Q. What are the forces exerted by the expanding bubble?

There are some good technology books that cover the physical, optical, electrical, thermal, etc. properties of glass and I am attaching a list of those to this email.  If these titles aren't available at your local library then you can request a copy through your library's interlibrary loan department.  If we have a second copy, we will gladly loan that at no charge to your library, for your use.

I'm not a scientist, mathematician, or a glassblower, so I did ask this question of a scientist from Corning, Inc. Here is his description of the forces at work:

"Bubbles form, generally, from the balance between the gas pressure inside the bubble, pushing outward, and the surface tension of the glass, which tries to prevent the bubble from expanding.  When the amount of pressure overcomes the surface tension the bubble grows.  

The shape that growing bubble takes depends upon how uniformly the molten glass expands in response to that pressure.  Hotter glass is softer (scientists say it has a “lower viscosity”) which means it is easier to push around.  Colder glass is stiffer (higher viscosity).  If the glass surrounding the bubble is hotter and softer in one place than another the bubble will push outward more into that hotter glass.

Gaffers use this behavior to make different shapes on a blow pipe: using tools like the wet, wooden blocks a gather can be set to a uniform temperature so the bubble grows into a sphere.  By specifically cooling the sides of the bubble (with a wet newspaper pad, for example) it expand toward the end of the gather, making a longer bubble.  By cooling the end (perhaps with a wooden paddle) the bubble will spread to the sides.  If the bubble is expanded without taking care to control the temperature, so that a particularly soft area expands too far, the bubble will stretch the glass until it is so thin that it will burst through the wall of the gather."

What factors control the final bubble shape? Contributing factors, among others, may be, according to the source linked below from Corning, Inc.:

temperature

viscosity of the glass

surface tension of the glass

internal pressure of the gas in the bubble

initial shape of the bubble.

https://www.math.wpi.edu/MPI2008/Corning/corningMPI2008.pdf

Many of these texts on the attached bibliography provide mathematical formulas for the process of forming glass.  I hope this helps! Please let me know if I missed the mark with your question. I wasn't entirely sure what you were referring to by "popping the bubble".  If I did misconstrue your question, I'll be glad to get back to the Corning scientist with any questions.