Dimensional analysis

Dimensional analysis is a mathematical tool often applied in physics, chemistry, and engineering to simplify a problem by reducing the number of variables to the smallest number of "essential" parameters. Systems which share these parameters are called similar and do not have to be studied separately.

The dimension of a physical quantity is the type of units needed to express it. For instance, the dimension of a speed is length/time and the dimension of a force is mass×length/time². In mechanics, every dimension can be expressed in terms of length, time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to chose one or the other set of fundamental units. Every unit is a product of (possibly fractional) powers of the fundamental units, and the units form a group under multiplication.

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. The two sides of any equation must have the same dimensions. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is often achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.

The above mentioned reduction of variables uses the Buckingham π-theorem as its central tool. 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 units 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.

Two systems for which these parameters coincide are called similar; they are equivalent for the purposes of the equation, and the experimentalist who wants to determine the form of the equation can choose the most convenient one.

The π-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 isn't present). Multiplication of physical units is then represented by vector addition within this vector space. The algorithm of the π-theorem is essentially a Gauss-Jordan elimination carried out in this vector space.

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. The variables involved are: the speed, density and viscosity of the fluid, the size of the body, and the force. Using the algorithm of the π-theorem, one can reduce these five variables to two dimensionless parameters: the drag coefficient and the Reynolds number. The original law is then reduced to a law involving only these two numbers. To empirically determine this law, 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.

Worked example

Consider Einstein's well-known equation E = mc². As stated above, the two sides of any equation must have the same dimensions. We can check this as follows.

E is energy, which has units of mass × length² / time². (This is because energy = force × length, and force = mass × acceleration, and acceleration = length / time².)

m is mass, which is a unit on its own.

c is speed, which has units of length / time.

The left-hand side, E, therefore has units of mass × length² / time².

The right-hand side, mc², has units of mass × (length / time)² = mass × length² / time².

The two sides therefore have the same dimensions.

References

Barenblatt[?], G. I., "Scaling, Self-Similarity, and Intermediate Asymptotics", Cambridge University Press, 1996
Bridgman, P. W., "Dimensional Analysis", Yale University Press, 1937
Langhaar, H. L., "Dimensional Analysis and Theory of Models", Wiley, 1951
Murphy, N. F., Dimensional Analysis, Bull. V.P.I., 1949, 42(6)
Porter, "The Method of Dimensions", Methuen, 1933
Boucher and Alves, Dimensionless Numbers, Chem. Eng. Progress, 1960, 55, pp.55-64
Buckingham, E., On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis, Phys. Rev, 1914, 4, p.345
Klinkenberg A. Chem. Eng. Science, 1955, 4, pp. 130-140, 167-177
Rayleigh, Lord, The Principle of Similitude, Nature 1915, 95, pp. 66-68
Silberberg, I. H. and McKetta J. J., Jr., Learning How to Use Dimensional Analysis, Petrol. Refiner, 1953, 32(4), p179; (5), p.147; (6), p.101; (7), p. 129
Van Driest, E. R., On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems, J. App. Mech, 1946, 68, A-34, March
Perry, J. H. et al., "Standard System of Nomenclature for Chemical Engineering Unit Operations", Trans. Am. Inst. Chem. Engrs., 1944, 40, 251
Moody, L. F., "Friction Factors for Pipe Flow", Trans. Am. Soc. Mech. Engrs., 1944, 66, 671

External links:

http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf

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