5.4.1. Basic Concepts

Try these first

1. 1.

   What is a stream function?

   Hint

   Think about how it connects to velocity.

2. 2.

   How does the flow velocity relate to the stream function?

   Hint

   Consider the relationship with x and y.

3. 3.

   What is the primary purpose of a stream function in fluid mechanics?

   • To visualize flow only
   • To derive velocity components
   • To represent pressure changes

   Hint

   Think about its role in analyzing flow.

4. 4.

   True or False: Stream functions can be defined only for incompressible flows.

   • True
   • False

   Hint

   Consider the broader applicability of this tool.

5. 5.

   Derive the u and v velocity components for the stream function φ = x^2 + e^y, and analyze the direction of flow.

   Hint

   Use partial derivatives and recall the relationship between velocities and stream functions.

6. 6.

   Consider a rotating cylinder with stream function φ = r^2 sin(θ) in cylindrical coordinates. Derive v_r and v_θ and discuss flow behavior characteristics.

   Hint

   Recall cylindrical coordinate system relationships for deriving velocities.