1.4. Lecture-11

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 introductory segment, the professor welcomes students and introduces the focus of the last lecture in the fluid mechanics series. The key topic is the conservation of momentum, a fundamental principle that will be explored in detail during the lecture.

Imagine a soccer player striking a ball. The force exerted by the player changes the momentum of the ball, demonstrating conservation of momentum as the ball rebounds off the player's foot and travels in a new direction.

So, starting with the last slide, where we left in the last lecture and the we derived the general form of Reynolds transport theorem here.

The Reynolds Transport Theorem is a crucial mathematical tool in fluid mechanics that relates the change in quantity (like momentum) within a control volume to the flow of that quantity across the boundaries of the volume. The professor indicates that this theorem will be crucial for deriving various conservation equations.

Think of a river flowing into a lake. The Reynolds Transport Theorem would help analyze how the volume and momentum of water flowing into the lake from the river changes as it enters the larger body of water.

So, now 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 the mass of a closed system will remain constant over time, regardless of the processes acting inside the system. The professor explains that the concepts derived from the Reynolds Transport Theorem can be adapted to express mathematical equations that embody the principle of mass conservation.

Consider filling a bathtub. As water flows out of the faucet, it also flows into the tub. The total amount of water does not change, illustrating conservation of mass as the inflow equals the outflow.

So, we can simply write the velocity, average velocity as... V bar is the spatially averaged velocity normal to the cross section, this is important to note.

The continuity equation describes how the velocity of fluid changes in different parts of a system, ensuring that the mass flow rate remains constant. For incompressible fluids, the equation applies that the product of cross-sectional area and velocity at one section equals that at another section.

Think of a water hose. If you cover the end of the hose with your thumb, the water velocity increases. This demonstrates how the speed of a fluid changes in a varying cross-section to maintain a constant flow rate.

So, now the linear momentum equation...

The linear momentum equation relates the forces acting on a fluid to the change in momentum of that fluid. The professor emphasizes the importance of this equation in real-world applications, such as the behavior of water moving through pipes or jets.

Imagine a basketball moving towards a wall and bouncing back. The change in momentum of the basketball from its approach to the rebound can be analyzed using the principles of linear momentum.

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

In a steady-state scenario, the properties of the fluid such as velocity and density do not change with time at any point within the flow field. Here, the forces acting on the fluid, like pressure and gravity, must balance out to maintain equilibrium within a control volume.

Consider a suspension bridge where the cables are under constant tension. The forces of the cables, gravity on the bridge deck, and wind resistances work together in a steady state to maintain the bridge's stability, akin to fluid forces in a control volume.

So, some examples of conservation of mass...

In this example, the professor discusses how conservation of mass is applied to a reservoir. The rate at which water flows out of the reservoir can be used to determine how quickly the water level drops over time, illustrating the practical utility of the conservation principle.

Imagine monitoring the water level in a swimming pool while draining it. By observing how fast the water is drained, you can accurately predict how long it will take for the water level to recede to a certain point, demonstrating the conservation of mass.

Now, we can solve a practice problem...

The professor wraps up by indicating that various real-life problems using linear momentum, conservation of mass, and the Reynolds transport theorem can be solved. Such practice enhances understanding and application of the principles learned.

Solving real-world problems, like calculating how different angles of a water jet affect its trajectory, helps reinforce concepts from the lecture and makes learning interactive and practical.

--