8.1.6. Hydrostatic Equations

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).

This chunk introduces the concept of pressure in a fluid that is at rest, indicating that pressure can be defined as a function of its spatial coordinates (x, y, z). Essentially, it sets the foundation for understanding how pressure behaves in a static fluid scenario, where no motion is occurring, and hence shear stress is non-existent.

Imagine a still pond. The pressure at any point below the surface can be thought of as depending on the distance from the surface. It varies with depth, but at any given depth, it's consistent across the pond, just like the pressure at point P is consistent in the fluid's control volume.

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.

This section explains that when dealing with a fluid at rest, one can assume a control volume for analysis. Within this control volume, shear stress is absent, meaning that the forces acting on it are purely normal forces (due to pressure). This is crucial because it simplifies the analysis of fluid behavior.

Think of a glass of water sitting on a table. The water is calm; there are no waves or currents. The only forces acting on the bottom of the glass are the weight of the water above and the atmospheric pressure acting on the surface. This is similar to how pressure acts in the control volume.

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.

In a fluid at rest, gravity exerts a force on the fluid. This force is responsible for the pressure gradients observed within the fluid. The pressure at a certain point in the fluid (P(x, y, z)) reflects the influence of this gravitational force acting over the fluid's volume. The body force is often expressed as the product of fluid density and gravitational acceleration, indicating how pressure increases with depth due to the weight of the fluid above.

Consider diving into a swimming pool. The deeper you go, the more pressure you feel. This increased pressure is due to the weight of the water above you—the force of gravity acting through the water’s mass.

pressure acts normal to the surface. So that what we can consider pressures act there normal to the surface. Then we need to define now as a pressure as a x, y, z is a pressure field what we are looking it which is a function of x, y, z.

This part discusses how pressure acts specifically on the control surfaces of the fluid. The use of normal force implies that pressure exerts a force perpendicular to any surface in contact with the fluid. As we continue, we focus on how pressure can be assessed as a function of spatial coordinates, allowing us to model how pressure distributes across different points within the fluid system.

Think about holding a beach ball underwater; the water presses in from all sides. The pressure you're resisting from the water is acting perpendicular to your hand, demonstrating how pressure acts normally to surfaces.

I can use Taylor series to approximate distance of this surface which is a far away from. The pressure at point on the surface can be expressed as...

Here, the discussion shifts to using mathematical tools, such as Taylor series, to model how pressure changes concerning distance from a specific point (like the center of the control volume). This approximation helps us understand pressure variations across a surface in a more manageable form, which can be particularly useful for applications in engineering and physics.

Imagine trying to estimate the height of a tree from a distance. By knowing the height at a nearby point and using simple geometry (like a triangle), you can predict the height further away. Similarly, Taylor series helps estimate how pressure behaves at different points based on known values.

If this is the pressure is acting on the surface, similar way I can get it what will be pressure is acting. I know this pressure multiply the area over these simplified control volumes, I will get the force component. That what is acting along the y directions.

This section explains how to calculate the force exerted by pressure acting on a surface area. By multiplying the pressure by the area of the control surface, one can derive the resultant force component. This principle is essential in hydrostatics and aids in understanding how forces interact within a fluid at rest.

If you've ever pressed down on the bottom of a children’s swimming pool with your hand, you've felt the force of the water pressing back against your palm. The pressure of the water multiplied by the area of your hand gives you the total force you're resisting.

As the fluid is rest, the sum of the force should be equal to zero. That the vector is equal to zero and that field what we will get it this part and is finally the ∇...

In this part, it is emphasized that in a static fluid, the sum of all forces acting on a control volume must equal zero to maintain equilibrium. This leads to the derivation of hydrostatic equations, which describe how pressure varies with depth in a fluid. These equations simplify the analysis of forces in a hydrostatic environment and are foundational for fluid mechanics.

Think of a balloon filled with water. If it’s not moving, the forces (the water pressing against the inside of the balloon and the balloon’s elasticity pushing back) balance out perfectly. If they did not, you'd see changes—equal forces result in stability.

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.

This section introduces gauge pressure and vacuum pressure concepts. Gauge pressure is the pressure reading that is above atmospheric pressure, while vacuum pressure is below atmospheric pressure. It discusses the importance of establishing a reference point (datum) for pressure measurements, which can significantly influence interpretations of pressure in various scenarios.

When you inflate a tire, the gauge pressure tells you how much air is in the tire above the surrounding atmosphere. If you measure vacuum pressure in a sealed container, you're determining how much lower the pressure is than the atmospheric pressure outside.

If you need to compute it what will be the absolute pressure, then it is a very easy. You just use whether the gauge pressure or this the vacuum pressure that difference what we can get it that what you just add with atmospheric pressure to get the absolute pressure.

Here, the process for calculating absolute pressure is clearly laid out. To find absolute pressure, you can either add gauge pressure to atmospheric pressure when the gauge pressure is above atmospheric levels, or subtract the vacuum pressure from atmospheric pressure when it is below. This combination helps in understanding the total stress exerted on an object by the fluid.

Consider a soda can. When you open it, the pressure inside (initially higher than atmospheric because of carbonation) is what you feel when the gas rushes out. The absolute pressure would be the combination of how much that pressure exceeds the atmospheric pressure.

Now as we derive pressure distribution equations which in vector forms and let we simplify that equations...

This part concerns the process of deriving pressure distribution equations, which can be expressed in vector forms. When gravity is aligned with a certain coordinate system, it leads to the conclusion that pressure only varies along that particular axis (commonly the z-axis), reinforcing the idea that pressure remains constant across horizontal planes in a static fluid.

Think of a column of water. If you go from the top down, the pressure increases due to the weight of the water above. However, if you look at any horizontal layer of that water, the pressure remains the same regardless of where you take that measurement—much like a level surface of a bathtub full of water.

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