It is interesting that you say this, because after spending a lot of time and experimenting with various coil builds, I kept looking at a motor-less magnetic stirrer on the market that just had me wildly curious as to how they pulled it off, so I bought it ... and took it apart ... and was kind of surprised to see that at its core, it used an arduino chip to control everything, though the surrounding electronics was far more vast than I ever thought would be necessary and as it turns out, I was able to use just the coils that they put in the device with my own code using the Teensy 4.1 and I was able to get the same results that they provided with the unit ... albeit without the liquid crystal display and their very accurate ability to select an RPM value that spun the pill at that speed. Also, the problem with using a highly viscous liquid in the beaker produced the same problem for my driver as it did for theirs... it lost magnetic coupling at roughly the same speed which was slow enough that I could never create a vortex with either my driver or theirs.
Anyway, what is especially interesting about your comment, is that in the commercial product, they in fact use steel cores and they then attach to the cores, a steel plate that focuses the magnetic field up to the center of where the pill will spin ... here are some pictures...
Here is the main board that came with the unit
I'm assuming that the JP1 header that has no pins, could be used to get at the Arduino which makes me wonder if there is a way to get the code from it and decompile it somehow...
... and in my search to build this thing before seeing this design, no one ever mentioned the use of steel cores, so I defaulted to ferrite cores and never even thought of using a means of focusing the magnetic field closer to the pill as this design does and as you suggested. Also, their coils were MUCH larger than the ones I was experimenting with, which also makes sense in terms of generating a stronger field.
Ultimately, I think that if I built even larger coils, and used three sets of two coils per set ... each pair sitting opposite the other (one North the other South) and used three drivers for the sets and if I worked out the code just right, I think I could pull this off and get a vortex in a viscous solution by doing what I mentioned where the "center" coil pairs would have the strongest voltage applied to them at any point in time, then the next ahead of that pair (or the "leading" coil pair) would be primed with a voltage lower than the core and the tailing pair would have a voltage even lower than the leading which in my view would generate a magnetic field where each end of the pill would have a very strong and stable magnetic field that would make it hard for the pill to decouple from ... assuming I could keep that consistent voltage applied throughout the totality of the spin... but I haven't really thought that through yet nor even remotely tried to experiment with that idea... for now its just a thought.
For the time being, however, I'm going to try using different "stir bars" as there are many different styles out there and the two that look the most promising to me are these:
Those seem like they would offer the best stability inside a thick solution...
Where I've been using one that looks just like this
And it quickly "spins out" in this solution.
But good call on the steel cores and the guidance of the magnetic field as that seems to be the best way to accomplish this task.