1.5. Basics of Fluid Mechanics-II

Welcome students, this is going to be the last lecture for the basics of fluid mechanics 2. Where we are going to see the conservation of momentum in more detail.

In this introduction, the professor sets the stage for the final lecture in the fluid mechanics series. The focus will be on the conservation of momentum, a fundamental principle in fluid mechanics used to analyze the motion of fluids.

Think of a car colliding with a wall. The momentum of the car before the impact is conserved to some extent by the wall. This principle of conservation of momentum helps us understand how fluids behave under various conditions.

Now, we are moving from a system to a control volume. We will see how when we apply the system to mass, what is going to happen.

The Reynolds Transport Theorem is a fundamental principle in fluid mechanics. It helps us relate the change of properties of a fluid within a specific control volume to the flux of those properties across the boundaries of that volume. This theorem allows engineers to analyze different aspects of fluid flow.

Imagine a water tank with a pipe draining water. The Reynolds Transport Theorem helps us understand how the amount of water inside the tank changes over time based on how much water flows in and out.

So, conservation of mass, how can we apply what we have learned in Reynolds transport theorem to the conservation of mass?

The conservation of mass states that mass cannot be created or destroyed in an isolated system. This principle can be expressed mathematically using the Reynolds Transport Theorem. The equation ultimately shows that the mass leaving a control volume minus the mass entering equals the rate of change of mass within that volume.

Picture a swimming pool with a constant input of water from a hose. The conservation of mass would illustrate how the water level does not change if the flow out of the pool is equal to the flow into it.

Continuity equation for constant density and uniform velocity... V A₁ = V A₂ = Q.

The continuity equation is derived from the conservation of mass and states that the product of cross-sectional area (A) and fluid velocity (V) must remain constant along a streamline for incompressible fluid flow. This leads to the conclusion that if the area decreases, the velocity must increase and vice versa.

Think of a garden hose. When you put your thumb over the end, the cross-sectional area is reduced, and the water shoots out faster. This is a practical illustration of the continuity equation.

Now, we are going to see some examples of conservation of mass...

Applying conservation principles allows us to solve real-world problems involving fluid flow. For instance, calculating how fast the water level in a reservoir drops based on the flow out can be analyzed through a series of equations derived from mass conservation.

Imagine filling a bathtub and having water drain out of a hole at the bottom. Using the principles discussed, we can determine how quickly the water level drops based on the drain's size and the water flowing from the tap.

So, now the linear momentum equation. An example here is if you shoot a water jet onto a wall...

The linear momentum equation facilitates the understanding of how forces act during fluid interactions. When a water jet strikes a wall, the change in momentum can be used to analyze the forces exerted on the wall.

Consider a high-pressure water cannon. As the water hits a surface, the force and momentum change directions. Understanding this helps in designing structures to withstand such impacts.

Now, the question is, what are the forces acting on fluid in the control volume...

To analyze fluid systems, one must account for all acting forces such as pressure, weight, and shear forces. The relationship established among these forces allows for a comprehensive understanding of fluid behavior and equilibrium.

Imagine holding a balloon underwater. The buoyancy from the water, the weight of the water inside, and the force you're applying all interact to maintain the position of the balloon.

An example we are going to see is the reducing elbow using this equation...

This example illustrates how to apply the principles of conservation of momentum and mass to real-world systems like a pipe elbow through which fluid flows. Engineers can calculate the forces acting on the elbow to ensure structural adequacy.

Think of a garden hose that bends at a right angle. When water flows through, the angle influences the force on the hose. Understanding these forces prevents damage and ensures the hose can handle the pressure.

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