11.3.2. Pressure Diagrams and Control Volumes

Let me take this example. So you have the free surface and you have a curved surface and you have the liquid which is having the density let be the rho. If it that conditions, what is the basic difference of this point is here that if you take any point as you know it the pressure is a scalar quantity and will have a the pressure component will be normal to the point where you were considering the surface. In that case, what it happens when you have a curved surface, the directions of the pressure that what changes from the point to point.

In this chunk, we explore the interaction of pressure with a curved surface in a fluid. Pressure is a measure of the force exerted by a fluid on a surface, and it acts perpendicular (normal) to that surface. However, on a curved surface, the pressure is not uniform; it varies depending on the curvature and the depth in the fluid. This situation necessitates breaking down the pressure force into components that can be analyzed through a Cartesian coordinate system (x, y, z). By resolving these components, we can perform calculations to find the total force acting on the surface.

Imagine pressing a balloon with your hand. If you press evenly across the surface, it feels simple; however, if you press a curved section with your fingers, each finger can apply a different amount of pressure depending on where and how you're pressing. Similarly, the pressure on a curved surface in a fluid changes from point to point, which requires us to analyze each area individually.

We can simplify that ones like we can resolve the curved surface into two-part horizontal and vertical components okay. Like here what the diagrams have shown it, it is a curved surfaces and that is a projected surface on the verticals and on that the horizontal force is acting it and verticals are in the downward directions.

To calculate the total pressure forces acting on a curved surface, we can break the curved surface into two components: horizontal and vertical. This decomposition allows for clearer analysis of forces because we can subsequently calculate the net vertical and horizontal forces separately. For example, the distribution of pressure along the vertical side of a curved shape increases with depth in the fluid, while the horizontal components depend on the shape's projection on a vertical plane. By identifying the areas related to these forces, we can calculate the resulting forces acting on the control volume.

Think of a water fountain. The water sprays upwards from a curved surface, and if we imagine slicing that fountain in halves, we could analyze the upward forces pushing water against gravity (vertical) and the sideways forces from the water that spreads out (horizontal). This helps us understand how much pressure each part of the fountain must withstand.

If I draw the free body diagram, there will be the free surface and the top of the free surface there will be weight due to the air. Then there will be weight above this curved surface W1 and W2. That is a weight of the liquid what we have. Now if I draw this is my control volume and the pressure diagrams okay.

In engineering and physics, free body diagrams are crucial for visualizing the forces acting on an object. In this case, we first consider the weight from the air acting on the free surface of the liquid, in addition to the weights of the liquid above the curved surface. When we combine these forces with pressure diagrams, we can derive the total forces acting on the control volume. This process illustrates how external weights and internal pressures influence the behavior of the liquid.

Picture a submarine submerged in the ocean. The submarine has to balance the weight of the water pressing down on it (like W1 and W2 in our example) while also managing the air pressure above. By comparing these forces visually on a diagram, we can see the balance needed for the submarine to maintain its position underwater.

This is what my control volume and if I draw the pressure diagrams, I can draw the pressure diagrams, which will be the triangular shape. The linearly will increase this at z we go further down and down.

When we examine the pressure within a control volume, we often find that it increases linearly with depth in the liquid, forming triangular pressure diagrams. These diagrams help us understand that as we delve deeper into the fluid, the pressure grows stronger due to the weight of the fluid above. The area represented by these diagrams can be used to compute the total force acting on a surface, enabling better predictions about how the surface will behave under varying conditions.

Envision a diver going deeper underwater. As they descend, they notice that the water pressure against their body feels greater. If we track this pressure increase with depth using a triangular diagram, we can predict the strength of the pressure the diver feels and the forces acting on their chest over time.

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