8.2.2. Atmospheric Pressure as a Datum

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. So there is no shear stress component on this fluid plane. Only this normal stress which is as equivalent to the pressure will act over this control surface. Over the control surface only the pressure is going to act it because the shear stress is zero okay.

This chunk focuses on how pressure behaves in a fluid that is not moving (at rest). In such a fluid, shear stress— which is a force that causes layers of fluid to slide past one another—is absent. Instead, the only type of stress present is normal stress, which is related to pressure. When we analyze a specific volume of this fluid, known as a control volume, we can say that the pressure at any point within the fluid can be defined by the function P(x, y, z), where these variables represent spatial coordinates. Hence, the pressure acts perpendicularly on the surfaces of the control volume, illustrating that the fluid is acting under a state of hydrostatic equilibrium, where forces are balanced, and thus the fluid itself remains static.

Imagine a glass of water sitting perfectly still on a table. Inside this glass, every drop of water is pushing down due to its weight. The pressure at the bottom of the glass increases with the height of water above it because of this weight. However, the water is not moving, and there are no forces trying to make it slide or flow. Here, the weight of the water creates a pressure that is uniform at any horizontal level within the glass, illustrating how pressure in fluids at rest works.

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. The gravity force what will be there? It will be, ρgA volume control. 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 LxLyLz which is very simple things. The control volumes, the volume is this much and the unit is ρg and ρ stands for here the density and g stands for accelerations due to gravity.

In this part, we look into the derivation of pressure in a control volume further. Here, we designate the pressure at the center of our control volume as P(x, y, z). The influence of gravity is introduced, which acts as a body force. This body force generates pressure because gravity pulls on the mass of fluid. The force of gravity (body force) can be calculated as: Force = weight = density (ρ) × gravitational acceleration (g) × volume (which is LxLyLz). In summary, there are two types of forces acting: surface forces, caused by pressure, and body forces, caused by gravity. This reveals how both pressure and gravity interplay in determining the force acting on a body of fluid.

Imagine a tall water tower. The water at the bottom feels the weight of all the water above it. This weight creates pressure at the base. If you were to take a slice of this tower (your control volume), you'd be able to calculate how much force is pushing down due to gravity on that column of water, and how this relates to the pressure felt at the base of the tank.

Now 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. There is two conditions from where you have to measure the pressure. It could be absolute vacuum point where the pressure is equal to zero okay, theoretically it is a zero pressure. You have a vacuum where the zero pressure and you are measuring it okay.

This section introduces two essential concepts in pressure measurement: gauge pressure and vacuum pressure. Gauge pressure is the pressure relative to the atmospheric pressure. This means if you measure pressure in a tire with a gauge, you're noting how much pressure is in the tire above the surrounding atmospheric pressure. Vacuum pressure, on the other hand, measures pressure above a vacuum, meaning that gauge pressure can be thought of when measuring with atmospheric pressure as a reference point. Therefore, the choice of reference point—whether it be absolute zero pressure (vacuum) or atmospheric pressure—affects how we quantify pressure.

Think about measuring tire pressure using a gauge. If the gauge reads 30 psi, it tells us how much higher the pressure inside the tire is compared to the atmospheric pressure outside it. If you were at sea level, that's one point of reference. If you took the same tire to the top of a mountain, not only would the outdoor atmospheric pressure change but so might the reading of the gauge—illustrating how pressure measurements depend on the reference point chosen.

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, If you make it at the scalar level as you can just split that equations, you will have a three equations. One will be the z direction, another will be the y direction, another will be the x direction.

This section discusses how one can derive and simplify equations that describe pressure distribution in a fluid. When considering terms like gravity as a vector, we can derive equations for pressure variation in three dimensions (x, y, and z). However, when the fluid is at rest, these equations can simplify significantly, primarily affecting the variable in the vertical (z) direction as opposed to horizontal directions since pressure is constant on any horizontal surface when the fluid is at rest—thus, we can express changes in pressure simply with respect to z.

Think about swimming in a pool. If you are at the surface of the water, the pressure you feel is quite low. However, as you dive down to greater depths, you feel more and more pressure—not because of horizontal changes or movements, but strictly because of your depth—showing how pressure builds linearly in the vertical direction under the influence of gravity.

If you consider a horizontal plane, at that horizontal plane you will be pressure will be the constant. That what these equations indicate for us. Now if you pressure only varies along the z directions, that means if I take from this point and if I just integrate this because pressure become now only a function subject okay. Pressure is only functions of the z, only in the vertical directions.

This chunk reinforces the principle that, when considering a fluid at rest, pressure remains constant across horizontal planes. As you go deeper into the fluid, the pressure changes linearly with depth, as indicated by our earlier equations. This reliance on depth helps to inform us that changes in pressure can be modeled mathematically when we relate them explicitly to vertical position (z), thereby allowing for integration to solve for pressure at different depths with clear functional relationships.

Picture a water tank filled to various levels. If you measure the pressure at the bottom of the tank, it will be the same no matter if you measure it from one end to the other—say on opposite sides of the tank at the same depth! This consistency showcases how highlighted pressure changes only with depth (z) and remains constant on the same depth (horizontal plane).

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. We neglect that pressure. Then after that we draw the pressure distributions as it is a relative pressure difference is only a function of z of a linear function only. So 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.

In this section, we explore how pressure changes from the free surface of a fluid down to the bottom of a body of water, primarily focusing on how atmospheric pressure at the free surface may often be neglected—this happens because, compared to the pressure due to the weight of the liquid column, its effect is often minimal. This leads us to conclude that pressure can almost be treated as a linear function of depth (z). Therefore, in practice, the pressure at various depths can be predicted simply based on the height of the fluid column above that depth multiplied by its weight per unit volume.

Consider an aquarium filled with water. The pressure on a fish swimming near the water's surface is very slight because of the minimal weight above it. But as the fish swims deeper, it can feel a significant increase in pressure due to the weight of all the water columns above it, highlighting the increasing pressure with depth while often neglecting the tiny force from the weight of the atmosphere.

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