6.2. Finding Pi Terms

So, step 1 is, you have to list all the variables that are involved in the problem. In our case, we know that listing the variables was, one is pressure per unit length, something that needs to be find out. Then there is a diameter D, there is the density ρ, then there is viscosity µ and the velocity V. So, first step we have done. We have listed all the variables that are involved in the problem.

In this step, we begin the process of dimensional analysis by identifying all relevant variables involved in the problem. For example, in the pipe flow scenario, we need to identify pressure per unit length, diameter, density, viscosity, and velocity. Listing these variables allows us to understand which factors are at play in the system we're analyzing.

Think of this step like preparing for a cooking recipe. Before you start cooking, you need to identify the ingredients you need. Similarly, here we identify the 'ingredients' of our problem which will help us understand the 'recipe' for solving it.

Second step is, we have to express each of these variables in terms of basic dimensions, very simple to do. So, we have to write down the dimensions of all these 5 variables. So, velocity is LT- 1 here, µ is FL - 2T, so, delta pl is FL – 3. These dimensions, if you recall, we had written some slides ago when we were trying to explain it through experimental procedure and the D is L. Whatever remaining it is ρ, it is FL – 4 T 2.

We proceed to specify each variable's dimensions based on fundamental physical quantities: mass (M), length (L), and time (T). For instance, velocity has the dimension of length per time, or LT^-1. Viscosity may be expressed as force per area times time (FL^-2T). This step is crucial as it provides us the foundation to create dimensionless numbers later in the analysis.

Imagine building a structure. Just as you need to measure lengths and widths in consistent units like meters or feet, here we convert our variables into a standard measure using basic dimensions. This ensures that all our 'building blocks' fit together correctly in our final analysis.

Now, step 3 says, determine the required number of pi terms. So, in our, in this particular case, the basic dimensions are F, L, T. So, that means basic dimensions are 3 total. Then the number of Pi terms are the number of variables, 5 minus the number of basic dimensions, 3. So, there should be 2 Pi terms for this case.

For this step, we use Buckingham's Pi theorem to determine how many dimensionless groups, or Pi terms, we can derive from our variables. In our case, we have 5 variables (pressure, diameter, density, viscosity, velocity) and 3 fundamental dimensions (F, L, T). According to the theorem, the number of Pi terms will be the number of variables minus the number of dimension types, resulting in 2 Pi terms.

Think about how a jigsaw puzzle works. Each piece represents a variable. Sometimes, pieces are too similar and don't fit together, just like dependent variables in our analysis. Determining how many distinct pieces (or Pi terms) we can create enhances our ability to solve the overall puzzle or problem at hand.

Now, step 4, we have to select a number of repeating variables, where the number required is equal to the number of reference dimension. So, repeating variables will be equal to the number of reference dimension. In this case, we have how many references dimension? 3.

We need to choose variables to serve as 'repeating variables'. The quantity of these must match the number of reference dimensions we derived earlier. For pipe flow, with 3 reference dimensions (F, L, T), we can select 3 repeating variables which should be dimensionally independent.

Choosing repeating variables is similar to selecting key ingredients for a cake. You need the right combination to ensure the cake rises correctly, just as we need the right combination of variables to analyze a physical scenario accurately.

Now, step 5 is, we have to form a Pi term. Now, how is that is formed? We have to form a Pi term by multiplying one of the non-repeating variables by the product of the repeating variables each raised to an exponent that will make the combination dimensionless.

This step involves creating the actual Pi terms from our selected variables. We take a non-repeating variable and multiply it by products of the repeating variables raised to powers that make the units cancel out, leading to a dimensionless outcome.

Imagine mixing colors to create a new shade. You take a base color (non-repeating variable) and mix it with others (repeating variables) in proportions (exponents) that yield a specific desired color (dimensionless term). This balancing act is similar to forming our Pi terms.

Now, the step 6 of a general procedure is, we have to repeat this step 5, if there are more number of repeating variables that are left.

If there are remaining non-repeating variables after forming the first Pi term, we need to repeat the process to form additional Pi terms. This ensures we account for all variables in our dimensional analysis.

Think about collecting all your candies for a batch of cookies. If you only used some types initially, you'd have to repeat the process to include all candy flavors for a truly diverse batch. In the same way, we ensure all non-repeating variable contributions are included.

Now, the step 7 is, you have to check all the resulting pi terms to make sure that they are dimensionless.

After forming each Pi term, it is essential to verify that they are dimensionless. This involves confirming that the compounded dimensions of each Pi term ultimately reduce to a neutral dimension, reinforcing our analysis is robust.

This is analogous to checking your bank account balance after every transaction. Just as you want to ensure every entry is correct and balances out, ensuring our Pi terms are dimensionless confirms the integrity of our dimensional analysis and the reliability of our findings.

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