 Consider a sphere in cross-flow where the drag force is related to this sphere's diameter as well as the fluid's velocity, density, and viscosity. I want to determine a dimensionless relationship governing this problem. To do that, I'm going to use my Buckingham Pi theorem, which is going to require that I first start by identifying all the physical variables involved. So I'm going to abbreviate my drag force as an F subscript D. And I'm saying that's a function of diameter, the sphere's diameter, and then the fluid's velocity, density, and viscosity. I'll abbreviate velocity with a V, density with a rho, and viscosity with a mu. Those five variables, one dependent for independent, are what I list for step one. Drag force, diameter, velocity, density, viscosity. For step two, I want to list the dimensions of each of those variables. Drag force is going to be the force dimension. Force would be mass times length over time squared. Diameter would be length. Velocity would be length per unit time. Density would be mass per length cubed. Viscosity would be mass per length times time. If you don't know these off the top of your head, you can always work through them by thinking through an example unit. For example, if we measure the force in Newtons, I recognize that a Newton is a kilogram meter per second squared. Kilogram is a mass length per time squared. Step three is to figure out a J value, the number of repeating parameters. My rule of thumb is to start with the number of dimensions that appear. If you can't make it work, reduce by one. I have mass length and time appearing, so I'm starting with three. That means I have three repeating variables, which means two non-repeating variables. If there's a pi group for each non-repeating variable, that means I will have two pi groups. For step four, I need to select the repeating parameters that do not form a pi group in and of themselves. This is a matter of trial and error. It's kind of an art more than a science and selecting the ones that you want the first time correctly. Generally speaking, I try to avoid selecting things that are dimensionless already or that vary from each other by just an exponent of their dimensions, and density and velocity are good things to choose as a general rule of thumb, especially over surface tension or viscosity. So over here, I'm going to semi-arbitrarily select diameter, velocity, and density. I'm leaving out my drag force because that's my dependent variable. I'm leaving out viscosity because I like to avoid viscosity when selecting repeating parameters. Then for step five, I go through my non-repeating parameters one at a time and build a pi group for each. So pi group one would use the drag force as the non-repeating parameter, and then diameter to the A, velocity to the B, density to the C. I have to figure out values of A, B, and C that leave pi group one as a unitless proportion. Generally speaking, I like to write that out in terms of dimensions and then solve it like a chemical reaction. You'll note that in this example, I'm writing them out in a numerator and a denominator. In the previous example, I had written everything in the numerator with things that would have been divided as negative terms. It doesn't really matter whichever way is easier for you to think about it. Then I set up an equation for each dimension, build an equation for that dimension, and then solve those equations for my number of variables. Since I have three repeating parameters, that means I will have three equations and three unknowns. That is, by the way, one of the reasons that starting with a J value of the number of dimensions that appear is useful. So zero is equal to one. I have mass appearing also in that last term, so one plus C. For length, I have zero is equal to one plus A plus B minus three times C. For time, I have zero on the left. I have negative two in the first term. I have negative B appearing in the third term and no other times. Three equations, three unknowns. I'll start with mass. C is equal to negative one. Then I can say B is equal to negative two from the time equation. And then plugging both of those into length, I can say zero is equal to one plus A plus negative two minus three times negative one. So I have one plus A minus two plus three. So one minus two plus three would be positive two. I'm like 60% sure. Therefore, A is equal to negative two. Therefore, pi group one can be written out as the drag force multiplied by diameter to the negative two times velocity to the negative two times density to the negative one. Or drag force times, excuse me, or drag force divided by density times velocity squared times diameter squared. So one of the dimensionless forms is the drag force divided by density times velocity squared times diameter squared. I can repeat the process for the second pi group using the other non-repeating parameter. Pi group two, I have viscosity this time. Viscosity multiplied by diameter to the E times velocity to the F times density to the G. And then I want mass to the zero, length to the zero, time to the zero is equal to viscosity is mass per length times time. Then length to the E, then length to the F per time to the F, and then mass to the G per length to the three times G. Then I build an equation for each dimension. For mass terms, I have zero is equal to one plus G. For length, I have zero is equal to negative one plus E plus F minus three times G. For time, I have zero is equal to negative one minus F. Solving those three equations for three unknowns in my head, what could go wrong? I'll start with the mass equation G is equal to negative one and then I move on to F, excuse me, then I move on to the time equation where I see that F is equal to negative one as well. And then in the length equation I'm writing zero is equal to negative one plus E plus negative one minus three times negative one. So that would be positive three minus one, just positive two minus one, just positive one. Therefore E is equal to negative one. So I can write up pi group two as being viscosity divided by diameter times velocity times density. As a fun fact while we're here, this dimensionless proportion is a proportion that we run into a lot. So frequently in fact that we have a special name for it. And by convention, we typically write it the other way around. Diameter times velocity times density over viscosity. The reason we do that, because by the way, writing the numerator over the denominator or the denominator over the numerator will both yield a unitless proportion. The only advantage for one or the other is that you would have a direct correlation instead of an indirect correlation or vice versa. Here, writing it as diameter times velocity times density over viscosity is more commonly a direct correlation and therefore we have developed a common rule of writing it out that way for consistency. It is the established custom. And the name we give this is the Reynolds number. Specifically this is the Reynolds number with respect to diameter. But more importantly, that's our second pi group. So I could write my relationship between the two as pi group one is equal to a function of pi group two. Therefore drag force over density times velocity squared times diameter squared is a function of and then I could write either viscosity over diameter times velocity times density or diameter times velocity times density over viscosity. So note as a result of simplifying this relationship from four independent variables to one independent variable. We can develop a correlation between the two with four fewer orders of magnitude. We could run say 10 experiments for each of the independent variables up here to try to get a sense of how each of them affects the drag force. But that means we have to have 10 experiments where we vary diameter, 10 experiments where we vary velocity, 10 experiments where we vary density and 10 experiments where we vary viscosity. Those 10 variations are going to occur for each of the other variations, which means we have 10 to the fourth power experiments to run in order to see how each of them affects the drag force. And that is a lot of experiments and 10 is not that many data points from which to draw a strong correlation. Compare that to over here we could reach the same amount of confidence by running just 10 experiments where we vary diameter and velocity and density and viscosity. And as a result we see how that total affects the drag force. And as a result we can see how those things that aren't really as independent as they seemed affect the drag force.