15.5. Conservation Principles in Fluid Mechanics

Today, we will talk about how we can derive the conservation of mass. As I said earlier, when you have the control volume, through the control surface there is mass influx coming into the control volume, going out from the control volume. Similar way, momentum flux comes into the control volume, also goes through the other surface as momentum flux going out from that. Similar way, we can think this energy flux comes into the control volume and goes out of this thing.

In fluid mechanics, conservation of mass states that mass cannot be created or destroyed in a closed system. In this context, we consider a control volume — a defined space through which fluid can flow. Mass can enter this control volume (influx) or leave it (outflux). The principle outlines that the total mass within the control volume remains constant unless there is a net mass entering or leaving. The same concept applies for momentum and energy.

Think of a bathtub filled with water. If you pour water into the tub, that’s the influx, increasing the mass of water in the tub. If you drain some water from it, that’s the outflux, reducing the mass. If the amount of water pouring in equals the amount draining out, the total mass in the tub remains constant.

There are three types of control volume: fixed control volume, moving control volume, and deformable control volume.

Control volumes can be classified into three categories: fixed, moving, and deformable. A fixed control volume does not change its position over time, providing a stable reference frame. A moving control volume, such as a ship at sea, travels with a certain velocity through the fluid, necessitating adjustments to the analysis based on that motion. Lastly, a deformable control volume changes its shape over time, like a balloon being squeezed and expanded – affecting how fluid enters and exits through its surfaces.

Imagine a soccer ball — this represents a fixed control volume, sitting still on the ground. Now visualize a boat moving through a river — this is akin to a moving control volume. Lastly, consider a flexible balloon being inflated and deflated — this represents a deformable control volume, where the surface through which the fluid (air) passes is constantly changing.

If you look at this equation, let us look at this. 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.

The conservation of mass equation can be derived from considering the rate at which mass enters or exits the control volume. The net mass change in the control volume is equal to the mass that enters minus the mass that exits. This can be mathematically expressed as the integral of density times velocity across the control surface, showing how fluxes of mass are related to changes in mass within the volume over time.

Consider a factory producing widgets. The mass of widgets coming in (raw materials) and going out (finished products) must balance out. If more raw materials are added each hour than the widgets produced, the factory is effectively storing more mass, much like how we calculate the net change in mass within a control volume.

Now, let us go to very interesting applications. As you know, it Indian Space Research Organisation launched the satellite which is called MOM programmes, that means Mars Orbiter Mission programme. Fluid mechanics knowledge allows us to design the trajectory from Earth to Mars orbit.

Conservation principles in fluid mechanics are not just theoretical; they have real-world applications, such as in spacecraft design. The trajectory of a satellite like the Mars Orbiter Mission is calculated using principles of mass conservation, among other factors, to ensure that the spacecraft accurately reaches its destination by accounting for fluid interactions such as atmospheric drag.

When an arrow is shot from a bow, its trajectory is influenced by gravity and air resistance. Similarly, when designing the path for a satellite, engineers use fluid mechanics to predict how the satellite will move through various fluid environments, adjusting its path for the best outcome — just as an archer might adjust their aim for wind.

If I consider moving control volume like a ship which is moving with a velocity V. So, then, the concept will be the same. Only here we will talk about the relative velocity component.

In a moving control volume, such as a ship, the analysis of fluid motion requires us to consider the relative velocity, which is the difference between the fluid velocity and the control volume's velocity. This allows us to better understand how the fluid interacts with surfaces of the control volume as it moves, ensuring accurate calculations for mass flow rates and other properties.

Imagine swimming in a river. When you swim downstream, the current aids your motion, making it easier to go faster relative to the bank. On the other hand, if you swim upstream, you must counteract the river’s current. This relative motion is crucial for determining how much effort you need to exert, similar to how engineers calculate fluid interaction with moving bodies.

--