4.3.2. Mass Conservation Equations

Now let us apply the mass conservation equation where inflow minus outflow is equal to mass stored. In this case, we can simply consider:

Outflow = Inflow

For incompressible flow:

ρ * A₁ * v₁ = ρ * A₂ * v₂

Where ρ is the density, A is the cross-sectional area, and v is the flow velocity.

The mass conservation equation states that the mass that enters a volume must equal the mass that leaves it, plus any change in mass stored within the volume. In this equation, we can see the relationship between the inflow (mass entering) and outflow (mass exiting), which must balance out. For incompressible flow (where density remains constant), we express this as the product of density, cross-sectional area, and flow velocity equals constant across different points in the system. This is crucial for understanding fluid dynamics.

Think of a water hose connected to a tank. When you turn on the faucet, water flows in at a specific speed (v₁) and fills the tank through the hose (A₁). At the same time, water can flow out of the tank (A₂) through a different outlet at another speed (v₂). The amount of water coming in must match the amount of water leaving the tank (assuming no leaks), which is the essence of mass conservation.

Applying Reynolds transport theorem to relate the change of momentum in control volume:

∂(mass flux) = ∑(outflux) - ∑(influx)

Reynolds transport theorem connects the flow of mass and momentum in control volumes which can change with time. This theorem allows us to express how momentum within a defined zone changes as a consequence of the inflow and outflow of fluid. Essentially, it states that the rate of change of momentum within the volume is equal to the net momentum flux leaving or entering that volume. This provides a robust framework for analyzing forces acting on fluid systems.

Imagine a crowded subway train. As people enter (influx) and exit (outflux), the pressure and crowding inside the train changes. If many people push against you while trying to enter the train, that's akin to how inflow affects momentum. The change in crowd density (momentum) reflects all the actions (pushing, holding, pulling) that are happening in the system.

Here, we consider the change in momentum flux within a control volume. This involves considering both influx and outflux momentum thrust, specifically in terms of fluid dynamics:

Force = Pressure × Area.

When analyzing forces acting on a control volume from fluid flow, particularly under pressure, we use the relationship that states force is equal to pressure multiplied by area. Here, the forces acting on the control volume can be broken down into their components, allowing us to calculate the total force experienced due to the inflow and outflow of fluids. This is vital in engineering applications like designing pressure vessels or understanding aircraft behavior during flight.

Consider an inflatable balloon. When you apply pressure to the balloon’s surface (like squeezing it), you create forces that will influence how it expands or contracts. The pressure on the outside (force) works against the internal pressure of the air (influx) and how much air is being expelled (outflux) when you release the balloon. This concept clearly illustrates how fluids interact with surfaces and the resulting forces at play.

Consider a water jet impinging normal to a flat plate moving to the right with velocity Vc. The equation states:

Force required to keep the plate moving at constant velocity is calculated using:

Inflow = Outflow and considering density, jet velocity, and area of the jet.

In this example, a water jet strikes a moving plate. The force needed to continue moving the plate at a consistent speed can be calculated using the principles of mass conservation and momentum equations. By ensuring that the inflow of water (mass being hit by the jet) equals the outflow (mass leaving the plate area), we establish a balance that allows us to calculate the resisting force from the jet striking the plate.

Imagine a firefighter aiming a hose onto a moving fire truck. The force of water hitting the truck can push it backward unless additional force (like a driver pressing on the gas) keeps it moving forward. It’s similar to how jets impact surfaces: inflowing water generates an opposing force that must be overcome for motion to remain constant.

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