Remember how they used to deduct points in high school or university if you did not specify units in your final answer? The reason is that almost every formula or quantity has a unit of measurement—a dimension in physical terminology—and it indicates the type of information required. Yet it turns out that sometimes dimensionless variables actually describe the physical system under investigation better. So what are dimensionless variables, and how are they related to almost every experimental setup in engineering?

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Units of measurement are not particularly adored by science students (“Why? Why do we have to write units in the final answers?”). One possible reason is that they expose mistakes: if you tried to find the velocity of a ball or the time it takes for a chemical process to occur and the resulting units are not meters per second or a unit of time, respectively, you probably went wrong along the way or used the wrong formula. But if the final answer is given in percent (for example, the efficiency of a system), what are the units then? Percentages have no units, yet you still obtained a result that teaches you something about the system you studied.

In almost every physical equation most terms carry some unit, whether they are variables (such as position or velocity) or constants (such as the speed of light or the gravitational constant). Physical equations describe how different mechanisms affect the state of a system. For example, the flow equations [1] describe how a fluid responds to various forces, and the equation of motion describes the position of a body given its velocity and initial position. At first glance, therefore, it seems that all physical equations depend on a specific system of units (SI or CGS for the students among us [2,3])—meter, second, kilogram, and so on.

The dependence of an equation on units creates a fundamental problem in physics and engineering: suppose we are designing an airplane, and to test the design we built a scaled-down model and placed it in a wind tunnel at the same wind speed and temperature. The model performed as expected, but when we built the full-scale airplane it failed to take off. Why did this happen? When we moved from the model scale measured in centimeters to the real scale measured in tens of meters, only some of the dimensional quantities in our equation changed—the size and weight—while others did not: the air density and wind speed. Therefore, for the enlarged model we get different equations in which only some variables differ, and what worked for the small model may not work for the real airplane.

This raises the question: can we write equations that are independent of scale or of a particular unit system? More than a hundred years ago, Lord Buckingham proved that any physical system can be represented mathematically as some combination of dimensionless variables [4]. This highly important theory, now called the “Buckingham π theorem,” shows that physical systems are independent of the unit systems we choose. In addition, it helps us immensely in simplifying equations and eliminating dependence on external constants. The resulting equations are usually much “nicer” and provide a great deal of information with minimal effort. From this theory, families of dimensionless variables were developed that dramatically changed experimental physics and engineering. Each such dimensionless number expresses a ratio of several dimensional quantities arranged so that all the units cancel out.

Now we get to the real punch line! If we pick two different physical systems—for example, a scale model of an airplane versus the real airplane—that are described by the same equations, and we make sure to match the dimensionless numbers, we will find that the systems are physically identical. Thus, results obtained for the small model will also work in the real system. Thanks to dimensionless numbers, one can perform laboratory experiments that describe much larger systems, and vice versa. Following up on the previous example, we can test a tiny airplane model in a wind tunnel only centimeters to a few meters in size and still accurately describe a real airplane tens of meters long [5]—provided we adjust other variables such as velocity, temperature, etc., so that the key dimensionless numbers of the system remain unchanged. Alternatively, one can build a banana-sized robot that accurately captures the motion of microscopic bacteria [6].

How wild is that! (Applause, please.) In fact, by definition of the theory, different phenomena are linked only to the dimensionless quantities, not to the dimensional variables, so one can only imagine their importance to science. In practice, within physics alone there are hundreds of dimensionless variables and constants [7], many of them in the fields of fluid flow and heat transfer.

If you are curious how this relates to you, you might be surprised to find that you probably know a few of them:

• Basic reproduction number, R (epidemiology)—the average number of people one carrier infects during the infectious period.

• Mach number—velocity relative to the speed of sound in the medium.

• Reynolds number (flow)—the ratio between inertial forces and viscous (friction) forces in the medium [8].

Now you know what dimensionless variables and constants are and what scientific significance is. Thanks to them, laboratory experiments can elegantly and accurately describe different systems. One can only wonder how scientists ever managed without them.

Hebrew editing: Shir Rosenblum-Man
English editing: Elee Shimshoni

References:

1. Navier–Stokes equations
2. SI unit system
3. CGS unit system
4. Buckingham π theorem
5. Wind-tunnel experiments
6. Paper by Prof. Yizhar Or (Technion) on modeling microscopic swimmers
7. List of dimensionless quantities
8. Post on boundary layers