Dimensionless products and complete set of dimensionless products
Recall that in dimensional analysis the product(ibid. 3.) is given by the expression
x1k1 ⋅ x2k1
⋅ … ⋅ xnkn
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
To be more precise, π is used to symbolize the complete set of dimensionless products πi
of a function.
where j = 1 to n the total number of independent variables, and
the coefficients aj, bj, …, gj 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
This is referred to as the dimensional matrix of exponents.
Independent dimensionless products
The products π1, π2, …, πp are said to be
independent, if other than h1 = h2 = … = hp = 0
there are no constants h1, h2, …, hp such that
π2h2 ⋅ … ⋅
πphp ≡ 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.
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,
…, hp 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,
h2ki'' + … +
hpkip = 0, for i = 1, 2, …, n.
Notice that for a matrix of two rows linear dependency is equivalent to a proportion between two rows.
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, …, πp be independent is that the
rows in the matrix of exponents be linearly independent.
Hypothesis: Dimensionless products are independent. That is,
π2h2 ⋅ … ⋅
πphp ≡ 1 iff, hi = 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