A one-dimensional analysis of slender axisymmetric viscous liquid jets is considered. A set of one-dimensional models is derived by substituting a truncated Taylor series in the radial coordinate into the Navier–Stokes equations and boundary conditions at the interface. The relative error, defined as the order of magnitude of the neglected terms divided by the order of the retained ones, is small if the dimensionless wave number k is small enough. The Lee slice model is generalized to take into account viscosity, the relative error being k2. A new model having a parabolic radial dependence for the axial velocity is developed, with a relative error k4. The Cosserat model comes from the introduction of the mean axial velocity into the previous one, but an inconsistency arises from neglecting some viscous terms of the same order as those retained. A new model for the mean axial velocity is derived. It conserves the same inertial contribution but avoids the above-mentioned problem by estimating the involved terms instead of neglecting them. Therefore the relative error is k4 for any value of viscosity. Linear stability analysis is performed for the infinite jet. Results are compared with the exact linear solution given by Lord Rayleigh. The main features predicted in the derivation of the one-dimensional models manifest themselves in the linear case.

The normal-mode linear analysis of an axisymmetric infinite capillary jet is generalized to account for arbitrary initial conditions. An exhaustive study of the dispersion relation reveals the parametric behaviour of all eigenvalues and their corresponding normal modes. The two capillary modes (dominant and subdominant) are found to be necessary and sufficient to describe any possible non-recirculating initial conditions. An infinite set of other modes accounts for initial conditions with recirculating velocity field. The predictions of the normal-mode analysis are contrasted against previous computations of the initial-value problem, previous experiments, and our own one-dimensional numerical simulations. Contrary to the claim of some authors, the normal-mode analysis accurately predicts the initial transient with non-exponential growth of the disturbance amplitude observed in previous works. Simple and accurate formulae for the duration of the initial transient are deduced, with emphasis on improving the growth-rate measurement.

The growth of perturbations on a capillary jet issuing from a circular nozzle in the Rayleigh regime is experimentally investigated. Electrohydrodynamic sinusoidal stimulation is employed to this end, along with two independent methods to obtain growth rates of the linear regime with the best accuracy so far. The first method exploits the correlation between the stimulation voltage and the breakup time measured with the help of stroboscopic images of the jet. The second method is an analysis of the spatial evolution of perturbations through a local jet-shadow-width photometry, with careful avoidance of the initial transient and the final nonlinear stages. Experiments conducted with ink allow the application of both methods, as the liquid is opaque. They give consistent results, with very small statistical errors, with respect to the expected theoretical dispersion relation, once the dynamic surface tension is adjusted. The adjusted value is in accordance with an estimate made from drop-dynamics experiments also reported here. By dealing with a simpler liquid (aqueous solution of NaNO3), we are able to compare results from the first method against the theoretical predictions without adjustment of any parameter. The agreement is again excellent. Possible sources of systematic errors in this kind of measurements are identified and procedures for avoiding them are designed.

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AbstractSurface 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.