5.1. Categories of Variables

So, the 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.

The first step in dimensional analysis is to identify and list all relevant variables in the problem you are studying. In this case, the important variables include pressure per unit length (which we aim to calculate), diameter (D), density (ρ), viscosity (µ), and velocity (V). Each of these variables plays a crucial role in understanding pipe flow.

Think of it like preparing a recipe. Before you start cooking, you first gather all the ingredients you will need. Similarly, in dimensional analysis, gathering all variables is the first step towards solving the problem.

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. So, this is step 2.

In the second step, we take each variable from our list and express it using fundamental dimensions: Length (L), Time (T), and Force (F). For instance, velocity (V) is expressed as L/T (length per time), viscosity (µ) is expressed as F/(L²T), pressure drop per unit length (Δp/l) is expressed as F/L³, and density (ρ) is F/(L⁴T²). This step allows us to understand how each variable behaves in a dimensional sense.

Imagine you're trying to describe a car's performance. You might say it moves at a certain speed (velocity), has a certain weight (mass, which relates to force), and uses a specific type of fuel (which influences efficiency). By breaking down the car's performance into these basic dimensions, you can better analyze and compare its performance.

After that, we have to determine the unique number of pi term. So, what we have seen? The unique number of pi terms. How? So, going to tell you now, first I will tell you and then do all the. So, first we have to know what is k, k is total number of variables. So, in our case, 1, 2, 3, 4, 5, or we can count here, as well, 1, 2, 3, 4, 5. So, k was 5. And what our minimum numbers of reference dimensions that are there? So that you can see by looking at the dimensions of this variable. So, we have length L also is there, T is also there, F is also there. Is there any other term? All this 5 of these terms has basic dimensions length T and F, F, L, T. So, that means, suppose if there was no F in any of the 5 variables, then reference dimensions could have been 2. Here, L is also there, T is also there, F is also there, so, r will be 3. So, the number of Pi terms is going to be k – r, according to the Buckingham Pi theorem.

In this step, we determine the number of Pi terms by first identifying the total number of variables involved (k) and the minimum reference dimensions (r). In our example, we have 5 variables (k = 5) and 3 reference dimensions (length, time, and force), giving us a total of 2 Pi terms (Pi = k - r = 5 - 3). This calculation is essential to proceed with dimensional analysis using Buckingham's Pi theorem.

Think of this like figuring out how many teams you can form from a group of friends. If you have 10 friends but only want to form teams based on their interests (like sports, music, and science), you will be limited to the number of interests you choose. Here, the interests represent the reference dimensions.

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. So, number of repeating variables that will be there in all the dimensionless Pi terms will be 3.

The purpose of selecting repeating variables is that they must correlate with the number of reference dimensions. In our analysis, since we identified 3 reference dimensions, it follows that we must select 3 repeating variables to form our dimensionless Pi terms. These repeating variables must be independent of each other, meaning one cannot be derived from another to ensure diverse representation of effects in the analysis.

Imagine creating a survey to understand student preferences in school subjects. If you want to gauge interest based on different categories (like Math, Science, and Arts), you would need to ensure that each category is distinct to get a comprehensive understanding of student preferences.

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.

In this step, we form the Pi terms by taking one non-repeating variable and multiplying it by the product of the repeating variables raised to unknown powers. This process aims to ensure that the resulting term is dimensionless, meaning it has no units associated. For our case, this would involve defining a relationship that encapsulates the dynamic interactions among the selected variables and leads to a dimensionless expression.

Think of this as creating a recipe where you mix specific quantities of different ingredients to achieve the perfect taste without letting one flavor dominate. Each ingredient’s contribution needs to be appropriately measured to maintain balance, similar to how we balance the variables in a Pi term.

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