a proper verb that describe the formation of Taylor cone in polymer science

The core solution and shell solution are combined. They combine through contact and are coaxially ejected. (I don't believe they are actively mixed, so to speak of mixing would be somewhat misleading.)

If the two solutions don't suitably adhere or aren't effectively joined or if the less viscous core solution isn't successfully entrained with or attached to or integrated with or encapsulated by or supported by the more viscous shell solution, then the former might detach and drip.

(Note that these individual descriptions—a lack of adhesion, a lack of entrainment, a lack of encapsulation—describe different defect modes; select the one that best fits your intended meaning. Most broadly, you can say that they don't usefully combine.)

In that case, the shell solution, apparatus, and/or procedure cannot accommodate that particular core solution.

I simply googled formation of Taylor cone in polymer science and got this:

"...this causes dripping of the two solutions followed by an intermittent jet from the sheath with an occasional incorporation of the core."

https://www.sciencedirect.com/topics/engineering/taylor-cone

I have no idea what a Taylor cone is but I suggest that this is a suitable way to discover the usual terminology. Another way is to use Google Scholar with a well-crafted search term.

...a regular Taylor cone must form from the two solutions,

Therefore, the core solution failed to resolve with the shell solution to form a Taylor cone. As a result the core solution started dripping.