7.5.1. Single Independent Variable

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 talk about a fluid being at rest, it means that there is no movement in the fluid. In this state, the velocities of the fluid particles are zero. A significant simplification occurs because we only need to consider how pressure changes within the fluid. Velocity gradients, which can create shear stress, are not present when the fluid is at rest. Thus, the only forces acting on the fluid are due to pressure and gravity.

Imagine a glass of water sitting on a table. The water is not stirring or moving; it is completely still. In this scenario, we only need to consider how deep the water is (which affects pressure) and the force of gravity acting on it. This is similar to how a dam holds still water behind it, where the pressure exerted at the bottom depends solely on the height of the water above it.

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, the absence of velocity means there are no velocity gradients. According to Newton's second law, this translates into no shear stress acting on the fluid at all. In practical terms, this means if you were to draw an imaginary box within the fluid, the forces due to fluid motion (shear forces) would not act upon the walls of that box. Thus, the analysis of fluids at rest becomes much simpler because shear stress can be ignored.

Consider a thick slab of ice on a pond. The ice remains still with water underneath. If the ice has a flat surface, there is no push or glide occurring between the water and the ice. This implies no shear forces are acting on the ice from the water, just like with a still fluid where shear stress is nonexistent. Imagine how complicated it would be to analyze movement if the ice were constantly shifting!

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.

When analyzing a fluid at rest, we only consider two main forces: the gravitational force acting downwards and the hydrostatic pressure acting upwards and sideways. The pressure generated from the fluid itself must balance the force of gravity for the fluid to remain at rest. Therefore, understanding how these forces interact and equate is crucial for designing structures like dams and tanks.

Think about water at a consistent level in a swimming pool. The weight of the water exerts pressure on the bottom of the pool, which must be balanced by the structural integrity of the bottom surface. Analogously, if the pool were to suddenly empty, the balance of these forces would change, indicating how crucial understanding gravity and pressure is in fluid mechanics.

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. So there is a vertical surface, there is a inclined surface. So we need to know it what is the fluid pressure is going to act on this dam, on these vertical surface and also the inclined surface.

The example of a dam illustrates the principles of fluid statics. When water is stored in a dam, the pressure at any point on the dam wall depends on the height of the water above it. As such, the pressure at the base of the dam where the water is deepest will be greater than the pressure higher up. This concept helps engineers design safe and strong dams that can withstand the pressures exerted by the water.

Consider a high-rise building with a water tank on the roof. The water pressure at the base of the tank is dictated by the height of the water column directly over it. Engineers need to understand this pressure to make sure the tank’s structure is strong enough to handle the immense force it experiences due to gravity, analogous to how a dam must manage the weight and pressure of water.

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