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## Wednesday, September 2, 2015

### How to Stir a Cup of Tea

You have just made yourself a cup of tea. On top of this cup you have poured some milk. Now what?

Both tea and milk are themselves admixtures. Tea consists out of water molecules, mostly, and various more complex molecules such as caffeine, which give it its flavor and effect. Similarly, milk consists out of, again, water molecules, and various complex molecules, such as lipids, which give it its flavor and effect.

Even if the original black tea and milk were perfectly mixed, the contents of your cup will be inhomogeneous: different molecules will have different concentration at different points in the cup. The milk molecules will tend to be concentrated on top and the tea molecules on bottom.

But you'd like the liquid in your cup to be approximately homogeneous because that way the first sip from the top of cup and the last sip from the bottom will have all have equal concentrations of the various molecules, approximating the perfect ratios you set by choosing the ratio of tea of milk initially. (If there are desirable chemical reactions between the molecules of your two original admixtures, that is an additional reason to seek homogeneity for in a homogeneous liquid the rate of those reactions will be higher. But that is not the case, I believe, for our example.)

Given time, and a temperature above absolute zero, statistical mechanics will take care of the problem for us. For it is one of its laws that, over time, systems will tend to go to configurations with larger state spaces and the state space of a homogeneous liquid is larger than that of an inhomogeneous liquid. That is just what we call diffusion.

Now, even given infinite time, diffusion will not achieve perfect homogeneity in the presence of a gravitational gradient (such as the one we experience at rest with respect to the surface of the earth, which is where we generally prefer to drink our tea). Molecules with higher specific density will tend to sink to the bottom of the gradient, pushing molecules with lower specific density to the top. But, unless those differences in density are large or the gradient very strong, diffusion will ordinarily achieve a reasonable approximation of homogeneity. However, this explains the commonly observed phenomenon that some perfectly mixed liquids will, when left alone over sufficient periods, tend to separate themselves again.

The problem with diffusion is that it takes time, particularly over macroscopic distances (like the height of our cup) and we are impatient. So we stir the cup with a spoon in hope of achieving homogeneity faster. But perfect conventional stirring—spinning the spoon through the cup at constant angular velocity and direction—is worse than useless for this purpose.

To see why it is useless, consider your cup from a frame of reference rotating along with the contents of the cup. In this frame of reference the spoon and the contents are at perfect rest! And at rest, all that happens is diffusion, the same as if you did nothing. So perfect conventional stirring will not speed up homogenization one bit.

To see why it is worse than useless, remember that in the rotating frame you have induced (everywhere except on the axis of rotation) an additional horizontal gravitational gradient on top of the vertical gradient provided by earth's gravity. As the direction of these two gradients is perpendicular, the magnitude of their sum will be even larger than that of the original vertical gradient. So fast, perfect stirring will tend to dehomogenize your tea, rather than to homogenize it. That is why centrifuges—really no more than very fast perfect stirring—can be used to separate the components of liquids.

Now, your conventional stirring is probably not quite perfect. There will be some slight variations in speed, in particular as you start and stop it. There will be some friction between the liquid and the interior surface of the cup and the surface of the spoon. Both of these will induce some complicated flows within the liquid, called turbulence, which will bring volumes with different concentrations closer together and speed up diffusion. Significantly, your stirring implement remains of fixed orientation with respect to you and hence will rotate with respect to the frame of liquid. If your stirring implement is not symmetrical (and a spoon isn't) that will cause more turbulence. (However, a symmetrical stirrer, such as a stirring stick won't and is hence the worst implement for its stated purpose.)

But while all of these factors help, any approximation to perfect stirring, the worst possible method, is very unlikely to be the best method.

So how should one stir one's tea? So as to create maximum turbulence (consistent with not spilling the tea) and hence as different from perfect conventional stirring as possible. Alter the direction of the stirring as much as conveniently possible. Reverse direction after every rotation. And be sure not to stir your tea at centrifuge speed.