Rotating fluid, curl and suspended object rotation

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SUMMARY

The discussion centers on the relationship between the angular velocity of a suspended object in a rotating fluid and the fluid's curl. The fluid's velocity field is defined as F(x,y,z) = yi - xj, resulting in a curl of -2k. The user proposes that an infinitely light ball placed in this fluid would rotate at an angular velocity of -1 rad/s, but later concludes that the angular velocity of the ball is actually half of the fluid's curl, leading to a final angular velocity of -1 rad/s.

PREREQUISITES
  • Understanding of fluid dynamics concepts, specifically curl and velocity fields.
  • Familiarity with angular velocity and its mathematical representation.
  • Basic knowledge of vector calculus and its applications in physics.
  • Concept of drag forces in fluid mechanics.
NEXT STEPS
  • Study the mathematical derivation of curl in vector fields.
  • Explore the principles of angular momentum in rotating systems.
  • Learn about the effects of fluid drag on suspended objects in motion.
  • Investigate real-world applications of rotating fluids in engineering and physics.
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Students and professionals in physics, particularly those focused on fluid dynamics, mechanical engineers, and anyone interested in the behavior of objects in rotating fluids.

luca-deltodesco
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I'm trying to figure this out.

Say you have a cylinder of perfectly rotating fluid, so that it's velocity field is:
F(x,y,z) = yi - xj
which has curl -2k

assuming there is 'infinite' fluid drag and you have an 'infinitely' light ball which you place into the fluid at any point (let's say it's centre).

then how fast would the ball be rotating?

In my head I equate this to having a ball cemented into a rotating cylinder of concrete (or simply an object sat ontop of a spinning disc) and it would have an angular velocity of -1rad/s.

but at the same time, it would seem to me that the spinning object should have an angular velocity equal to the curl of the fluid at that point, and of course -1 =/= -2
 
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... okay nevermind:

tl;dr from wikipedia's 'intuitive interpretation' of curl:

ang. velocity of the ball is 'half' the curl >.>
 

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