Editing Dimensional analysis (section)

=== Huntley's extensions ===
Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank <math>m</math> of the dimensional matrix.<ref>{{harv|Huntley|1967}}</ref>

He introduced two approaches:
* The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have L<sub>x</sub> represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.
* Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.

==== Directed dimensions ====
As an example of the usefulness of the first approach, suppose we wish to calculate the [[trajectory#Range and height|distance a cannonball travels]] when fired with a vertical velocity component <math>v_\text{y}</math> and a horizontal velocity component {{tmath|v_\text{x} }}, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then {{mvar|R}}, the distance travelled, with dimension L, {{tmath|v_\text{x} }}, {{tmath|v_\text{y} }}, both dimensioned as T<sup>−1</sup>L, and {{mvar|g}} the downward acceleration of gravity, with dimension T<sup>−2</sup>L.

With these four quantities, we may conclude that the equation for the range {{mvar|R}} may be written:
: <math>R \propto v_\text{x}^a\,v_\text{y}^b\,g^c .</math>

Or dimensionally
: <math>\mathsf{L} = \left(\mathsf{T}^{-1}\mathsf{L}\right)^{a+b} \left(\mathsf{T}^{-2}\mathsf{L}\right)^c</math>
from which we may deduce that <math>a + b + c = 1</math> and {{tmath|1=a + b + 2c = 0}}, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.

However, if we use directed length dimensions, then <math>v_\mathrm{x}</math> will be dimensioned as T<sup>−1</sup>L<sub>{{math|x}}</sub>, <math>v_\mathrm{y}</math> as T<sup>−1</sup>L<sub>{{math|y}}</sub>, {{mvar|R}} as L<sub>{{math|x}}</sub> and {{mvar|g}} as T<sup>−2</sup>L<sub>{{math|y}}</sub>. The dimensional equation becomes:
: <math>
  \mathsf{L}_\mathrm{x} =
    \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{x}}\right)^a
    \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{y}}\right)^b 
    \left({\mathsf{T}^{-2}}{\mathsf{L}_\mathrm{y}}\right)^c
</math>
and we may solve completely as {{math|1=''a'' = 1}}, {{math|1=''b'' = 1}} and {{math|1=''c'' = −1}}. The increase in deductive power gained by the use of directed length dimensions is apparent.

Huntley's concept of directed length dimensions however has some serious limitations:
* It does not deal well with vector equations involving the ''[[cross product]]'',
* nor does it handle well the use of ''angles'' as physical variables.

It also is often quite difficult to assign the L, L<sub>{{math|x}}</sub>, L<sub>{{math|y}}</sub>, L<sub>{{math|z}}</sub>, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?

Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.

==== Quantity of matter ====
In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (''inertial mass''), and mass as a measure of the quantity of matter. '''Quantity of matter''' is defined by Huntley as a quantity only {{em|proportional}} to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition.

For example, consider the derivation of [[Poiseuille's Law]]. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables:
{| class="wikitable"
! Symbol !! Variable !! Dimension
|-
| <math>\dot{m}</math> || mass flow rate || T<sup>−1</sup>M
|-
| <math>p_\text{x}</math> || pressure gradient along the pipe || T<sup>−2</sup>L<sup>−2</sup>M
|-
| {{mvar|ρ}} || density || L<sup>−3</sup>M
|-
| {{mvar|η}} || dynamic fluid viscosity || T<sup>−1</sup>L<sup>−1</sup>M
|-
| {{mvar|r}} || radius of the pipe || L
|}

There are three fundamental variables, so the above five equations will yield two independent dimensionless variables:
: <math>\pi_1 = \frac{\dot{m}}{\eta r}</math>
: <math>\pi_2 = \frac{p_\mathrm{x}\rho r^5}{\dot{m}^2}</math>

If we distinguish between inertial mass with dimension <math>M_\text{i}</math> and quantity of matter with dimension <math>M_\text{m}</math>, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:
: <math>C = \frac{p_\mathrm{x}\rho r^4}{\eta \dot{m}}</math>

where now only {{mvar|C}} is an undetermined constant (found to be equal to <math>\pi/8</math> by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield [[Poiseuille's law]].

Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements he postulated for it. For a given substance, the SI dimension [[amount of substance]], with unit [[Mole (unit)|mole]], does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable.