless products.

Finally, because the above system of solutions possess no more than (n − r) linearly independent solutions, it is impossible to have more than (n − r) independent dimensionless products.

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Non-Dimensionless Products (p:7) ➽

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