A Diffuse Interface Model for the Analysis of Propagating Bulges in Cylindrical Balloons

During the inflation of a cylindrical rubber balloon, it can be observed that the homogeneous cylindrical configuration quickly becomes unstable. A bulge is then formed while the pressure drops suddenly. Upon further inflation, the bulge propagates while the pressure remains constant. In earlier work, the value of the pressure at this plateau has been calculated based on Maxwell's construction for the coexistence of phases. In this talk, I will push the analogy between bulges in rubber balloons and phase transitions further. I will show that the details of the bulge formation and propagation can be captured accurately by a one-dimension model similar to the diffuse interface model introduced by van der Walls in the context of liquid-vapor phase transitions: in our model, the energy depends both on the tube radius and on its gradient. This model will be justified from a non-linear membrane model by a formal expansion. I will also compare numerical solutions of this model with solutions of the original non-linear membrane model, and show how our model can be used to make analytical predictions on the bifurcation loads and on the post-buckling behavior of the tube, while accounting for the finite tube length.