Dimensional analysis

Dimensional analysis is a mathematical tool often applied by engineering sciences to reduce the complexity of a problem by reducing the number of variables to the smallest number of "essential" parameters.

The term "dimension" is here used in a different meaning from that in mathematics: 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?, the dimension of a force is Mass*Length/Time?². In mechanics, every dimension can be expressed in terms of Length, Time, and Mass.

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful equation, the dimensions of the two sides must be identical. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is typically achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.

The above mentioned reduction of variables uses the [pi theorem]? as its central tool. This theorem describes how an equation involving several variables can be equivalently rewritten as an equation of fewer dimensionless parameters, and 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 then 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.

A typical example of such dimensionless parameters is the drag coefficient and the [Reynolds number]? in fluid dynamics. If a moving fluid meets a body, it exerts a force. The resulting relationship can be expressed in terms of these two parameters alone; instead of experimenting on huge bodies with fast flowing fluids, one may just as well experiment on small bodies with slow flowing, more viscous fluids, because these two systems are described by the same set of dimensionless parameters.

References

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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

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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