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Understanding Curl and Divergence of Vector Fields in Fluid Dynamics

This section reviews the concepts of curl and divergence in vector fields, particularly within fluid dynamics. We explore how a velocity vector field, represented by F, behaves in terms of rotation and divergence. The example provided illustrates a diverging vector field that is irrotational, as well as a rotating field with non-zero curl. Differential identities involving curl and divergence are also discussed, highlighting their importance in computing vector field properties. These concepts are essential for understanding fluid behavior and dynamics.

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Understanding Curl and Divergence of Vector Fields in Fluid Dynamics

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  1. MA 242.003 • Day 57 – April 8, 2013 • Section 13.5: • Review Curl of a vector field • Divergence of a vector field

  2. Section 13.5Curl of a vector field

  3. “A way to REMEMBER this formula”

  4. “A way to REMEMBER this formula”

  5. “A way to REMEMBER this formula”

  6. “A way to REMEMBER this formula”

  7. “A way to REMEMBER this formula”

  8. “A way to REMEMBER this formula”

  9. (continuation of example)

  10. Let F represent the velocity vector field of a fluid. What we find is the following: Example: F = <x,y,z> is diverging but not rotating curl F = 0

  11. All of these velocity vector fields are ROTATING. What we find is the following: F is irrotational at P. Example: F = <x,y,z> is diverging but not rotating curl F = 0

  12. All of these velocity vector fields are ROTATING. What we find is the following:

  13. All of these velocity vector fields are ROTATING. What we find is the following: Example: F = <-y,x,0> has non-zero curl everywhere! curl F = <0,0,2>

  14. (See Maple worksheet for the calculation)

  15. Differential Identity involving curl

  16. Differential Identity involving curl Recall from the section on partial derivatives: We will need this result in computing the “curl of the gradient of f”

  17. The Divergence of a vector field

  18. The Divergence of a vector field

  19. The Divergence of a vector field

  20. The Divergence of a vector field

  21. The Divergence of a vector field Then div F can be written symbolically as:

  22. The Divergence of a vector field Then div F can be written symbolically as:

  23. The Divergence of a vector field

  24. The Divergence of a vector field

  25. So the vector field

  26. So the vector field Is incompressible

  27. So the vector field Is incompressible However the vector field

  28. So the vector field Is incompressible However the vector field Is NOT – it is diverging!

  29. Differential Identity involving div

  30. Differential Identity involving div

  31. Differential Identity involving div Proof:

  32. (continuation of proof)

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