Today, we will explore the conservation of mass. When we apply the Reynolds transport theorem, it helps us establish the continuity equation. Can anyone tell me what the continuity equation implies?

Exactly! We express this mathematically as the integral of density multiplied by velocity over the cross-sectional area. Can anyone remember what this represents?

That's correct! Let’s remember this with the acronym 'MASS'—Mass Always Stays Steady! Now, what happens if we have a variable density?

Yes! Very well summarized. Let’s recap: for a steady flow with constant density, the continuity equation helps us maintain that balance. Next up, let’s discuss specific applications.

Now that we understand mass conservation, let’s move to the conservation of linear momentum in fluid systems. Who can explain how momentum applies when a jet of water strikes a wall?

Great observation! This is where Newton's second law fits in, stating that force equals the change in momentum over time. Can someone recall how we mathematically express this?

Absolutely right! By integrating these quantities, we can relate the forces acting on the control volume to the mass and velocities involved. Let’s process this with a practical example next.

In turbo machinery, these conservation principles are vital. Can anyone think of where they see these concepts in real-life applications?

Exactly! In turbines, for instance, the flow of fluid is manipulated to maximize energy transfer, adhering to our conservation laws. Let’s break down an example involving a simplified turbine system.

You would balance the forces using our conservation equations, considering inflow and outflow rates along with pressure forces.

Precisely! It's crucial to connect this theory to practice. And remember, understanding these foundations helps in engineering applications. We will solidify these ideas in further exercises.