Thanks vpl, IRstuff and msquared48.
Crane was helpful in getting to grips with what I should be looking for and I have made progress since starting to read it.
However, while I am getting closer to solving the problem, I think the long tube with an diameter of less than 1mm is affecting my results. I have a hand pump which feeds into the long tube. The tube widens out into the balloon. I have called it a balloon but it does not display the properties of a balloon. It is already expanded into shape and then a vacuum is pulled on the whole system.
I would like to initially solve the problem assuming that there are no forces acting on the outside other than atmospheric pressure, and the balloon does not resist inflation. When I get this much done I will then have to look at the case where the balloon begins to expand as the pressure increases.