Gallium, as a safer liquid metal than mercury, has two major drawbacks: the surface rapidly acquires a layer of dull oxides, and when in contact with surfaces it tends to leave a thin layer behind. I've been doing some reading to see if I can work around this problem, and I came across this enjoyable article (which may be behind a paywall).
It's titled "The determination of the viscosity of liquid gallium over an extended range of temperature", by K. E. Spells, and it's from 1936. Clearly gallium was a fairly newly-available substance: "The first question which arose was that of choosing a method suitable for viscosity-determinations with a substance of which, owing to its high cost, only 2 or 3 cm3 were available", and "The only value for the density of liquid gallium given in the literature is for a temperature near its melting-point, but experiments have been made in the laboratory by W. H. Hoather to determine the density of gallium over a wide range of temperature, and my thanks are due to him for allowing me to use some of his results before publication."
The article goes into quite some detail on the construction of a viscometer that will work at 1100°C; in places it feels more like a (well-funded) instructable than the multi-million-dollarinstrumentationpapers I'm more used to reading. But they talk about the oxidation and the wetting issue.
The key point seems to be that if you can suppress the oxidation, no wetting happens. But suppressing the oxidation is not at all easy. They add a little dilute hydrochloric acid and find it is able to remove the oxide (but as I understand it, if you left it long, the hydrochloric acid would be consumed and the oxidation would resume). For the viscosity measurements, they need to remove the hydrochloric acid regardless, so they pumped all the air out of the vessel (which tidily evaporates the hydrochloric acid, and they confirm that the chlorides boil off when they heat the gallium). Normally in a viscometer you push the liquid around with gas pressure, so they tried filling the space above the gallium with inert gases. They tried carbon dioxide, nitrogen, hydrogen, and argon, and report "[n]o gas could be found [...] in which the behaviour of the gallium was satisfactory." I suspected carbon dioxide might be a problem, since magnesium is greedy enough for oxygen that it can pull it out of carbon dioxide, but I don't really understand what was happening with nitrogen or hydrogen, let alone argon. It would also be nice if they explained what was unsatisfactory about the gases — was there just a change in the meniscus, or was there still wetting? Oxidation?
In the end, they settled on vacuum, pushing the liquid around by tilting their apparatus (including the furnace), and they got nice measurements of the viscosity of gallium. They mention that these measurements span the largest temperature range of any liquid viscosity curve then available, and compare them to a theoretical model of viscosity, which does fairly well (perhaps coincidentally, the model is by the scientist listed as having communicated this to the journal, probably Spells' supervisor).
For my purposes, though, this is something of a setback; I had envisioned building a sealed little fountain under carbon dioxide, so that it wouldn't oxidize and therefore it wouldn't wet and one could watch it through the glass container. But if I can't prevent the container from being wetted by the liquid, it will shortly become opaque and I will have to open it to see what's going on, allowing oxidation. My initial experiment with carbon dioxide was not very impressive, but I'm not sure how effective my baking-soda/vinegar system was at flushing the oxygen out of the container (I'm not sure where I'd buy dry ice). If nitrogen would work, I could probably arrange something with somebody down the hall who has a great big tank of liquid nitrogen. I also saw a claim that it might be possible to use some commercially-availablesilane coatings to passivate the glass (though these experiments are not promising).