Dimensional Analysis

Dimensional analysis is a mathematical tool often applied in physics, chemistry, and engineering to understand physical situations that are so complicated that it is difficult or impossible to derive the underlying differential equations. More often, it is used by undergraduates to check the correctess of derived equations. The dimensions of a physical variable is associated with symbols, such as M, L, T which represent mass, length and time, each raised to rational powers. For instance, the dimension of the physical variable, speed, is distance/time (L/T) and the dimension of a force is mass×distance/time² or ML/T². In mechanics, every dimension can be expressed in terms of distance (which physicists often call "length"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another other set of dimensions. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, Q, where Q represents quantity of electrical charge. The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or microns; but a length always has a dimension of L. Dimensional symbols, such as L, form a group: there is an identity, 1; there is an inverse to L, which is 1/L, and L raised to any rational power p is a member of the group, having an inverse of 1/L raised to the power p. There are conversion factors between units; for example one meter is equal to 39.37 inches, but a meter and and inch are both associated with the same symbol, L. In the most primitive form, dimensional analysis may be used to check the correctness of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat connot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is 0.477. The mass of a rat cannot be equal to the length of a flea. It should be noted that very different physical quantities may have the same dimensions: work and torque, for example, both have the same dimensions, M L2T-2,. An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct! The Buckingham p-theorem forms the basis of the central tool of dimensional analysis. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n-m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown. The p-theorem uses linear algebra: the space of all possible physical units can be seen as a vector space over the rational numbers if we represent a unit as the set of exponents needed for the fundamental units (with a power of zero if the particular fundamental unit is not present). Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of the p-theorem is essentially a Gauss-Jordan elimination carried out in this vector space.

Worked Example

A typical application of dimensional analysis occurs in fluid dynamics. If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and not completely understood) law. We might suppose that the variables involved under some conditions may be assumed to be the speed, density and viscosity of the fluid, the size of the body (expressed in terms of its frontal area A), and the drag force. Using the algorithm of the p-theorem, one can reduce these five variables to two dimensionless parameters: the drag coefficient and the Reynolds number. That this is so becomes obvious when the drag force F is expressed as part of a function of the other variables in the problem:

This rather odd form of expression is used because it does not assume a one-one relationship. Here, f is some function (as yet unknown) that takes five arguments. We note that the right hand side is zero in any system of units; so it should be possible to express the relationship described by f in terms of only dimensionless groups. There are many ways of combining the five arguments of f to form dimensionless groups, but the Buckingham Pi theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by:

and the drag coefficient, given by

Thus the original law involving a function of five variables may be replaced by one involving only two:

where f is some function of two arguments. The original law is then reduced to a law involving only these two numbers. Because the only unknown in the above equation is F, it is possible to express it as:

or
F = ?Au²f(Re).

Thus the force is simply ?Au2 times some (as yet unknown) function of the Reynolds number: a considerably simpler system than the original five-argument function given above. Dimensional analysis thus makes a very complex problem (trying to determine the behaviour of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number. The analysis also gives other information for free, so to speak. We know that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project. To empirically determine the Reynolds number dependence, instead of experimenting on huge bodies with fast flowing fluids (such as real-size airplanes in wind-tunnels), one may just as well experiment on small models with slow flowing, more viscous fluids, because these two systems are similar.