7.4.2. Control Volume Considerations

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.

When we discuss fluids at rest, we examine situations where the fluid does not flow or change position, leading to simplified mathematical modeling. At rest, the velocity of fluid particles is zero. Because there is no movement, we only need to focus on how pressure varies within the fluid. In this simplified scenario, we can ignore the effects of velocity and temperature changes.

Think of a glass of water standing still on a table. The water is not moving, so we don’t have to worry about any currents or ripples affecting its pressure. We only need to be concerned with how much pressure pushes down from the water above and how that pressure influences the glass.

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.

In fluid mechanics, when we analyze forces acting on a control volume, we note that with fluid at rest, shear stress vanishes because the fluid isn't moving. Hence, the main forces acting within the control volume are due to pressure and gravity. In simple terms, if you consider a bottle filled with water standing still, the only forces at play are the weight of the water and the pressure exerted on the walls of the bottle by that water.

Imagine a swimming pool. When you jump into the pool, the water moves and creates waves and pressure differences. But if the pool is filled and no one is in it, the water simply exerts pressure against the pool walls and the bottom, and there are no waves or currents. This is similar to how forces balance out when the fluid is at rest.

So that is the very easy problems what we have now. And what the two forces we have? The gravity force and the force due to the pressure distribution.

When analyzing fluids at rest, we primarily consider two forces: gravity and pressure distribution. The pressure distribution is influenced by the fluid's column above each point. Essentially, the deeper you go into the fluid, the higher the pressure due to the weight of the fluid above. For example, at the base of a dam, the water pressure is significantly higher than at the surface. This pressure must be calculated to ensure the dam can withstand the force without failing.

Consider a bathtub filled with water. The water exerts pressure on the bottom of the tub due to the weight of the water above it. The deeper you go (like having more water in the tub), the more pressure you feel against the bottom. This is why when you try to dive deep in a pool, you'll feel the pressure on your ears—it's the water above pushing down.

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.

Pascal's Law asserts that in a fluid at rest, pressure applied to any part of the fluid is transmitted equally in all directions. This principle helps us understand how pressure changes across different points within the fluid and allows us to calculate how the pressure influences forces acting on surfaces within the fluid. By analyzing the pressure functions based on depth, we can derive crucial engineering applications, such as designing hydraulic systems.

Think of squeezing a balloon filled with water. When you apply pressure on one end of the balloon, the water moves and transmits that pressure across the balloon evenly, causing it to bulge out on the opposite end. This illustrates how pressure operates uniformly throughout the mass of the fluid.

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