15.5.4. Steady Flow Mass Conservation Equation

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.

This chunk introduces the concept of mass conservation in a control volume. A control volume is a defined region in space where we analyze the behavior of fluids. In this context, the mass influx refers to the mass entering the control volume through its control surface, and the mass outflux refers to the mass leaving it. Additionally, the flow of momentum and energy is discussed, indicating that similar conservation principles apply. Essentially, mass conservation means that the mass flowing into the control volume should equal the mass flowing out, assuming there are no additional mass sources within the volume.

Imagine a water tank filled with water through a pipe. As water flows into the tank, it should also flow out through another pipe at the bottom, maintaining a steady level. If more water enters than exits, the tank fills up (i.e., an increase in mass storage), and vice versa. This water tank scenario illustrates the principle of mass conservation, where the amount of water in the tank remains balanced depending on the flow rates.

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.

In this part, the lecture discusses the derivation of the mass conservation equation, distinguishing between system-level and control volume-level equations. The mass conservation equation relates the rate of change of mass within the control volume to the mass flux across its control surfaces. At steady state, the equation simplifies, typically showing that the mass entering the volume equals the mass leaving it. The lecturer emphasizes recognizing how mass and momentum fluxes cross through control surfaces — the mathematical foundation that leads to the conservation principles used in fluid mechanics.

Consider a bus moving with passengers getting on and off at various stops. Each time a passenger enters the bus, it's equivalent to an influx of mass, and when a passenger exits, it's an outflux. By tracking how many passengers get on and off, we ensure that the total number of passengers is accounted for, similar to ensuring that mass conservation is maintained in our fluid control volume.

Now, if you look if I have the velocity vectors, okay? Let us consider I have a control surface like this. If the n vector is like this, the perpendicular vector to the control surface is like that, if I consider one simple case, the velocity is either parallel or opposite.

This chunk covers the assumptions we make for steady and incompressible flow while deriving the mass conservation equation. Steady flow means that fluid properties at any point do not change over time. Incompressible flow assumes that the fluid density remains constant. Under these conditions, the conservation of mass can be simplified because the density does not vary. The lecturer talks about velocity vectors' interactions with control surfaces, clarifying that if the normal vector (n) of the control surface is aligned with the fluid velocity, calculations become simpler since only the scalar product of velocities needs consideration.

Imagine you have a smooth water slide (control surface) and water (fluid) flowing down it. If the slide is straight (fixed surface) and the water flows consistently, the speed of the water remains the same (steady). If it were splashing out (density variation), calculation would become more complex. Thus, the streamlined slide shows how steady and incompressible flow works together, resulting in easier mathematical descriptions.

Now coming to a very interesting (()) (13:15) 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.

This section shifts the discussion towards moving control volumes, such as a ship in motion. The key point is that when analyzing a control volume that itself is in motion, we need to consider the relative velocity between the fluid and the control volume. The relative velocity is the difference between the velocity of the fluid and the velocity of the control volume itself. This concept allows us to apply the steady flow mass conservation principle while accounting for the motion of the control volume, making it necessary to adjust our calculations for fluid entering or leaving it.

Think about a swimmer in a flowing river. If the swimmer moves upstream against the current, their speed relative to the river bank is their swimming speed minus the river speed. To calculate how much water flows past them while swimming, we must consider this relative speed, just like we must consider the relative velocity in moving control volumes for accurate mass conservation calculations.

The net change of the mass in this control volume should be equal to the change of mass storage within the control volume. That is what the concept is you are looking at?

The final part of the discussion reinforces the fundamental concept of mass conservation: the net change of mass in a control volume should equal the change in mass stored within that volume over time. For steady-state flow, this means that the mass entering the control volume equals the mass exiting. If there's no accumulation or depletion of mass over time in the control volume, the inflow and outflow must balance out. This principle is crucial for solving a wide range of practical problems in fluid mechanics.

Consider a bank account where you deposit and withdraw money. If you continuously deposit the same amount that you withdraw, your account balance remains unchanged (like steady flow). In fluid mechanics, the principle is the same: the mass that flows in equals the mass that flows out, maintaining balance over time.

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