Solving Smoke Ring Mysteries: Exploring Fluid Motion

[AttilaTheNun]

Hi all,

I'm trying to answer the following questions related to a project on smoke rings.

We are in an introductory class, so this project is more for fun and just applying the little we have learned about point, rigid body, and fluid motion to something less understood like vortex rings.

Essentially we have contsructed a smoke ring device that, guessed it, blows smoke rings. I have already experimentally shown that by blowing two different sized rings, you can find that the larger ring moves at a slower pace than the smaller (radius) ring.

However, I am having trouble answering the following questions:


8. Explain mathematically why the speed of propagation of the
ring should vary roughly as the reciprocal of its radius.

9. Show (from calculations) that the air in the center of the
ring is sucked through at a speed of 4 times the propagation
of the ring itself (it's like some weird flying, sucking thing!)


Any help is greatly appreciated, even just a quick explanation off the top of your head.

thanks

 

[mark726]

Hi there,

I would be happy to help answer your questions about smoke rings. First, let's start with the concept of vortex rings. These are circular patterns of rotating fluid, such as smoke or water, that are created when a fluid is pushed through a small opening or nozzle. The motion of the fluid and the resulting vortex ring is governed by the principles of fluid dynamics, specifically the conservation of momentum and energy.

Now, let's address your first question about the relationship between the speed of propagation and the radius of the ring. As you have observed, the larger ring moves at a slower pace than the smaller ring. This is because of the conservation of momentum - as the ring expands, its mass increases, which requires more force to maintain the same speed. Mathematically, this can be represented by the equation v = rω, where v is the speed of propagation, r is the radius of the ring, and ω is the angular velocity. As the radius increases, the angular velocity decreases, resulting in a slower speed of propagation.

For your second question, let's consider the air in the center of the ring. As the ring expands, it creates a low pressure area in the center due to the conservation of energy. This low pressure area acts as a suction, pulling air towards it at a speed of 4 times the propagation speed of the ring. This can be calculated by using the Bernoulli's equation, which describes the relationship between pressure, velocity, and elevation in a fluid.

I hope this helps to answer your questions and provide a better understanding of the physics behind smoke rings. Keep exploring and experimenting - that's what science is all about!