7.2.2. Fluid Statics Concepts

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. That means when fluid is at rest conditions only we need to know it how the pressure variations is going over these fluid domains. That is what the simplified problems.

When we discuss fluids, especially for fluid mechanics, it’s important to understand what happens when fluids are not in motion, which is the case here. In a fluid at rest, the velocity of the fluid is zero; thus, there are no changes or gradients in velocity. This allows us to simplify our analysis significantly, focusing solely on how pressure within the fluid varies. Since there's no flow, we can assume that the density of the fluid remains constant (for incompressible fluids) and we don't need to account for any changes resulting from temperature variations. The only thing we need to be concerned about is how pressure changes across the different points in the fluid.

Consider a swimming pool during a calm day. The water is still, and if you dip a stick into the pool, it won't cause any ripples to form. This is similar to our idea of fluid at rest—where only the pressure in different parts of the pool matters, not how the water is moving.

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.

In fluid mechanics, shear stress is the force per unit area that causes one layer of fluid to slide past another. However, in a static fluid (fluid at rest), there is no movement or velocity gradient between fluid layers, which means no shear stress occurs. Applying Newton's second law in this context, if there is no acceleration, then there cannot be any forces resulting in shear. Therefore, when analyzing fluids at rest, we don’t have to consider these shear stresses, simplifying our calculations.

Think of how a stack of pancakes behaves when you cook them on low heat without stirring. The layers remain stationary and do not slide over each other. In the same way, when a fluid is at rest, the different layers of fluid just sit still without exerting shear forces on one another.

So what the two forces we have? The gravity force and the force due to the pressure distribution. So the 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.

In a stationary fluid, the two primary forces acting on it are the downward gravitational force and the upward force resulting from pressure acting throughout the fluid. For an element of fluid at rest, these forces are balanced. This means that the pressure forces within the fluid counteract the weight of the fluid above it, allowing us to describe the state of the fluid without considering mass transfers or external work affecting the fluid’s state.

Imagine a glass of water sitting on a table. The weight of the water pressing down is balanced by the pressure exerted by the water at the bottom of the glass. This simple relationship defines the static state we observe—the forces are in perfect equilibrium.

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.

When engineers design structures like dams, they must consider the fluid static principles to ensure safety and stability. For instance, if a dam is 100 meters high and the water level is at 90 meters, the pressure exerted on the dam’s walls is significant and can be calculated using fluid statics. Understanding how pressure varies with depth allows engineers to design dams capable of withstanding these pressures without collapsing.

Think of how you would lean against a full water balloon—when it’s full, you can feel the pressure against your hand. Similarly, if a dam or any container is filled with water, the pressure at the bottom and along the sides significantly impacts the construction design.

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