Drinking (tea) with mathematicians

This morning, I made myself some tea in one of the matching mugs from my dinner set, waited an inordinately long time for it to start to cool down, and then downed the rest of it in a few gulps because it was quickly becoming too cold to drink. Not the most satisfying tea-drinking experience, but that’s what I get (I think) for throwing narrow cylinders.

So I started to wonder what shape of mug, made out of clay of constant thickness, would be ideal for drinking hot beverages. Specifically: I would like to pour myself some tea, wait a fixed time t for the surface of the tea to cool down to a drinkable temperature before beginning to sip, and then sip at a fixed (and preferably constant) rate until the tea is gone, with the tea remaining at a constant temperature.

It seems like this should be doable; at the very least, I don’t think it’s not a ridiculous thing to ask for: the surface of the tea cools much faster than the tea below, and so there should be a way to sip tea from the appropriate vessel in such a way that each sip exposes another layer of tea that cools down to the temperature of the previous layer, just as I’m drinking it.

It also seems clear that the top should be wider than the bottom, as the tea at the bottom will not be completely insulated by the tea above. Also, the heat transfer through the mug cannot be ignored: the specific heat capacity of clay is around a third of that of water, and only a third more than that of air.

…that’s as far as I got: I don’t have the physics background to set up this problem, though I spoke with someone who did, and he came up with some rather grisly equations that I don’t have the calculus background to solve. I also don’t know what simplifications could reasonably be made. But if anyone does, and can come up with a reasonable shape of mug that is conducive to tea-drinking, I’ll make you such a mug.

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