3. Steps in Dimensional Analysis

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.

In the first step of dimensional analysis, it's crucial to identify and list all the variables relevant to the specific problem at hand. In this example, we're dealing with a problem relating to pipe flow, and the key variables include: pressure per unit length, diameter of the pipe (D), density of the fluid (ρ), viscosity of the fluid (µ), and the velocity of the fluid (V). This step lays the foundation for the analysis, as understanding what quantities influence the problem is necessary for the next steps.

Imagine you're trying to bake a cake. Before mixing ingredients, you need to list what you need: flour, sugar, eggs, baking powder, and milk. Similarly, in dimensional analysis, identifying what you need to analyze helps you understand the problem better.

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–4T2.

In this step, we convert each variable we listed in step 1 into its basic physical dimensions. This is essential for dimensional analysis because it allows us to identify the relationships among the variables based on their dimensions. For example, we express velocity (V) as length (L) per time (T), viscosity (µ) as force (F) over length squared and time, and density (ρ) as force per length cubed and time squared. These dimensional expressions will help in forming dimensionless groups later.

Think of basic dimensions like the fundamental ingredients in a recipe. Just as flour, sugar, and eggs are essential components of a cake, L, T, and F are the key elements of physical analysis. Converting each ingredient into its simplest form helps you understand how they work together in the final product.

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.

In this step, we calculate how many dimensionless groups, called Pi terms, can be formed from our variables. Using the Buckingham Pi theorem, the number of Pi terms (which represent independent groups of variables) can be found by subtracting the number of fundamental dimensions (length, time, force) from the total number of variables identified in step 1. In this case, we have 5 variables and 3 basic dimensions, resulting in 2 Pi terms.

Imagine trying to organize your closet. You have five types of clothes, but only three shelves to store them. The number of unique ways (or arrangements) you can organize them corresponds to the Pi terms in dimensional analysis. Just as you find efficient ways to maximize space, you find relationships in your variables.

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.

In this step, we identify which variables will serve as the repeating variables that will appear in all dimensionless Pi terms. The number of repeating variables corresponds to the number of basic dimensions. Since we identified three basic dimensions in step 3, we need three repeating variables. Importantly, these repeating variables must be independent of each other to ensure we do not create any redundant groups.

It's like creating a three-member committee to represent all aspects of a large team. Each committee member should bring unique perspectives to ensure comprehensive representation, similar to how repeating variables should be distinct to provide a full picture in dimensional analysis.

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 µltiplying 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 start creating the actual Pi terms by combining one of the non-repeating variables with the repeating variables. Each repeating variable is raised to a power that we will determine to ensure that the entire expression is dimensionless. This process may involve some trial and error as we adjust the powers of the repeating variables.

Think of it as mixing ingredients in the right proportions to create a balanced dish. You add one primary ingredient (the non-repeating variable) and sprinkle in the necessary spices (the repeating variables) to achieve a harmonious flavor (dimensionless term).

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. So, we have to repeat step 5 for each of the remaining repeating variables.

This step emphasizes that if there are still repeating variables not yet used, we follow the same process as in step 5 to create additional Pi terms. This may involve using different non-repeating variables with the chosen repeating ones to derive more dimensionless groups.

Consider creating multiple recipes using the same set of ingredients. Just like you can make variations of dishes by altering the main ingredient and spices, in dimensional analysis, you derive new relationships by using different combinations of repeating and non-repeating variables.

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

In this final step, it’s critical to verify that all the Pi terms constructed in the previous steps are indeed dimensionless. This ensures that the relationships derived from these expressions are valid and applicable to the problem being analyzed. If any terms are not dimensionless, adjustments need to be made.

It’s like completing a puzzle; you check to ensure every piece fits correctly without any gaps. If one piece doesn’t fit, the whole picture is off. Verifying dimensionlessness ensures that your mathematical relationships are cohesive and valid.

Now, in the end, express the final form as a relationship among Pi terms and think about what it means.

In the last step, we formulate a clear mathematical relationship between the Pi terms we have derived. This expression highlights how different physical phenomena are interconnected. Understanding this relationship is fundamental to applying the results of dimensional analysis in practical scenarios.

Imagine tying together all the strings of a musical instrument to create harmony; this final expression represents the harmony between different physical variables in your problem, showing how they interact to influence outcomes.

--