5.5. Derivation of Stream Functions

Try these first

1. 1.

   What is the definition of a stream function?

   Hint

   Think of how it relates to velocity and flow.

2. 2.

   Define incompressible flow.

   Hint

   Consider how fluids behave under certain conditions.

3. 3.

   What does a stream function primarily do?

   • Select components of velocity
   • Visualize streamline patterns
   • Calculate pressure gradients

   Hint

   Remember how streamlines relate to velocity.

4. 4.

   True or False: Incompressible flow maintains constant density.

   • True
   • False

   Hint

   Think about how fluids behave in different scenarios.

5. 5.

   Analyze a two-dimensional incompressible flow field defined by the stream function ψ = x²y - y³. Derive the velocity components and verify the continuity equation.

   Hint

   Use partial differentiation and remember continuity principles.

6. 6.

   Evaluate the flow around a rotating cylinder modeled by the stream function ψ = r²θ. Can you describe the behavior of the flow and identify any critical points?

   Hint

   Visualize fluid paths around a moving object.