7.4.1. Pressure Field Analysis

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 refer to a fluid at rest, we're discussing a situation where the fluid is not moving. In this condition, the velocity of the fluid is zero, simplifying our analysis. Since there is no motion, we only need to consider how pressure varies within the fluid. Additionally, the density remains constant for incompressible fluids, which further simplifies our calculations.

Imagine a glass of water sitting still on a table. Since the water is not moving, we can easily analyze the pressure at different depths in the glass using basic principles. The deeper you go, the greater the pressure due to the weight of the water above.

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 a static fluid, there is no change in velocity at different points, which means that there are no shearing forces acting on the fluid. Since shear stress arises from velocity gradients, a constant velocity (in this case, zero) means that shear stress is also zero. This allows us to conclude that when fluid is at rest, there are no shear stresses acting on any surfaces in contact with the fluid.

Think of honey sitting in a jar. If you don't stir the honey, the layers remain evenly distributed with no sliding between them. This is similar to how pressure varies in a static system, where there's no movement causing any shear.

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

When analyzing a fluid element at rest, we primarily consider two forces: the gravitational force acting downwards and the pressure force acting perpendicular to the surface. The pressure force is a result of the pressure distribution within the fluid, while gravity pulls the fluid down due to its weight. At equilibrium, these forces balance each other.

Consider a swimming pool filled with water. At any given depth in the water, the pressure force from the water above balances the downward gravitational force. This equilibrium is essential for understanding how structures like dams and underwater pipelines function.

Based on these the pressure load we can design this dam structure. That is what the examples for fluid at the rest.

When dealing with hydrostatic pressure, engineers need to calculate how much pressure is exerted on surfaces by fluids at rest, such as in dams. This helps ensure that structures can withstand the forces applied without failing. The pressure increases with depth, governed by the equation P = ρgh, where ρ is the fluid density, g is gravity, and h is the height of the fluid.

Like assessing how heavy water exerts pressure on the walls of a swimming pool, engineers use similar principles to calculate the pressure exerted by the water behind a dam wall, ensuring that the structure remains stable and safe.

Second thing is that fluid act as a rigid body motions.

In some cases, such as when a tank holding fluid is accelerated, the fluid behaves as if it were a rigid body. As the tank accelerates, the fluid surface will tilt, maintaining equilibrium while experiencing a pressure distribution similar to that found in static fluids. Even though the tank moves, there are no shear stresses acting in the fluid, allowing us to apply the principles of hydrostatics.

Picture riding in a car that suddenly accelerates. You may feel yourself being pushed back into your seat due to the forces acting on you. Similarly, the fluid in the acceleration tank equally experiences forces but remains at rest relative to the tank and adjusts its free surface.

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.

In a fluid at rest, pressure varies depending on the position within the fluid, typically increasing with depth. Since the fluid is static, we can express pressure as a function of spatial variables (like height), eliminating time from consideration. Taylor series can be used to approximate pressure values at different heights, although in many cases, we focus on the primary term of the series, neglecting smaller contributions from higher order terms.

Imagine measuring the pressure at different depths in a pool using a pressure gauge. At the surface, the pressure is lowest and gradually increases as you go deeper, reflecting how position affects the pressure experienced within the fluid.

Now let come it to very basic law is called Pascal. As you when you fluid is rest let us consider is that the there will be a normal stress acting on any plane.

Pascal's Law states that in a static fluid, pressure is transmitted equally in all directions. This means that if you apply pressure on a fluid in a confined space, that pressure will increase uniformly throughout the fluid. As we analyze a control volume, we can expect that the normal stresses due to pressure are equal in all directions, confirming that pressure is indeed a scalar quantity.

Think of a balloon. If you squeeze it at one point, the pressure goes up throughout the entire balloon equally, demonstrating Pascal's Principle. This principle is vital in hydraulic systems where force applied in one area can be used to create a larger force elsewhere.

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