Dimensional analysis

Dimensional analysis is a mathematical tool often applied by engineering sciences to simplify 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/time². 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 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 and shape of the body, and the force. Using the algorithm of the pi theorem, one can defined two dimensionless parameters from these variables: 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, one may just as well experiment on small models 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
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• Porter, "The Method of Dimensions", Methuen, 1933
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