8.1.2. Forces Acting on the Control Volume

Now if I go for the next ones that how to get the pressure field when fluid is at the rest. That means I am just looking the what could be the functions of the P, P = P (x, y, z). If it that it is now again I am considering a very simple case, let us consider a control volume like this okay. This is what my control volume. As I said it when the fluid is at rest the shear stress become zero.

This chunk introduces the concept of analyzing a control volume in fluid mechanics. It begins by clarifying what needs to be calculated: the pressure field when the fluid is at rest. When the fluid is stationary, it is emphasized that shear stresses are non-existent, which means that the only forces acting on the control surfaces of this volume are normal stresses, or pressures. This situation leads to the simplification of analyzing how pressure behaves across the control volume.

Imagine a balloon filled with water that is not being squeezed. The water inside does not move, similar to a fluid at rest. The pressure inside the balloon thus acts equally in all directions, like how the normal stress acts on the control surfaces of the fluid volume.

At that point, the gravity force is acting it which is the body force component. The gravity force what will be there? It will be, unit weight multiply the volume of the control volume.

In this chunk, the analysis includes gravitational forces acting on the control volume. The text states that the gravity acts as a body force, meaning it influences the entire volume of fluid rather than just specific points. Body force is calculated as the unit weight (density × gravity) multiplied by the volume of the control volume, highlighting how gravity affects the pressure distribution within the fluid.

Consider a diving board. When a person stands on the board, their weight exerts a downward force throughout the entire board, similar to how gravity exerts a force throughout the entire control volume of fluid.

Now if you look it over this control volume, all these pressure is going to act it as the Pascal law says that pressure acts normal to the surface. So that what we can consider pressures act there normal to the surface.

This passage discusses Pascal's law, which states that pressure applied to a confined fluid is transmitted undiminished in all directions throughout the fluid. In the context of the control volume, the pressure acting on each surface of the volume is normal to that surface, meaning it is perpendicular and does not generate shear forces. This understanding is crucial for analyzing how fluid pressure balances the forces on the control surfaces.

Think of a sealed plastic bag filled with water. If you squeeze one side of the bag, the pressure increases at that point and pushes outward equally in all directions, just as Pascal's law describes—it conveys how pressure acts on surfaces of the bag.

Then we need to define now as a pressure as a x, y, z is a pressure field what we are looking it which is a function of x, y, z I can use Taylor series to approximate distance of this surface which is a far away from.

This chunk introduces the concept of using the Taylor series expansion to approximate pressure values at various points in the control volume based on its coordinates (x, y, z). The Taylor series is a mathematical tool that allows us to express a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This approach helps derive the pressure distribution around the centroid point within the volume.

Consider a gentle hill where the height of the hill can vary gradually. By measuring the height at a specific point (your location), and knowing how steep the hill is (derivatives), you could predict the heights of distant points on the hill using a simplified formula like the Taylor series.

If this is the pressure is acting on the surface, similar way I can get it what will be pressure is acting. I know this pressure multiply the area over these simplified control volumes, I will get the force component.

In this passage, the relationship between pressure and force component is reinforced. By multiplying the pressure acting on a control surface by the area of that surface, we calculate the net force acting due to pressure. The text emphasizes that this relationship establishes gradient relations and will allow for the analysis of the pressure distributions, enabling a final equation that balances these forces effectively across the control volume.

Imagine holding a balloon. The pressure of the air inside pushes on the walls of the balloon. The larger the balloon's surface area, the stronger the total force pushing out against your hand when you squeeze it, similar to how pressure works over the surface area of the control volume.

Now the point is what we are going to discuss is that gauge pressure and vacuum pressure. Components now is coming to measure the pressure.

In this part, the concepts of gauge pressure and vacuum pressure are explained. Gauge pressure refers to the pressure relative to atmospheric pressure, while vacuum pressure is the pressure below atmospheric levels. Understanding these terms is crucial for practical applications, as they define how we measure and interpret pressure in various scenarios. Gauge pressure is essentially the measurement of pressure above the surrounding atmosphere, which is typically the context in fluids applications.

Think of a car tire pressure gauge. When you check the air pressure in your tire, the gauge shows you how much pressure is in the tire compared to the atmospheric pressure outside. This is gauge pressure—indicating whether the tire is properly inflated above the ambient atmosphere.

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