13.. Fluid Mechanics

Welcome all of you to this lectures on dimensional analysis. This very interesting lectures what today I will cover it talking about how to do the experiments in fluid mechanics to characterize the probable. So very interesting topic and this topic I will cover it in 2 lectures.

This introduction sets the stage for the discussion on fluid mechanics and dimensional analysis. Dimensional analysis is a technique used to study how various physical quantities relate to one another. It involves examining the dimensions of these quantities in order to draw insights from experiments in fluid dynamics.

Imagine you're baking a cake. Understanding how much flour, sugar, and eggs you need (the quantities involved) and their relationships is similar to understanding how physical quantities interact in fluid mechanics.

So I will talk about Dimensionless groups, Dimensional Homogeneity principles, Buckingham’s pi theorem, Dimensionless groups what is used in fluid flow that is what I will introduce to you then we will discuss some example problems based on Buckingham’s theorem and we will have a summary.

Dimensional homogeneity indicates that the equations governing fluid mechanics should remain consistent regardless of the units used to express the physical quantities. This principle ensures that the left-hand side of an equation has the same dimensions as the right-hand side, promoting the universality of the equations derived from fluid studies.

Consider a recipe that works whether the measurements are in cups or liters. The underlying relationships in the recipe (like flour to water ratios) remain consistent, similar to how equations in fluid mechanics maintain dimensional homogeneity.

So basic dimensions what we have mass length and the time these are 3 basic dimensions mass, length and time. So what we have when we would do a theoretic analysis or the experiment analysis can be made universal okay that is what exactly highlighting here independent of its specific locations using a diverse group to express them.

The three basic dimensions in fluid mechanics are mass (M), length (L), and time (T). These dimensions serve as a foundation for expressing other physical quantities, such as velocity and acceleration. By using these dimensions, we can create dimensionless groups that help simplify complex fluid flow problems.

Think of basic dimensions as the primary colors of paint: red, blue, and yellow. Just as you can mix primary colors to create other colors, you can combine fundamental dimensions to derive new fluid properties.

Now if you look at the fluid properties what we have some of the fluid properties if you know it is related to the dimensions okay that is the length area and the volume. So it is just a dimension and geometric dimensions so it has only unit in terms of the length okay.

Fluid properties like area, volume, velocity, and acceleration are expressed in terms of the basic dimensions of mass, length, and time. For example, velocity is defined as the distance covered per unit of time, while area and volume relate directly to the dimensions of length. Understanding these relationships is crucial in fluid mechanics.

Imagine measuring a swimming pool. The length and width give you the area, while adding depth gives you volume. Just like in fluid mechanics, understanding how these measurements relate helps you grasp the overall properties of the pool.

So that means what we have to look at that for any physical political properties is and all so somewhere it follows this dimensional homogeneous concept. ... So the drag force is a function which we do not know it okay D is the functions of the diameters V velocity of the flow and the density and dynamic viscosity.

When designing experiments to measure fluid properties, it is essential to ensure the relationship between various physical parameters is clear and consistent. For example, drag force depends on cylinder diameter, flow velocity, and properties like density and viscosity. Dimensional analysis helps to relate these parameters without needing to conduct numerous experiments.

It's like planning a car test drive. You don't need to drive every car under every condition. Instead, you analyze what factors (like engine size or weight) will affect performance and only test the most relevant ones, just as dimensional analysis allows a streamlined approach to fluid experiments.

Mow let us commit how to do that is Buckingham’s theorem it is a very simple theorem concept is that so we will have a number of independent dimensionless groups ... Now I have to develop the dimensional groups of that...

Buckingham's Pi theorem offers a method for determining the number of independent dimensionless groups in a problem, which can greatly simplify fluid mechanics analysis. This theorem states that if you have 'n' variables and 'm' fundamental dimensions, the number of dimensionless groups is 'n - m'. This process reduces complex problems to more manageable analyses.

Think of organizing your work tasks by categorizing them. If you have 10 tasks (like variables) but only three categories (dimensions), you reduce them to manageable groups that can be addressed more efficiently, similar to how the theorem simplifies fluid problems.

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