Linear oscillations of axisymmetric viscous liquid bridges

Small amplitude free oscillations of axisymmetric capillary bridges are considered for varying values of the capillary Reynolds number C^-1 and the slenderness of the bridge L\Lambda. A semi-analytical method is presented that provides cheap and accurate results for arbitrary values of C^-1 and L\Lambda; several asymptotic limits (namely, C << 1, C >> 1, L << 1  and  |p-L| << 1C\ll 1, C\gg1, \Lambda\ll 1 \ \rm{and} \ |\pi-\Lambda|\ll 1 ) are considered in some detail, and the associated approximate results are checked. A fairly complete picture of the (fairly complex) spectrum of the linear problem is obtained for varying values of C and L\Lambda. Two kinds of normal modes, called capillary and hydrodynamic respectively, are almost always clearly identified, the former being associated with free surface deformation and the latter, only with the internal flow field; when C is small the damping rate associated with both kind of modes is comparable, and the hydrodynamic ones explain the appearance of secondary (steady or slowly-varying) streaming flows.