Capillary Stability of a Hollow Inviscid Cylinder

The application of Rayleigh's treatment of capillary stability to the case of an inviscid liquid bounded by two concentric circular cylindrical surfaces leads to a characteristic equation from which the following predictions can be made. The system is stable for disturbances that do not have cylindrical symmetry. It is stable (unstable) for disturbances with cylindrical symmetry when the wavelength of the disturbance in axial direction is less (greater) than the circumference of the inner surface. There are always two modes of deformation. In the special case of a thin-walled cylinder these two modes are symmetric and antisymmetric, respectively. When the wavelength of the disturbance is varied in such a manner that there is a transition from stable to unstable conditions, then at the same time there is a transition from symmetric to antisymmetric mode or vice versa. It is shown, further, that when one mode is stable and the other unstable, the stable mode is a deformation for which the total surface area of the system is increased by the disturbance, while the unstable mode corresponds to decreased surface area.