14.3.3. Simplification in Steady Incompressible Flow

The conservation of momentum which is required for you to know it, how the fluid particles are moving it, what could be the force exerting on that, what could be the velocity. Similar way, we can understand the energy conservation which plays a major role for us when the fluid comes from one location to other locations, how much of work is done by the fluid or into the fluid.

Similar way, whether there is heat transfers happening which you can feel it, if there is a temperature gradient there will be heat transfer either to the surrendering of the systems or into the systems or out of the system, that is what we can do. So, to summarise this, that means, we all know that there are three energy conservation principles that we follow in solid mechanics when you consider as a system.

Same concept also we can use at the system levels to solve the problems, conservation of mass, conservation of linear momentum which is Newton’s law, and the conservation of energy which is the first law of thermodynamics.

In this chunk, the importance of conservation principles in fluid dynamics is emphasized. Specifically, it describes three main conservation laws: conservation of mass, conservation of momentum, and conservation of energy. Conservation of mass states that mass cannot be created or destroyed, hence the mass flow into a system must equal the mass flow out of that system. Conservation of momentum, based on Newton's laws, explains how forces act on the fluid particles, affecting their velocity and direction. The conservation of energy states that energy can neither be created nor destroyed, only transformed. Therefore, understanding these principles is essential for analyzing fluid flow.

Think of a water pipe: water flowing into the pipe at one end (mass inflow) must equal the water flowing out at the other end (mass outflow). If you have a consistent water pressure, the velocity of the water flowing in must correspond to the velocity flowing out, regardless of how much water is in the pipe. If you add a pump (the force), the flow rate and the energy in the system changes, impacting the overall water pressure and flow directions.

Now, let me define two types of properties that we have; one is called extensive property and the other is the intensive property. The extensive property which is considered as proportional to the amount of mass. When you apply extensive properties, that means, you are the properties which are proportional to the amount of mass. That means, as mass increases you will have extensive properties going to increase.

Mass decreases, the extensive property decreases. It is proportional to the mass. For example, as we discussed it in three basic laws, we talked about mass conservation, momentum conservation, and energy conservation. So, m will be the mass conservation part, momentum and energy conservation. But when you look at the intensive properties, that means it is independent of mass, that means, which is denoted as specific values like velocity or specific energy. For energy conservation the extensive property will be the one, but in the case of the momentum, but intensive property will be the velocity vectors.

This chunk introduces two essential concepts: extensive and intensive properties. Extensive properties, such as mass and total energy, change depending on the size of the system; if the mass increases, so do extensive properties. Conversely, intensive properties, like density or temperature, do not depend on the system size. For instance, the temperature of water remains the same whether you have one liter or a thousand liters. Understanding the difference helps in analyzing how certain characteristics of fluids change with volume and mass.

Imagine a birthday cake: the total amount of frosting is extensive; if you double the size of the cake, you need double the frosting. However, the flavor of the frosting is intensive; whether your cake is small or large, the frosting retains its same flavor. This difference helps us understand fluid behaviors in varying situations.

Now, we will go to derive Reynolds transport theorem. The derivation of the Reynolds transport theorem are available in almost all the fluid mechanics books. The idea for me is to introduce the Reynolds transport theorem so that you can easily understand it. But the step wise derivations, if you are not understanding it, I could suggest you to follow any of the fluid mechanics books, F.M. White, Cengel Cimbala, or any other advanced fluid mechanics books, you can see the derivations of Reynolds transport theorem. Only the symbol of representation of extensive properties, intensive properties, either B or b or are used in different books in different forms.

Reynolds Transport Theorem is a fundamental equation in fluid dynamics that allows us to relate the change of a quantity within a system (such as mass or energy) to the quantities flowing across the boundaries of that system. This theorem bridges the gap between system and control volume analyses. It states that the rate of change of a property in a control volume can be expressed as the sum of the rate of change of the property in the system and the net flux of that property across the control surface.

Consider a bathtub as our control volume. If you fill the bathtub (influx of water) while also allowing some water to drain out (outflux), the change in water level in the bathtub depends on the rate of inflow and outflow. The Reynolds Transport Theorem helps calculate how fast water level changes given these rates, effectively linking the inflow and outflow to the water level inside the bathtub.

Now, coming to the third part. The first two terms we can write it as earlier you will have the influx and outflux. That part can be written as integrals of influx and outflux and if you combine it next what you are getting? This is total cross section, okay? That means net outflux of mass, momentum, or the energy flux going through the system, either influx or the outflux but as you integrate it it represents the net outflux that is going through this control surface.

Let us consider a steady incompressible flow, that means the flow does not change with time, the density changes in this case of the steady compressible flow. If that is the condition, your major part of this becomes 0, because if d by dt becomes 0.

In steady incompressible flow, the properties of the fluid do not change with time, meaning that any variations in flow (like velocity) remain constant throughout. This allows for simplifications in calculations since the time derivatives (change over time) of properties can be treated as zero. As a result, the fluid dynamics become much simpler since the only factors that need to be taken into consideration are the inflow and outflow of mass, momentum, and energy across the control surface.

Think about a garden hose constantly delivering water at a steady rate. If you maintain that rate (inlet), the water flowing out of the hose doesn't change (outlet) as long as no new water enters the system or leaks out. This steady state simplifies your calculations regarding how much water is in transit and the pressure required behind it.

With these things let me summarise today’s lecture. We discussed about system and control volume. We talked about fixed and deformable and moving control volume concept. The more important thing is that we derived the Reynolds transport theorem which can be used for fixed control volume, deformable control volume, moving control volume. And we also demonstrated the use of the simplification of the steady problems.

In summary, the lecture covered fundamental concepts in fluid mechanics, the distinctions between types of properties, and the significance of the Reynolds Transport Theorem. It emphasized the importance of understanding steady versus unsteady flows and how to apply these concepts to solve real-world fluid mechanics problems efficiently. The simplifications associated with steady incompressible flow make analysis much more manageable.

Understanding fluid dynamics is akin to learning to ride a bicycle. At first, it may seem complicated with all the mechanics (forces, balance, and direction), but once you grasp the basic principles and practice consistently, it becomes easier to maintain balance and control. Like in fluid mechanics, once you grasp the foundational concepts, analyzing flow becomes smoother and more intuitive.

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