Spatial modes of capillary jets, with application to surface stimulation

Surface stimulation of any physical origin (electrohydrodynamic, thermocapillary, etc.) has the goal of generating localized perturbations on the free surface or the velocity field of a capillary jet. Among these perturbations, only the axisymmetric ones are determinant for the jet breakup. Often, the stimulation is weak enough for a linear model to be applicable. Then, the stimulation can be described by means of the Green functions for stresses, both normal and tangential to the interface, the calculations of which are, in addition, uncoupled from the hydrodynamic variables. If a harmonic forcing is applied, these Green functions are combinations of the spatial modes whose associated poles lie inside the appropriate integration contour of the complex wavenumber plane. This is the motivation for a comprehensive enumeration and description of the spatial modes, which has not been done up to now. Modes familiar from a temporal analysis, the dominant and subdominant capillary modes and the hydrodynamic modes, are present, along with modes specific to a spatial analysis. Most of the latter have already been mentioned in the literature for inviscid jets, but not analysed. A mode not previously found is reported. In addition, a description of the velocity field associated with each mode is provided, as a tool to understand their physical origin and behaviour. The relative importance of each mode in both normal- and tangential-stress stimulations is discussed. Finally, the well-known merging of poles below a critical jet velocity, leading to absolute instability, is analysed in the light of the modal description.