6. Question on Buckingham Pi Theorem

So, we will start with a question, another question. So, the time period T of the water surface waves is known to depend on the wave length lambda, depth of the flow D, density of the fluid ρ and acceleration due to gravity g and surface tension sigma.

In the context of fluid mechanics, the Buckingham Pi Theorem is a valuable tool when analyzing how one variable depends on others in a system. Here, we examine the time period (T) of water surface waves and identify that it depends on several factors: wavelength (λ), flow depth (D), fluid density (ρ), gravitational acceleration (g), and surface tension (σ). This sets the stage for applying the Buckingham Pi Theorem to see how these variables relate dimensionally.

Think of it like baking a cake: the time it takes to bake may depend on several factors, such as the oven temperature and the size of the cake pan. Just as you would list out all the ingredients and variables that influence baking time, here, we identify all variables influencing the time period of water waves.

Using the Buckingham Pi theorem, obtained the dimensionless form of the functional relationship. This is a classical problem of Buckingham Pi theorem, which we are going to solve in the class today. First, we write down all the variables which already have been given. It says, time period T is known to depend on wavelength lambda, depth of the flow D, density of the fluid ρ, acceleration due to gravity g and surface tension sigma.

Once we have the variables identified, the next step is to express each variable in dimensional form. For instance, T is measured in time, λ and D in length, ρ in mass per unit volume, g in acceleration units, and σ in force per unit length. This allows us to transition from the physical variables to their mathematical counterparts, which is key for dimensional analysis.

Imagine measuring ingredients in cooking. You need to know the quantity in cups (T), tablespoons (λ and D), ounces (ρ), or degrees Fahrenheit (g) to accurately prepare your dish. Similarly, by listing the dimensions of each variable in our theorem application, we prepare for accurate formulation and analysis.

So, total 6 variables are there. Second step is listing all the dimensions and that is going to be, so, T is time period is time period T. Lambda is, so, this is actually lambda is wavelength L and the depth of the flow is also L, ρ is the density of the fluid. So, it can be, what we have chosen, we have chosen the basic dimension as M, L and T.

Once we list all variables, we find that there are 6 total variables (k=6). The fundamental dimensions are mass (M), length (L), and time (T). According to the Buckingham Pi Theorem, the number of dimensionless Pi terms (r) is given by k - r. Since we have 3 fundamental dimensions here, we have pi terms equal to 3 (6 - 3 = 3). This calculation helps us understand how many independent dimensionless forms we can create to represent our physical system.

In simple terms, think about creating a recipe for a new dish. If you have 6 ingredients but realize only 3 key components are necessary for the dish, you can focus on those to create several variations of your dish. The Pi terms in fluid dynamics help us do just that by only focusing on essential variables.

Now, we have selected the repeating variables. So, we have selected lambda, g and ρ as repeating variable.

To form dimensionless Pi terms, we need to select repeating variables that will be part of every Pi term we create. The repeating variables must be dimensionally independent, meaning they cannot be derived from one another. In this case, λ (wavelength), g (gravity), and ρ (density) are chosen as repeating variables due to their fundamental roles in the dynamics of fluid flow. This strategic selection is crucial for forming well-defined dimensionless groups.

Consider a music band where you have several instruments. You can choose the guitar, drums, and keyboard as the main instruments to give a good rhythm to the song. Similarly, choosing λ, g, and ρ as repeating variables sets a strong foundation to analyze the fluid system effectively.

Pi 1 will be multiplied by say first non-repeating. So, T multiplied by lambda to the power a, g to the power b, ρ to the power c, so this should be dimensionless.

Next, we derive the dimensionless Pi terms by combining our non-repeating variables (in this case T) with the repeating variables raised to certain powers, e.g., T * λ^a * g^b * ρ^c. The goal is to adjust the exponent a, b, and c such that the resulting Pi term is dimensionless. We achieve this by equating dimensions to zero across M, L, and T, leading to a set of equations to solve for the exponents.

It's like adjusting the ingredients in a smoothie recipe to ensure it tastes just right. You mix different fruits and liquids, tweaking each to get the flavor perfect without it becoming too sweet or too tart. Similarly, we tweak the powers of λ, g, and ρ to ensure our Pi term is dimensionless and accurate.

The last step is expressing Pi 1 or any Pi's as a function of the other 2.

Finally, after deriving all necessary Pi terms, the last task is to express the relationship among them, indicating how one dimensionless group behaves as a function of others. For instance, we may conclude that Pi 1 is a function of both Pi 2 and Pi 3, showcasing their interdependencies systematically and mathematically.

Think about the connection between various sections in a story. How one chapter leads to the next, showing the evolution of characters and plot. Similarly, the relationships between our dimensionless groups illustrate how the variables influence each other in the context of wave dynamics.

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