7.2.1. Recap of Fluid Mechanics Principles

We already know it we talk about a either a system approach or the control volume approach to solve the fluid mechanics problems. And whenever you solve the fluid mechanics problems as I said it in the last class, we generally look for three velocities three fields, velocity field, pressure field and the density field.

In fluid mechanics, we often address two main approaches: a system approach, which considers the entire system as a whole, and a control volume approach, which focuses on a specific volume in space where we analyze fluid behaviors. To solve problems in fluid mechanics, we look at three main fields: the velocity field, which describes how fast and in what direction fluids are moving; the pressure field, which illustrates how pressure varies in different parts of the fluid; and the density field, representing how dense the fluid is at various locations.

Imagine a large water tank (the system) connected to a pipe (the control volume). Understanding how water flows into and out of the tank (velocity) requires us to assess how the pressure differs at the bottom of the tank compared to the top, and how the density of the water might change with temperature.

But when fluid is incompressible, then we just look for the pressure field and the velocity field. So these two fields we can get it using this three different approaches as I discussed earlier.

For incompressible fluids, the density remains constant, which simplifies our analysis. In this context, we primarily focus on the pressure and the velocity fields. These fields can be derived using experimental methods (like wind tunnel experiments), computational approaches (computational fluid dynamics), or analytical methods to solve simpler fluid mechanics problems.

Think of a water balloon. When you squeeze it, the water doesn't compress; it simply moves. This behavior is similar to how incompressible fluids behave in many cases, where we mainly consider how fast the water flows (velocity) and how hard it pushes against the walls of the balloon (pressure).

So that is what we look it for the flow which is considered incompressible flow. And then we already discussed about that to adopt these three different approaches we have to follow integral approach, differential approach and the dimensional analysis.

When analyzing fluid flow, especially in incompressible scenarios, we utilize three key approaches: the integral approach, which focuses on the accumulated effects over a specified area; the differential approach, which analyzes changes at an infinitesimally small scale; and dimensional analysis, which helps us understand how different physical quantities relate to each other without solving complex equations directly.

Consider baking a cake (integral approach) where you measure the total amount of flour needed. If you were to analyze each individual grain of flour’s contribution (differential approach), it would be impractical. Instead, dimensional analysis helps you determine how much flour you might need based on the size of your cake pan.

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?

When a fluid is at rest, it simplifies our analysis significantly. In such situations, the velocity of the fluid is zero, which means that all shear stresses disappear. The only forces we need to consider are the gravitational forces and the pressure forces. Consequently, we primarily study how pressure varies across the fluid.

Visualize a calm pond; since the water is not moving, you can easily see how deep the water is (pressure) without worrying about waves or currents affecting your measurements.

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.

Because the fluid is at rest, we see that there are neither velocity gradients nor shear stresses acting on the fluid. Hence, when analyzing a control volume of fluid, we can conclude that the only forces acting are those due to pressure and gravity. The implication of this is that the pressure within the fluid can be treated as a scalar quantity.

If you have a bottle filled with honey that you stand upright, the pressure at the bottom of the bottle is solely due to the weight of the honey above it. No matter how you shake or tilt the bottle, if the honey is at rest, the pressure is static and predictable.

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.

Understanding hydrostatic pressure distribution is crucial, particularly in engineering applications like designing dams. The pressure exerted by a fluid at rest is not uniform; it increases with depth due to the weight of the fluid above it. Therefore, it is essential to calculate how much pressure acts on different surfaces of structures such as dams, where vertical and inclined surfaces handle different stress distributions.

Think about a fish tank; the deeper down you go, the heavier the water above pushes down—this is similar to how the pressure works on a dam. The bottom of the dam faces far greater pressure than the top where the water is shallower.

Second thing is that fluid act as a rigid body motions. Let us take an example here that I have a tank which is a half filled liquid is there.

When a fluid-filled tank accelerates uniformly, the fluid behaves as if it were a rigid body. This means that while the fluid is still at rest relative to the tank, the effective pressure distribution changes as the entire system accelerates. The surface of the fluid will tilt, resulting in a new equilibrium position even though no fluid is flowing.

Imagine riding in a car that suddenly accelerates. If you have a cup of water in your hand, the water surface will tilt backward despite appearing calm. This is due to the car's acceleration, causing the water to shift as a unit.

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 when a fluid is at rest in a confined space, any change in pressure applied to the fluid is transmitted undiminished throughout the fluid. This means if you apply pressure at one point in a confined fluid, it will be felt equally at every point within that fluid, showing that pressure is a scalar quantity equally distributed within the fluid.

Think about how a balloon works: if you squeeze one part of it, the pressure increases throughout the entire balloon; no matter where you press, the air inside responds uniformly due to Pascal's Law.

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