Juan G. Santiago - A First Course in Dimensional Analysis: Simplifying Complex Phenomena Using Physical Insight (The MIT Press)

This book is an introduction to a powerful tool known as dimensional analysis. The method is also termed similarity analysis, and the concept referred to as similitude. Dimensional analysis can be used to find patterns within and succinct explanations for physical phenomena. It is used to find functional relations that are independent of whether we have a large amount of each quantity or a small amount. It is a search for laws that are independent of measurement systems.

The history of dimensional analysis is linked with the history of science starting as early as the great English mathematician and physicist Sir Isaac Newton. Histories and developments of the subject are provided by Macagno (1971), Rott (1992), White (2016), Taylor et al. (2007), and Gibbings (2011), and only some material is summarized here.

In 1687, Isaac Newton described some rules of dimensional relations and analyses in the Principia. This included similarity in the trajectory shapes of bodies in motion, and indeed his theories of motion present a precise set of mathematical relations between the quantities of position, velocity, acceleration, momentum, and force. In around 1736, Swiss mathematician and physicist Leonhard Euler wrote extensive descriptions of units and dimensional reasoning, including the dimensions associated with Newtons second law, and so-called unit conversion factorsconcepts essential to consistent descriptions of the physical world.

In 1822, Jean-Baptiste Joseph Fourier clarified greatly the concept of physical dimensions, introduced the concept of dimensional homogeneity in mathematical relations, pointed out that equations describing physical phenomena are independent of the choice of system of units, and applied similarity to describe and unify heat flow. Lord Rayleigh (b. 1842) provided perhaps the most influential breakthroughs in dimensional analysis in studies of hydrodynamic drag, flow stability, sound waves, and light scatter from particles. He also strongly advocated dimensional analysis. He demonstrated that the principle of similitude yields elegant scaling laws and demonstrated the method using examples from Newtons law of gravitation, drag on spheres, heat flow in thin fins, vibrating drops, and the vibration of Aeolian harp strings. Rayleigh argued in favor of the importance and appropriate use of the method (e.g., as an initial analysis of a problem) (Rayleigh, 1915):

I have often been impressed by the scanty attention paid even by original workers in physics to the great principle of similitude. It happens not infrequently that results in the form of laws are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes consideration.

Rayleigh stopped short of teaching a formalized approach to dimensional analysis, but the French mathematician Aim Vaschy in 1892, and independently the Russian mathematician Dimitri Riabouchinsky in 1911, each formulated and then reported methods that we now call the Pi theorem (named after and introduced to most of the community by Edgar Buckingham, 1914). Since this time, dozens of texts and papers on similarity and similitude have been written. Modern similitude has been strongly leveraged by researchers in and for the field of fluid mechanics, but it has been applied successfully in many other fields. White (2016) provides references to examples in biomechanics, astrophysics, economics, chemistry, clinical medicine, social sciences, and plant genetics, among others.

We shall not attempt to review here all of the history of similarity methods or applications, but instead offer a few select milestone contributions to dimensional analysis as summarized in . Note the years listed in the table are approximate dates of contribution as suggested by Rott (1992), Gibbings (2011), and White (2016).

The last entry of is a method for performing dimensional analysis due to David Carl Ipsen (1960), a professor at the University of California at Berkeley. Ipsens method is a formal, step-by-step approach that we will primarily adopt in this book for the process of dimensional analysis.

In addition to Ipsens method, we will discuss a set of rules of thumb that in this text I propose as a guide to the integration of physical intuition and experimental observations with dimensional analysis. Enabling the combination of dimensional analysis and physical intuition is a major goal of this book.