Curl vs. circular orbit

Homework Help
1

What is the difference between a fluid which is rotating and one which in a circular orbit? Provide an example of both.

Would a vortex flow (u_r = 0, u_theta = gamma/(2 pi r) be an example of a rotating flow, circular orbit flow, both or neither?

2

This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
To address your question about the difference between a fluid that is rotating and one that is in a circular orbit:

  1. Rotating Fluid: A fluid is considered to be rotating if it has a non-zero vorticity (\nabla\times \vec{u}\neq 0), which means that the fluid elements are undergoing angular motion about an axis. An example of this is the solid-body rotation of oil in a torque converter when the engine and transmission RPMs match.

  2. Circular Orbit Fluid: A fluid in a circular orbit is moving along a circular path but does not necessarily have any local rotation. For instance, the fluid circling a drain in a sink or toilet usually moves in approximately a circular path but the fluid is typically not rotating.

Regarding the vortex flow described by the equation u_r = 0 and u_\theta = \frac{\Gamma}{2\pi r}, this represents an irrotational vortex. In this case, the flow is an irrotational flow with no vorticity – it is not rotating. The flow is irrotational everywhere except at the origin, where there is a point singularity and vorticity is infinite.

If you have any further questions or need clarification on specific points, feel free to ask!

In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:

Textbook sections:

  1. Chapter 9 Fundamentals: Curl and Vorticity (RELEVANT)
  2. Chapter 9.4: Finding Analytical Solutions for \psi and \phi Part 2 (RELEVANT)
  3. Chapter 9 Fundamentals: Circulation (RELEVANT)
  4. Chapter 9 Fundamentals: Magnus effect (RELEVANT)
  5. Chapter 9.4: Finding Analytical Solutions for \psi and \phi Part 1 (RELEVANT)
  6. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 2 (RELEVANT)
  7. Appendix E.1: Velocity gradients (RELEVANT)
  8. Chapter 5.1: Inviscid Incompressible Flow (RELEVANT)
  9. Appendix B.4: Vector Calculus Part 3 (RELEVANT)
  10. Chapter 5 Fundamentals: Eulerian and Lagrangian specification of fluid flow (MARGINALLY RELEVANT)

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