Subject: Re: Re: why do particles move towards the center of the cup instead of outer rim?

Date: Tue Sep 30 14:54:17 1997
Posted by Ricky J. Sethi
Position: PhD

Hi John,

Thanks for your question. This is an interesting question that you can also experiment with at home to see the physics in action! Okay, so you ask why do the tea leaves collect at the center of the cup, rather than the outer rim. I'm assuming here that this is after stirring the tea with a spoon. You've probably studied centrifugal force in school and that says that things should be pushed to the outside so why the heck do the leaves collect at the center??? Well, the answer brings fluid dynamics into play.

Looking at the problem one way, you can think of a whirlpool being established in the cup. And the whirlpool, just like the one established when you let water drain away in a tub, "sucks" things down to the middle. So although the moving water does experience centrifugal force, the whirlpool effects "overpower" the others on the tea leaves and pull them toward the center. You can see this by using a cereal bowl and filling it with water. Then, plop some leaves down on it in a random fashion. After some movement (because of the tiny water currents created by plopping the leaves into the water), they settle down in a more or less random distribution. Now, if you spin the whole cereal bowl (carefully!), the leaves will, indeed, move to the outer rim because the whole bowl, and everything in it, is moving in the same way (this is true as long as you don't spin it too fast and only have uniform rotation; otherwise you might introduce a vortex and those effects then take over, as described below). But, if you stop the bowl and allow the water to come to rest, and then use a spoon to set up a whirlpool (i.e., stir it), you'll see the fluid dynamic effects take over again and pull the leaves into the center.

Now for a little more physics. What happens when you stir the water? Let's assume that you've started stirring the water really well and have established a whirlpool, or vortex (as opposed to the uniform rotational field of flow, above; btw, you can see the circular streamlines and velocity for a vortex and get a very nice introduction to fluid dynamics in Ch.18 of Halliday and Resnick's Physics). As the water starts to circle, it is thrown to the outside, due to centrifugal force (to be more accurate, the force that it experiences is centripetal force=(mv^2)/r since centrifugal force is a so-called fictitious force). As it's thrown outside, it starts to move faster (when things move in a circle, the things that are nearer the outside move faster than things near the center; you can see this if you have a "old-fashioned" record player by putting two coins on it, one near the center and one near the outer edge and trace how long it takes them to complete the lengths of their paths). So the circling water near the outer rim is moving very fast and experiences bigger forces (since the force is proportional to velocity). But the water near the middle is moving slower because it's being dragged along the bottom and so ends up slowing down much more. Since the water near the bottom is moving so slowly, it exhibits a much lower centrifugal force and so there is little to no centrifugal force near the bottom (or middle). This is why you have a vortex: the water near the top is pushed out more (since there's higher centrifugal force there) and the water near the bottom is hardly pushed outward at all (since there's lower centrifugal force near the bottom to push it out).

Ok, now we have a nice vortex set up. So what about the tea leaves? Well, the water near the outside is deeper than the water near the center (keep that picture of a vortex or a whirlpool in your mind for this). Since the water is higher (deeper) near the outside, the pressure on the water near the outer edge is higher than the pressure on the shallower water near the middle of the vortex (pressure is directly related to the height of the water; that's why water tanks are kept on the roofs of buildings so that there'll be sufficient water pressure to give you a shower rather than a drizzle). This pressure difference then forces the water (and whatever particles might be in it) to flow along the pressure gradient from the outside, or top (higher pressure), to the middle, or bottom (lower pressure). If the water near the bottom (or middle of the vortex) were moving, the centrifugal force might be enough to counteract the pressure difference effects but since the bottom water is hardly moving, this secondary circulation is set up. And this secondary circulation takes any particles in the water along with it. And since the water slows down near the middle (or bottom) the particles are deposited (sedimented) at the bottom.

As the whirlpool starts to die out, it becomes "shallower" (i.e., it's not as deep) and so the outside water (and particles) are simply pushed near the middle and deposited there as the whirlpool continues it's death throes, eventually ending with most of the tea leaves near the center. I hope this helps. It's much easier to see this if you try it out at home and it makes for a cheap experiment, too (all you need is a cereal bowl and some tea leaves). Plus, it'll be easier to see the physics of what's going on when you do the experiment and then go back and read these ramblings... I mean, this explanation. Good luck and please feel free to contact me if you'd like something clarified further.

Regards,

Rick.