8.1.5. Force Components due to Pressure

Now if I go for the next ones that how to get the pressure field when fluid is at rest. That means I am just looking the what could be the functions of the P, P = P (x, y, z).

In this part, we explore how to define the pressure field in a fluid that is at rest. The notation P = P(x, y, z) indicates that pressure can vary at different points in a three-dimensional space, depending on the coordinates x, y, and z. This means that pressure is not uniform; instead, it can change based on where you are looking within the fluid.

Imagine a calm lake. The pressure at the surface of the lake (where you would dip your finger) is different from the pressure felt deeper underwater. The deeper you go, the higher the pressure due to the weight of the water above you. This concept is similar to understanding how pressure changes in various locations within any fluid.

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 becomes zero.

A control volume is a specific region in space that we analyze for pressure effects. In a fluid at rest, shear stresses (which arise from parallel layers of fluid sliding past each other) do not exist, meaning only normal stresses (force per unit area acting perpendicular to the surface) are present. This simplifies our calculations since we only need to consider the force acting due to pressure at the control surface.

Think of sitting in a swimming pool. When you're still, the water exerts pressure evenly on you from all sides, but if you start moving your arms, the water exerts different forces on your arms versus on the rest of your body. At rest, the pressure is constant, making calculations straightforward.

So we can have two force components. One is the surface force component and other is body force. The body force components if you look it that it will be unit weight multiply the volume of the control volume.

In our control volume analysis, we focus on two types of forces: surface forces and body forces. The surface force acts due to pressure on the walls of the control volume, while body forces arise due to gravity acting on the fluid within the control volume. The body force can be calculated by multiplying the unit weight of the fluid by the volume of the control volume.

Picture a box submerged in water. The water exerts an upward force on the box (body force), while the walls of the box feel the pressure from the water against them (surface force). The total effect of these forces will determine how the box behaves in the water.

The pressure at the centre of the fluid control volume is assumed to be P (x, y, z). At that point, the gravity force is acting it which is the body force component.

We define pressure at a specific point in the control volume as P (x, y, z). This point is usually at the center of the volume. The pressure at this point contributes to the gravity force, which acts as a body force since it influences the fluid's behavior due to its weight. The gravity force and pressure together will determine how the fluid will flow or behave under various conditions.

Consider the weight of the ocean. The pressure at the deepest parts contributes to the overall force acting on water layers above, influencing how currents behave or the types of organisms that can live at certain depths.

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.

According to Pascal's Law, any change in pressure applied to a confined fluid is transmitted undiminished throughout the fluid. This means that the pressure in our control volume acts perpendicular to the surfaces of the volume. Importantly, this characteristic leads to pressure increasing with depth due to the weight of the fluid above, creating higher pressures at lower depths.

When you dive into the ocean, you can feel a pressure increase on your ears the deeper you go. This is because of the water's weight above you, illustrating how pressure increases with depth according to Pascal's Law.

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

In measuring pressure, we differentiate between gauge pressure and absolute pressure. Gauge pressure measures the pressure relative to atmospheric pressure, while absolute pressure is the total pressure measured from a perfect vacuum (where no particles exist). Understanding this distinction is crucial for accurately calculating and reporting pressure in different situations.

An example can be found in car tires. When you check your tire pressure, the gauge shows you gauge pressure, which indicates how much air pressure is higher than the outside atmospheric pressure. However, absolute pressure would consider the atmosphere as a baseline, showing a lower number for the same tire.

As you consider the gravity in one of the coordinate axes most times, we align the gravity with the z-direction.

Typically, when analyzing pressure gradients, we align gravity along the vertical z-axis. This simplification allows us to analyze the pressure changes vertically more straightforwardly. Under normal conditions, it reveals that pressures do not vary in horizontal directions (x and y) while they vary in the direction of gravity (z). This creates a pressure gradient that increases with depth.

If you think of a tall tower of water—like in a water tank—pressure at the bottom is significantly greater than at the top due to the weight of the water above, illustrating this concept of pressure gradient with depth.

So if you consider a horizontal plane, at that horizontal plane you will be pressure will be the constant.

When observing a horizontal plane within a fluid at rest, we find that pressure remains constant across that plane. This is because, at any given depth, the weight of the fluid above is balanced and does not change, reflecting the idea that pressure varies with depth, but remains the same horizontally at any given level.

Picture the surface of a swimming pool: if you were to check the pressure at multiple points just below the same depth, it would be equal everywhere, illustrating how pressure remains constant across a level surface.

As we have derived pressure distribution equations which in vector forms and let we simplify that equations.

In real-world applications, understanding the derivations of pressure distribution equations is essential. These equations, often represented in vector forms, help simplify analysis across different dimensions. Recognizing how to adapt these principles allows engineers and scientists to accurately model fluid behavior for various systems.

Consider designing a dam: engineers rely on pressure distribution equations to predict how water will exert force on different parts of the structure, ensuring safety and effectiveness in real-world applications where human lives may be at stake.

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