18.13. Use in Fluid Flow – Laplace’s Equation

Try these first

1. 1.

   What does Laplace's equation describe?

   Hint

   Think about the kinds of flow it relates to.

2. 2.

   What does separation of variables do?

   Hint

   Recall the method we discussed in class.

3. 3.

   What is the form of Laplace’s equation?

   • ∂²ϕ/∂x² + ∂²ϕ/∂y² = 0
   • ∂²u/∂t² - c²∂²u/∂x² = 0
   • ∂u/∂t + ∂u/∂x = 0

   Hint

   Focus on the variables and their second derivatives.

4. 4.

   True or False: Neumann boundary conditions specify the potential value at the boundary.

   • True
   • False

   Hint

   Think about what is being controlled at the boundary.

5. 5.

   Given a rectangular domain with the following boundary conditions, set up Laplace's equation using separation of variables.

   Hint

   Recall how specific boundary values alter the separation.

6. 6.

   Analyze a fluid field behind a dam. How would you apply Laplace’s equation with respect to provided elevation head conditions?

   Hint

   Think about how potential translates to certain setups in dam engineering.