8.2.1. Definitions and Measurement

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.

In this section, we discuss how to determine the pressure field in a fluid that is at rest. When a fluid is stationary, the shear stress, which is the force per unit area acting parallel to the surface, is zero. This means that only normal stress, which is equivalent to pressure, acts on the control surface. Hence, we can express the pressure as a function of the coordinates x, y, and z.

Imagine a glass of water that has been undisturbed for some time. The water has no movement, and all the forces acting on the water molecules are balanced. This situation is similar to having no shear stress acting on the fluid's surface, where the only force felt is the weight of the water acting down due to gravity, which creates pressure at various depths.

There are 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, which is very simple things.

When analyzing the forces acting on a control volume of fluid, we identify two main types of forces: surface forces and body forces. Surface forces are due to pressure acting on the boundaries of the control volume, while body forces arise from the fluid's weight acting throughout the volume. The body force is calculated by multiplying the unit weight of the fluid by the volume of the control volume.

Think of a swimming pool. The pressure at the bottom of the pool is a surface force caused by the weight of the water above it. Similarly, the weight of the entire pool of water acts downwards, which represents the body force. Understanding these forces helps us predict how fluid behaves in different scenarios.

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, pressure applied to a confined fluid is transmitted undiminished throughout the fluid. In our control volume, this means that the pressure at any point within the fluid acts perpendicular to the surface it encounters. As this pressure distribution is established, we need to describe it in terms of spatial coordinates.

Consider a balloon filled with water. When you squeeze one part of the balloon, the pressure increases uniformly throughout the water, causing the balloon to change shape. This illustrates how pressure in a fluid transmits uniformly, obeying Pascal's law.

Pressure will vary along the z direction and as we go deeper into the fluid, the pressure increases linearly with depth. This is shown in the equation P = ρgz.

In a static fluid, the pressure increases linearly with depth due to the weight of the fluid above it exerting a force. The relationship is defined by the equation P = ρgz, where P is the pressure, ρ is the fluid density, g is the acceleration due to gravity, and z is the depth below the fluid's surface. This concept forms the basis for understanding pressure distribution in fluids.

Picture diving into a pool. As you swim deeper, you feel an increasing pressure on your ears. This increase in pressure corresponds with the depth of water above you, illustrating how pressure changes with depth using the equation for hydrostatic pressure.

The point is what we are going to discuss is that gauge pressure and vacuum pressure. Components now is coming it what is your datum to measure the pressure.

Understanding the difference between gauge pressure and absolute pressure is crucial. Gauge pressure is the pressure relative to the surrounding atmospheric pressure, whereas absolute pressure measures how much pressure is above a perfect vacuum. This distinction enables us to determine how systems respond depending on the reference point taken when measuring pressure.

Think of a car tire. When you check the tire pressure with a gauge that shows only the difference from the atmospheric pressure, that reading is called gauge pressure. If you were to add atmospheric pressure to that reading, you would get the absolute pressure inside the tire, which is important for understanding tire performance.

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