Dimensional Analysis - Extensions - Huntley's Extension: Directed Dimensions

Huntley's Extension: Directed Dimensions

Huntley (Huntley 1967) has pointed out that it is sometimes productive to refine our concept of dimension. Two possible refinements are:

• The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, we may have represent length 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 quantity is to be considered dimensionally distinct from mass as a measure of inertia.

As an example of the usefulness of the first refinement, suppose we wish to calculate the distance a cannon ball travels when fired with a vertical velocity component and a horizontal velocity component, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then, both dimensioned as, R, the distance travelled, having dimension L, and g the downward acceleration of gravity, with dimension

With these four quantities, we may conclude that the equation for the range R may be written:

Or dimensionally

from which we may deduce that and, which leaves one exponent undetermined. This is to be expected since we have two fundamental quantities L and T and four parameters, with one equation.

If, however, we use directed length dimensions, then will be dimensioned as, as, R as and g as . The dimensional equation becomes:

and we may solve completely as, and . The increase in deductive power gained by the use of directed length dimensions is apparent.

In a similar manner, it is sometimes found 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 quantity (substantial mass). 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

• the mass flow rate with dimensions
• the pressure gradient along the pipe with dimensions
• the density with dimensions
• the dynamic fluid viscosity with dimensions
• the radius of the pipe with dimensions

There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be and and we may express the dimensional equation as

where C and a are undetermined constants. If we draw a distinction between inertial mass with dimensions and substantial mass with dimensions, then mass flow rate and density will use substantial mass 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:

where now only C is an undetermined constant (found to be equal to by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law.