Hydrodynamic stability of high-viscosity cylindrical liquid bridges

The hydrodynamic stability of cylindrical liquid bridges is studied in the asymptotic limit when viscosity is paramount. The problem has been solved analytically in terms of a series of biorthogonal eigenfunctions (the Papkovich–Fadle functions), by using an orthogonalization method devised by Smith [Aust. J. Sci. Res. 5, 227 (1952)] for solving elasticity problems. It is found that the growth factor is real for all the values of the slenderness, leading to pure damping or growing motions for the stable or the unstable modes, respectively. This analysis is also valid when the viscous to capillary forces ratio is of order unity and the slenderness is close to the static stability limit. The results are compared with previous ones obtained from Rayleigh’s viscous model for jets or from Cosserat’s one-dimensional model, showing a very good agreement for long columns.