7.2. Summary of Previous Lectures

Now let us recap it, as of now what we learnt it. We already know it we talk about a either a system approach or the control volume approach to solve the fluid mechanics problems. And whenever you solve the fluid mechanics problems as I said it in the last class, we generally look for three velocities three fields, velocity field, pressure field and the density field. But when fluid is incompressible, then we just look for the pressure field and the velocity field. So these two fields we can get it using this three different approaches as I discussed earlier. One is experimental method, conducting the experiment in a wind tunnel, computational approach, which is a computational fluid dynamic now extensively used for very complex fluid flow problems or this analytical approach with a very simplified problem we can solve analytically to get the gross approximations of this pressure field the and the velocity field.

In this chunk, the professor recaps what has been learned in previous lectures about fluid mechanics. It emphasizes the difference between the system approach and the control volume approach for solving fluid mechanics problems. There are three main fields to consider: velocity, pressure, and density. When dealing with incompressible fluids, only the pressure and velocity fields are considered. The professor then identifies three main methods for analyzing these fields: 1) Experimental methods, where measurements are taken directly from physical experiments (like wind tunnels), 2) Computational methods that simulate fluid dynamics through complex algorithms, and 3) Analytical methods that involve solving equations for simpler problems, allowing for approximations of the velocity and pressure fields.

Consider a car's design process. Engineers might conduct experiments in wind tunnels (experimental method) to see how air flows around the vehicle. They can also use computer simulations (computational method) to predict fluid behavior under different conditions, and finally use mathematical formulas (analytical method) to simplify the analysis of forces acting on the car's body.

Now let us come to the very basic concept what we are talking about the fluid at rest. So the basically we are talking about now, the fluid at rest, okay? If it is a fluid is at rest, it is a very simplified problem now. Like as I said it any fluid flow problems we look at either the pressure field, the velocity field, or the density and temperature field. When the fluid is rest now, very simply way the velocity vectors becomes zero and if I consider incomprehensible the density is a constant and if the temperature is not very much I need not need a thermodynamics first laws to define the problems. Then only the left is that the pressure field.

This section introduces the concept of fluid at rest, which is a fundamental idea in fluid mechanics. When a fluid is at rest, it means there is no movement, and this greatly simplifies the analysis. The velocity vectors of the fluid are zero, and for incompressible fluids, the density remains constant. This means that factors like temperature can often be neglected for simplification. Therefore, the focus is shifted only to the pressure field, which is crucial in understanding how pressures vary throughout the static fluid body.

Imagine a glass of water sitting on a table. Since the water is not moving, there are no currents or waves; the velocity is zero. In this state, the pressure at any point in the water depends solely on the weight of the water above that point and is equal across all directions, which makes it easier to compute how the water exerts force on the bottom of the glass.

Since there is no velocity, there is no velocity gradient, definitely as Newton's second law says that there is no velocity gradients that means no shear stress. So when a fluid is at rest, there is no shear stress acting on that. So you can take a surface or take a control volume. Over that control surface you can define it the shear stress components become zero. That is very simplified now that any control volume you consider it over that control volume surface as the fluid is at rest conditions, there is no shear stress acting on the control surface.

When analyzing a fluid at rest, another critical aspect is the absence of shear stress. Shear stress occurs when there are velocity gradients, but in stationary fluids, these gradients don't exist. Therefore, if we analyze any surface within the fluid, the shear stress acting on that surface is zero. The forces we then analyze can be simplified to only look at gravity and pressure forces acting on the fluid. This fundamental idea allows us to make easier calculations regarding the behavior of the fluid without worrying about additional complexities of shear.

Think of a swimming pool filled with water. When no one is swimming, the water is still, and thus, the pressure on the bottom of the pool results solely from the weight of the water above it (gravity). Because the water is not moving, you don’t have to worry about currents or waves, making it easier to calculate the pressure at any point.

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