15.3.2. Control Volume Level Equations

Coming back to the Reynolds transport theorems, as I derived in the last class, one is the system level equations and the other is the control volume level equations.

In this part, we are reintroducing the concepts we learned in the previous lecture about Reynolds transport theorems. We differentiate between equations formulated at the system level and those at the control volume level. System level equations describe the behavior of a closed system, while control volume level equations allow us to analyze the flow of mass, momentum, and energy across a defined volume in space (the control volume).

Think of a car's fuel system as a closed system (like a system level equation) where we only look at how much fuel remains in the tank without considering its movement. In contrast, if we consider the entire car (the control volume), we can analyze how fuel flows in and out as the car drives, representing a dynamic analysis of the control volume.

B: extensive property of fluid (m, ṁ, E) b: corresponding intensive property (1, ŋ, e)

Here, we define extensive and intensive properties of fluids. Extensive properties are quantities that depend on the amount of fluid present, such as mass (m), mass flow rate (ṁ), and energy (E). Intensive properties are independent of the amount of fluid and include properties like density and specific energy. These distinctions are crucial for applying the Reynolds transport theorem since it allows us to relate these two types of properties to analyze fluid flow.

Imagine you have a glass of water. The total amount of water (mass) is an extensive property because it changes if we add or remove water. The temperature of the water, however, is an intensive property; it doesn't change based on the glass size or how much water there is.

If you look at this equation, let us look it. This is the system level equations. This is what is at the control volume level. So, one talks about how this particular mass or momentum flux is crossing through the control surface.

This chunk discusses how we derive the mass conservation equation from the Reynolds transport theorem. The equation relates how mass enters and exits a control volume through its boundaries (control surfaces). In essence, the change in mass within the control volume over time must equal the mass being transported across the surfaces of this volume, expressed mathematically as a balance between inflow and outflow.

Consider a bathtub with a plugged drain. If you are filling it with water (inflow) and the plug is intact, the water level (mass) rises. If you pull the plug (outflow), the water will drain out. The conservation of mass states that the change in water in the tub over time equals the water being added or removed.

That means the density vary but it is steady. So, once it is steady, if you look at this term, it becomes 0. Only what remains is the surface integration part which is quite easy now.

In fluid mechanics, flow can be categorized into steady and unsteady states. If flow is steady, properties like density do not change with time at any point in the control volume. In this case, the mathematical terms representing the time variations become zeros, simplifying the calculations. We focus on the surface integrals instead, making it easier to analyze mass flow across control surfaces.

Think of a river flowing steadily – the water level at a specific point remains constant over time (steady flow). However, if we were to look at a rapidly changing flood situation where water heights are constantly rising or falling (unsteady flow), we'd have to account for how the water isn't static, complicating our analysis.

Now coming to a very interesting applications. As you know it Indian Space Research Organisation launched the satellite which is called MOM programmes...

This section highlights the practical applications of fluid mechanics, specifically illustrating how control volume analysis is crucial in designing satellite trajectories and analyzing fluid behavior in various engineering projects. The Mars Orbiter Mission is cited as an example where understanding fluid dynamics allows engineers to calculate the correct trajectory and interactions with atmospheric forces, depending on principles derived from fluid mechanics.

When launching a satellite, engineers calculate the rocket's path similar to how we would throw a ball across a yard – we need to know how high to throw it (angle) and how hard (momentum) to ensure it lands in the right spot (orbit around a planet). By applying control volume analysis, engineers make these critical calculations that ensure successful missions.

Now coming to a very interesting (()) (13:15) if I consider moving control volume like a ship which is moving with a velocity V.

In analyzing moving control volumes, we must consider the relative velocity between the fluid flow and the moving control volume itself. By incorporating relative velocity into the equations, we can accurately describe the mass flux entering or leaving the control volume, essential for scenarios like a ship moving through water.

Imagine a swimmer moving through water. The swimmer has their own speed, and the water is also moving (like a current). The swimmer's effective speed regarding the bank of the river differs from their speed relative to the water - this is similar to how we calculate relative velocity in moving control volumes.

But in case you have a deformable control volume and also moving control volume, all the things you have are complicated now.

Deformable control volumes introduce complexity in analysis because both the fluid and control volume can change shapes and velocities with time and position. This requires careful consideration in calculations since both parameters will affect how we measure mass and momentum fluxes.

Consider a bag filled with flour that is being squished – as you push on it, the shape changes (deforms), and so does the pressure of the flour inside. Fluid dynamics involving deformable control volumes requires understanding both how pressure and volume change under these conditions.

Now, let us come to the, as I said it, we need to have three equations. One is conservation of mass, conservation of linear momentum or the angular momentum. And third is, as you know it, conservation of energy.

In fluid mechanics, three key conservation principles guide our analyses: conservation of mass, conservation of momentum, and conservation of energy. Understanding how these principles interrelate helps engineers and scientists solve a variety of fluid flow problems effectively. While solving practical problems, often we need only two of these equations, with mass conservation being crucial.

Think about cooking – when you bake, you have several ingredients (mass), and you must mix them (momentum) without losing anything (mass conservation). If you add heat (energy), it changes the physical properties of your mixture but, fundamentally, the quantities of each ingredient must match what you started with. Similarly, in fluid mechanics, these conservation laws ensure balance throughout any fluid system.

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