7.6.1. Mathematical Formulation of Pascal’s Law

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 fluid at rest, the pressure exerted on any part of the fluid is transmitted undiminished throughout the fluid. This means that any change in pressure applied at any point in a confined fluid will be felt equally throughout the fluid. This normal stress acting on any plane is crucial to understanding how fluids behave under pressure.

Consider a fully sealed balloon. When you press down on one part of the balloon, the pressure is felt throughout the entire balloon, causing it to change shape. This is a practical demonstration of Pascal's Law as the pressure is transmitted equally in all directions.

Here I am considering a fluid element or you can say it the fluid control volume element or fluid control volume. Now if you consider this is what my control volumes and along this perpendicular to this surface that is what my unit value.

When analyzing a fluid element at rest, we assume it is in equilibrium, meaning all the forces acting on it must balance out. The pressure acting on different surfaces will produce forces that must be equal to maintain this equilibrium. Therefore, the sum of the forces in any direction (x or z) must be zero. This leads to an understanding of how pressure varies within the fluid element.

Think of a car tire filled with air. The air pressure inside the tire pushes against the walls of the tire evenly, ensuring that it maintains its shape. If the pressure is not balanced, the tire could either deflate or burst, illustrating the importance of equilibrium of forces in fluids.

As the fluid is at rest, summation of forces in x and z directions must be equal to zero. ∑F_x = 0 = P_b Δz - P_b Δs sin θ; ∑F_z = 0 = P_b Δx - P_b Δs cos θ - ρgΔx Δz.

To derive the pressure relationships in a fluid at rest, consider the forces in both the x and z directions. These equations represent the balance of pressure forces acting on the surfaces of the fluid element and the weight of the fluid. This understanding leads to the formulation that allows us to relate the pressure at different points within the fluid. As Δs becomes infinitesimally small, we visualize the condition of uniform pressure in all directions.

Imagine a column of water in a tall glass. The pressure at the bottom of the glass is equal to the weight of the water above it and reflects the influence of gravity. If you were to take very small sections of the glass and apply the principles mentioned, you'd find that the pressure is consistently calculated based on the weight of the water, defending the equilibrium conditions.

That is what is the Pascal law. And since pressure is same in all the directions, so we can consider is a pressure is a scalar quantity and that what they must stated by Pascal’s law.

Pascal's Law indicates that pressure in a fluid at rest is isotropic—meaning it is the same in all directions. This leads us to treat pressure as a scalar quantity rather than a vector, simplifying calculations in fluid mechanics significantly. This scalar nature of pressure is fundamental for calculations involving forces, stress, and fluid dynamics.

A simple way to think about this is with a soccer ball. If you apply pressure on one point of the ball, the air pressure inside the ball works equally in all directions, keeping the ball firm. This illustrates how pressure acts uniformly in every direction within a fluid, enabling us to apply Pascal's principle effectively in engineering and real-world applications.

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