3.1.1. Lecture - 14: Conservation of Momentum: Example Problems

Welcome all of you for this course on fluid mechanics. In the last class we discussed about conservation of mass and momentum and its applications. To continue to that conservation of mass and momentum and its applications, today I will deliver a lecture on this topic and also I will solve some example problems to illustrate how we can use conservation of mass and momentum equations to solve real life problems.

In this introduction, the lecturer welcomes students to a fluid mechanics course and reviews previous lessons on the conservation of mass and momentum. The aim of today's session is to further explore these concepts through example problems, illustrating their practical applications in real-world scenarios.

Think of momentum conservation like a game of pool. When you strike a ball (momentum), it moves off and strikes another ball, transferring some of its momentum to that ball, causing it to move. In fluid mechanics, we study how fluids also behave similarly when they interact with surfaces and other forces.

Let us come back to what we discussed in the previous class that we discussed how we can approximate linear momentum equations from Reynolds transport theorems to specific cases like one inlet, one outlet, which is very simplified problems. And the momentum equation is vector equations. But many of the cases, we can solve problems in specified directions.

The lecturer emphasizes that in fluid mechanics, especially with momentum equations, simplifications are often made. For instance, problems with one inlet and one outlet can be treated as simpler cases. The momentum equations are vector equations that help describe how fluids move in different directions, and although they are complicated, focusing on specific directions can make these calculations manageable.

Imagine solving a puzzle. Sometimes, focusing on the edges first (inlet and outlet) makes it easier to see where the pieces fit. Similarly, simplifying the momentum equations to one inlet and outlet allows us to solve complex fluid dynamics problems more easily.

Most important qualities that we consider always is steady flow conditions. This is the approximation we do, the steady flow conditions where there is no change of the pressure or the velocity distributions with respect to time.

Steady flow conditions are crucial in analyzing fluid behavior. In steady flow, the pressure and velocity of the fluid remain constant over time at any given point. This simplifies calculations significantly, allowing engineers and scientists to apply conservation of momentum equations effectively without accounting for time-varying factors.

Think of a river on a calm day. If the water level and speed remain the same throughout the day, you’re observing steady flow. If the river’s flow changes rapidly from rain or dam openings, it becomes complex to predict how the water will behave, similar to fluctuating pressure in a fluid system.

So we can align the specific axis rotation in such a way that using only one equation we can solve the problems. So these are the simplifications for the one inlet, one outlet case, where the mass inflow will be equal to mass outflow.

In this section, the lecturer explains that by setting reference axes properly, they can simplify the momentum equations. In situations with one inlet and one outlet, they can equate mass inflow and outflow to make the calculations more straightforward.

Consider measuring traffic at a one-way street. If you know how many cars come in (inlet) must equal the number that exits (outlet), you can easily assess the flow rate without worrying about complex intersections.

When you apply the linear momentum equations, you have to draw control volume, the control surfaces. How to define this control volume and control surface whether it is a fixed control or movable control is essential.

Defining control volumes and surfaces is a critical part of applying momentum equations. A control volume is a specified region in space where we analyze the flow of fluids and the forces acting on them. Understanding whether these control volumes are fixed or movable helps make sense of how to apply the conservation laws correctly.

Think of a swimming pool as your control volume. If you want to understand how water flows in and out, you need to draw the edges of the pool and observe where water enters and exits. This will guide your understanding and calculations about water conservation in that area.

Also, we talk about the momentum corrections factors that means beta. How do you compute the beta value which is the momentum correction factors.

Momentum correction factors, referred to as 'beta', account for non-uniform velocity distributions in a fluid. When calculating momentum, it is crucial to understand how much the assumed average velocity deviates from the actual velocity across the flow area. Beta helps adjust calculations accordingly.

Imagine a marathon runner who averages 6 minutes per mile but sometimes speeds up or slows down. If you were to calculate their total time, you'd have to consider those variations instead of just the average. Similarly, the momentum correction factor beta adjusts our calculations to accurately reflect variations in fluid velocity.

Now let us come to today's lectures what I will talk about... we will solve one example, cases where no external forces are there specifically the spacecraft problems that we will solve.

The lecturer sets the stage for practical application by introducing example problems. These problems will help illustrate how to apply the theoretical concepts of momentum conservation in a scenario lacking external forces, akin to conditions experienced by spacecraft.

Think of a spacecraft traveling in the vacuum of space. Without external forces like air resistance, understanding momentum conservation becomes crucial to maintaining its trajectory. Performing these problems helps us understand the fundamental physics that governs these scenarios.

--