Dynamics of a viscous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: breakup and touchdown singularities

Abstract

The nonlinear dynamics of a viscous filament surrounded by a second viscous fluid arranged in a core-annular configuration when a radial electric field acts in the annular region, are studied analytically and computationally using boundary element methods. The flow is characterized by the viscosity ratio, an electric Weber number measuring the strength of the electric field, a geometrical parameter measuring the thickness of the undisturbed annular region, as well as a computational parameter that fixes the wavenumber of the undulations. Axisymmetric solutions are computed by direct numerical simulations in the Stokes limit for general values of the parameters when the two fluids have equal viscosities, and an asymptotic theory is carried out to produce a novel evolution equation for thin film dynamics valid when the undisturbed annular thickness is small and the viscosity ratio is of order one. It is established (in agreement with previous computations in the absence of electric fields) that a sufficiently thick annulus enables thread breakup while a sufficiently thin one (approximately one fifth of the undisturbed thread radius for the case of equal viscosities, for instance) suppresses pinching and drives the interface to approach the tube wall asymptotically without actually touching it. The present simulations show that the electric field affects the dynamics drastically in several ways. First, it promotes interfacial wall touchdown in finite time and a comparison between direct simulations and the asymptotic solutions are in fair agreement. Second, the electric field acts to suppress pinching in the sense that solutions that lead to jet breakup due to a thick enough viscous annulus are driven to wall touchdown. When pinching takes place we find that the ultimate pinching solutions are self-similar and recover the non-electrified ones to leading order for the range of parameters studied.

Dynamics of liquid films on vertical fibres in a radial electric field

“…The attraction between the electrode and the liquid interface becomes very large at the crest of the lower droplet, which squeezes the droplet into the singular shape. This phenomenon could also be observed in an electrified jet or thread (Wang et al 2009;Wang & Papageorgiou 2011). Results in figures 7 and 8 indicate that the electric field is destabilizing in the nonlinear regime, but it does not contradict with the linear theory.…”

“…Instead of considering the non-axisymmetric problem, they considered the axisymmetric case and derived a one-dimensional model to study the linear stability and nonlinear breakup of the conducting jet. Perfectly conducting liquid jets and threads surrounded by dielectric gases or dielectric liquids in the presence of a radial electric field were investigated by Wang et al (Wang, Mählmann & Papageorgiou 2009;Wang & Papageorgiou 2011). They proposed an asymptotic model under the long-wave approximation and investigated the electrostatic pressure's effect on the linear and nonlinear behaviour of the liquid jet/thread (Wang et al 2009;Wang & Papageorgiou 2011).…”

“…Perfectly conducting liquid jets and threads surrounded by dielectric gases or dielectric liquids in the presence of a radial electric field were investigated by Wang et al (Wang, Mählmann & Papageorgiou 2009;Wang & Papageorgiou 2011). They proposed an asymptotic model under the long-wave approximation and investigated the electrostatic pressure's effect on the linear and nonlinear behaviour of the liquid jet/thread (Wang et al 2009;Wang & Papageorgiou 2011). To capture the breakup of the liquid jet/thread in the vicinity of the singular event, they conducted a study by searching for self-similar solutions (Wang et al 2009;Wang & Papageorgiou 2011).…”

“…They proposed an asymptotic model under the long-wave approximation and investigated the electrostatic pressure's effect on the linear and nonlinear behaviour of the liquid jet/thread (Wang et al 2009;Wang & Papageorgiou 2011). To capture the breakup of the liquid jet/thread in the vicinity of the singular event, they conducted a study by searching for self-similar solutions (Wang et al 2009;Wang & Papageorgiou 2011). In these previous studies (Basset 1894;Taylor 1969;Saville 1971;Son & Ohba 1998;Collins et al 2007;Wang et al 2009;Wang & Papageorgiou 2011), the electrostatics was considered.…”

“…To capture the breakup of the liquid jet/thread in the vicinity of the singular event, they conducted a study by searching for self-similar solutions (Wang et al 2009;Wang & Papageorgiou 2011). In these previous studies (Basset 1894;Taylor 1969;Saville 1971;Son & Ohba 1998;Collins et al 2007;Wang et al 2009;Wang & Papageorgiou 2011), the electrostatics was considered. Conroy et al (2011) examined the influence of an AC electric field on the linear and nonlinear dynamics of a perfectly conducting liquid thread.…”