7.1.2. Concepts of Hydrostatic

Now let us come to the very basic concept what we are talking about the fluid at rest. So the basically we are talking about now, the fluid at rest, okay? If it is a fluid is at rest, it is a very simplified problem now. Like as I said it any fluid flow problems we look at either the pressure field, the velocity field, or the density and temperature field. When the fluid is rest now, very simply way the velocity vectors becomes zero and if I consider incomprehensible the density is a constant and if the temperature is not very much I need not need a thermodynamics first laws to define the problems. Then only the left is that the pressure field.

When we talk about fluid at rest, we refer to a state where the fluid does not exhibit any movement. In this scenario, the velocity of the fluid is zero. No flows or disturbances occur, meaning that there are no variations in velocity or shear stress acting on the fluid. Consequently, we only need to consider pressure variations within the fluid, because in the absence of fluid motion, density remains constant and temperature effects are negligible, simplifying our analysis to just the pressure field.

Imagine a glass of water sitting on a table. The water is perfectly still, with no waves or movement. In this case, we only need to think about the pressure the water exerts against the bottom of the glass due to gravity, and we don't have to consider any other factors, like the velocity of the water, because there's none to measure.

Since there is no velocity, there is no velocity gradient, definitely as Newton's second law says that there is no velocity gradients that means no shear stress. So when a fluid is at rest, there is no shear stress acting on that. So you can take a surface or take a control volume. Over that control surface you can define it the shear stress components become zero. That is very simplified now that any control volume you consider it over that control volume surface as the fluid is at rest conditions, there is no shear stress acting on the control surface.

In a fluid at rest, shear stress is non-existent due to the lack of velocity gradients. Essentially, shear stress arises because of the movement of one layer of fluid over another. With no movement occurring, there are no gradients to create this force. This simplification means we can analyze forces acting on a given control volume without having to account for any shear stresses, making it easier to determine the overall behavior of the fluid's pressure forces.

Think of a thick layer of honey sitting still in a jar. If you were to look at two layers of honey, one on top of the other, they would not slide past each other because the honey is at rest. In this situation, the only force acting would be due to gravity, pushing down on each layer, without any sideways movement creating shear stress.

What the two forces we have? The gravity force and the force due to the pressure distribution. So whatever the pressure distribution forces that what is equate with the gravity force. Very simple things now, and since is a fluid is at the rest, so you can say that there is no mass flux is coming into the any control volumes or the momentum flux or no external work done it. So this is the what the simplified case.

In a resting fluid, the only forces acting are due to gravity and pressure distribution. When we analyze these forces, we find that the pressure force must balance with the weight of the fluid acting under gravity. Since the fluid is at rest, there are no changes in mass or momentum within the control volume, simplifying our calculations because we are only focusing on these two forces.

Imagine a swimming pool filled with water. The weight of the water exerts pressure against the bottom of the pool due to gravity. If you dive into the pool and notice how you feel the pressure increase as you go deeper, this is exactly how pressure distribution balances with gravity in a body of still water. Each layer of water below you feels the weight of all the layers above it.

Like for example, if you take this dam, which is 100-meter-high and we have a reservoir, let you consider this is what 90 meter height from the bottom. That is what the water levels is 90 meter from the bottom. And you can understand it because of these in the reservoirs the fluid is at the rest conditions and that rest condition exerting the pressures on these surface.

In practical applications, such as with a dam or a tank, we need to understand how fluid pressure acts on surfaces due to hydrostatic conditions. Consider a dam that is 100 meters high with a water level of 90 meters. The pressure at different depths is not uniform; it increases with depth due to the weight of the water above, and this pressure must be considered when designing such structures to ensure they can withstand these forces.

A real-world analogy could be a large aquarium. As you fill it with water, you can feel the pressure against the sides of the glass increasing the deeper you go. Just like the dam, the bottom of the tank experiences greater pressure due to the layers of water above it pushing down. If the aquarium were not designed to handle this pressure, it could break, leading to a spill—just like how a dam could fail if it doesn't account for pressure distribution from the water it holds.

Second thing is that fluid act as a rigid body motions. Let us take an example here that I have a tank which is a half filled liquid is there. Let us consider it may have water and this tank is accelerated with a constant acceleration a. As you start accelerating this tank with a constant acceleration a, you can understand it, this free surface is going to change it.

When a fluid in a tank is accelerated, it behaves as if it were a rigid body in certain conditions. As the tank accelerates, the surface of the fluid adjusts to maintain equilibrium, forming a new surface angle depending on the acceleration. Even though the fluid is moving within the tank, there are no shear stresses acting within the fluid due to the uniform acceleration. Therefore, we can still apply the principles of hydrostatics to analyze the pressure distribution.

Think about a car filled with water in a cupholder. If you accelerate the car quickly, the water will slosh to the back of the cup, forming a new surface. Here, the water inside is moving but still behaves according to hydrostatic principles; we can calculate the angle of the water surface and the pressure at different points accordingly, treating it as if the fluid is in a static condition relative to the tank's motion.

Now whenever I am talking about that I am looking for a pressure field as the fluid is at rest. So basically I am looking at the pressure is a function of the positions and time is not there, as the fluid is at rest condition.

In hydrostatics, particularly when a fluid is at rest, we can express pressure as a function of position within the fluid. Since time is irrelevant in a static state, the pressure at a given point only depends on the vertical height of the fluid above that point. This principle leads to Pascal's Law, which states that pressure applied to any part of an incompressible fluid is transmitted undiminished throughout the fluid, meaning pressure is uniform in all directions.

A classic example of Pascal's Law is seen in hydraulic systems, like car brakes. When you press down on the brake pedal (applying pressure), this pressure is transmitted through the brake fluid to activate the brakes on the wheels, showing how a small force can be magnified to create larger forces in the braking system, making it an essential feature of modern automotive design.

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