3.1.9. Divergence Theorem

Try these first

1. 1.

   What does the divergence of a vector field measure?

   Hint

   Think about sources and sinks.

2. 2.

   What is mass flux?

   Hint

   Consider how density and velocity combine.

3. 3.

   What does divergence tell us about fluid flow?

   • Indicates fluid density change
   • Represents temperature
   • Measures pressure

   Hint

   Think about how fluid expands.

4. 4.

   True or False: The Divergence Theorem can only be applied to incompressible flows.

   • True
   • False

   Hint

   Consider the versatility of the theorem.

5. 5.

   Consider a cylindrical tank with varying cross-section. Derive the mass conservation equation by applying the Divergence Theorem, considering inflows and outflows.

   Hint

   Break down the regional inflow/outflow using the geometry of the cylinder.

6. 6.

   A fluid is flowing through a pipe with varying diameter. Analyze how the Divergence Theorem applies and derive the implications for velocity and density changes.

   Hint

   Use the principle of volume and area relation in flow.