Instability Of An Electrified Coaxial Jet

Coaxial electrospraying and coaxial electrispinning are new effective methods to produce micro-capsules and composite nano-fibres. They are associated closely with the instability mechanism of the coaxial liquid jet under electric fields. However, to our knowledge little has been studied on the instability behaviors of the coaxial jet under electric fields.Using the classical temporal instability analysis method, the instability of an inviscid coaxial jet under a radial and an axial electric field, respectively, as well as the axisymmetric instability of a viscous coaxial jet under a radial electric field in the outer driving case, is studied for the first time. The corresponding dispersion relations are derived and written in explicit analytical forms, and the eigenvalues are computed using numerical methods. The effects of the electric field, electrical conductivity, electrical permittivity and the other important parameters in hydrodynamics on the instability of the coaxial jet are studied. In the inviscid case, three different unstable modes, i.e. the para-sinuous mode, the para-varicose mode and the transitional mode, are found in the Rayleigh regime. In general, the radial electric field has a destabilizing effect on the para-sinuous mode and the para-varicose mode, while the axial electric field has a stabilizing effect on these two modes. Moreover, the axial electric field has a destabilizing effect on the transitional mode. According to the results of the outer driving and inner driving cases as well as of the axial electric field case, it is found that the electrical conductivity and permittivity of the liquids have a complicated and profound effect on the behavior of the coaxial jet. In the viscous case, there exist only two unstable modes, i.e. the para-sinuous and para-varicose modes. Liquid viscosity has a general suppression effect on the unstable modes. In most situations, the para-sinuous mode is the most unstable mode, indicating that the coaxial jet breaks up into composite droplets ultimately, like in coaxial electrospraying experiments. In the viscous case, the large and small relative electrical relaxation time limit cases are derived and calculated. Using the Chebyshev spectral collocation method, the instability of the coaxial viscous jet under a radial electric field as well as under the action of both radial and axial electric fields is calculated numerically for the outer driving case. The temporal growth rates for the axisymmetric and non-axisymmetric instabilities are calculated and compared with each other, and the effect of electric fields on them is studied. The calculation results show that for the non-axisymmetric instability there exists only one unstable mode, i.e. the non-axisymmetric para-sinuous mode, which is quite different from the axisymmetric case. Furthermore, this non-axisymmetric para-sinuous mode is induced by the radial electric field. In the long wavelength region, the non-axisymmetric para-sinuous mode is less stable than the axisymmetric para-sinuous mode, indicating that the non-axisymmetric mode is predominant. The radial electric field destabilizes the axisymmetric and non-axisymmetric modes to certain extent. However, the axial electric field stabilizes them to different extent. It is found that the non-axisymmetric unstable mode becomes dominant in the case of high viscosity or small surface tension, predicting the easy realization of coaxial electrospinning in experiments.The comparison of the experimental and theoretical results of the single viscous liquid jet is performed. The result indicates that instability analysis can predict the instability mode of jet and the effect of electric field on the transition of axisymmetric instability to non-axisymmetric instability qualitatively. However, quantitatively, theory coincides well with experiments only in the highly viscous case. In the slightly viscous case, instability analysis fails to predict accurately the most possible wavelength of the non-axisymmetric mode. In addition, the absolute/convective instability is studied, and the governing region in the parameter plane is declared. It is shown that the absolute instability may become predominant when the electric field or the surface tension is sufficiently large.The initial-value problem in electrohydrodynamics is studied for the first time. With the aid of the Laplace transformation, an analytical solution for the motion of the plane standing wave in the presence of free surface charge is obtained. Moreover, the motion of the cylindrical standing wave in the presence of free surface charge is solved using semi-analytical method. On the other hand, the energy growth of the plane standing wave is calculated according to the nonmodal stability theory. The phenomenon of energy transient growth is discovered. Finally, the government equations for the leaky dielectric model are rewritten in the form applicable to the direct numerical simulation.