Molten alumina is a bit weird, I'm not sure whether it's thick and sticky or thin and sticky. I guessed the first, the latter seemed more in tune with what happened - but I haven't worked it out. Getting the tube to blob and seal over was unexpectedly hard, or rather it didn't go as expected - but by no means impossible.
To stop the tube cracking, get an insulating firebrick, grade 26 or better (grade 23 won't do), drill a deep hole a couple mm larger than the alumina tube, and insert the tube so only a cm or two are showing. Surround the tube end at a distance of an inch or so with some extra bits of firebrick to form a hearth. It helps to be able to turn/rotate the tube/hearth.
Oxy/MAAP (real MAPP, not propylene), 1/4 by 1/8 Omegatite 450 99%+ impervious alumina thermocouple protection tube, j26 firebrick.
For reference, if anyone is interested.
Now, how do you weld miniscule 0.005" 0.0125mm tungsten/rhenium thermocouple wires together?
Anyone know of a source of tungsten tube, about 3mm OD, in 6" lengths? They make 1" lengths for fishermen...