8.3.2. Pressure Variation in a Horizontal Plane

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.

When discussing pressure in a fluid at rest, we start by defining the control volume where we analyze the fluid's behavior. In a control volume, the pressure, denoted as P, can be expressed as a function of the coordinates x, y, and z. It is important to note that, in this state, the fluid has no shear stress acting on it, meaning the only forces we consider are normal forces or pressures acting perpendicular to the surface of the control volume.

Think of a calm swimming pool. When no one is moving in the water, the pressure you feel at any point in the water (like at the bottom of the pool) is only due to the weight of the water above you, not due to any movement or waves, similar to how pressure acts in a fluid at rest.

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, ρgV. So we can have a 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 l_x.l_y.l_z.

In a control volume with fluid at rest, we can identify two types of forces acting on it: surface forces (due to pressure) and body forces (due to gravity). The body force is calculated by multiplying the unit weight of the fluid by the volume of the control volume. This highlights how gravity influences the fluid and consequently the pressure distribution within it. Further, the unit weight is represented as the product of the fluid's density (ρ) and the acceleration due to gravity (g).

Imagine a glass filled with water. The water exerts pressure on the bottom of the glass due to both the weight of the water above it and the atmospheric pressure exerted downwards. Similarly, in a tank of water, the deeper you go, the more weight of the water there is above you, leading to an increased pressure at lower depths.

So that we can have a linear variation of pressure from these free surface to this one and that what we get it here the linear variations of the pressure and you can look it that what is a unit weight into the z value. So these are very simple way. The pressure will vary, this is the free surface. At this point the pressure will be the atmospheric pressure, which most often we neglected that which is very less as compared to the liquid what we have generally we consider it.

When analyzing pressures in a fluid at rest, we discover that pressure remains constant along any horizontal plane. This invention arises from the fact that all horizontal points in a fluid are subject solely to the pressure from the fluid column above them. Consequently, as long as the fluid is at rest and not in motion, the pressure at any given horizontal level is uniform throughout. On the other hand, as we move vertically downward in the fluid, the pressure will increase linearly due to the weight of the fluid column above.

Consider the example of a water bottle. If you open the bottle, the pressure at any point on the surface of the water is the same, whether you are measuring at the top, middle, or bottom of the bottle. However, if you were to measure the pressure at various depths, you'd find that it increases deeper you go due to the greater weight of the water above.

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. Whether you have to make a absolute zero pressure, that means you have a vacuum. From there you are measuring the pressure, or you consider as local atmosphere, to measure the pressure.

In measuring pressure, we consider two types - gauge pressure and absolute pressure. Gauge pressure measures pressure relative to local atmospheric pressure, meaning it considers the surrounding atmospheric pressure as zero. On the other hand, if we measure from a vacuum reference point, that's the absolute pressure. When calculating pressures in practical scenarios, we often consider atmospheric pressure as a reference because most systems operate in atmospheric conditions.

Think of a bicycle tire gauge. When you check the pressure in your tire, it shows the pressure above atmospheric pressure—this is gauge pressure. If the gauge said your tire is at 30 psi, it means the tire pressure is 30 psi more than the atmospheric pressure surrounding it.

Now as we derive pressure distribution equations which in vector forms and let we simplify that equations which earlier we consider the acceleration due to gravity is a vector which will have, and then if you align the gravity in one of the coordinate axis most times that we do along the z directions.

When deriving pressure distribution equations, we often simplify the equations by aligning the gravitational force with one of the coordinate axes, typically the z-axis, which represents vertical depth. When gravity is aligned with the z-direction, we have a consistent framework to analyze how pressure varies with depth in the fluid. Thus, the gradients of pressure can be easily calculated, leading us to understand the behavior of pressure at various depths within the fluid.

Imagine a swimming pool where the depth of the water influences the pressure felt by someone diving into the pool. The deeper you go, the more pressure you experience, which is directly attributable to the weight of the water above you. This analogy reflects how pressure varies with depth in a static fluid.

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