Instability of a leaky dielectric coaxial jet in both axial and radial electric fields

The temporal linear instability of a coaxial jet with two immiscible Newtonian liquids in both the axial and radial electric fields is studied. The outer liquid is supposed to be a leaky dielectric and the inner liquid a perfect dielectric. The eigenvalue problem for both axisymmetric instability and helical instability is formulated and solved using the spectral method. Different from axisymmetric instability, for helical instability there is only one unstable mode, i.e., the helical mode, located in the long wave region. The axial electric field is found to have a strong stabilization effect on both the axisymmetric and helical modes, and the radial electric field has a great destabilization effect on them. The competition between the axisymmetric and helical instability under the action of the axial and radial electric fields is calculated. The boundary curve separating the stabilization and destabilization regions of the parasinuous mode, the neutral stability curve of the helical mode, and the boundary curve between the dominant regions of the axisymmetric and helical instability are plotted on the Q0-Eu plane and Pi-Eu plane, respectively (Q0 is the dimensionless surface charge density; Eu is the electrical Euler number representing the characteristic tangential electrostatic force; and Pi=Q0;2Eu is the dimensionless parameter representing the characteristic normal electrostatic force). In general, when surface charge density is small, the helical mode is stable, and the parasinuous mode is dominant; however, when surface charge density is sufficiently large, the helical mode is destabilized and becomes dominant in jet instability. Liquid viscosity influences the predominance of the helical mode significantly. Although liquid viscosity decreases the growth rates of both the axisymmetric and helical modes, it suppresses the axisymmetric instability much more than the helical instability, and therefore favors the realization of the helical instability in experiments.