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

Abstract

AbstractThe long-wave behaviour of perfectly conducting liquid films flowing down a vertical fibre in a radial electric field was investigated by an asymptotic model. The validity of the asymptotic model was verified by the fully linearized problem, which showed that results were in good agreement in the long-wave region. The linear stability analysis indicated that, when the ratio (the radius of the outer cylindrical electrode over the radius of the liquid film) $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\beta <e$, the electric field enhanced the long-wave instability; when $\beta >e$, the electric field impeded the long-wave instability; when $\beta =e$, the electric field did not affect the long-wave instability. The nonlinear evolution study of the asymptotic model compared well with the linear theory when $\beta <e$. However, when $\beta =e$, the nonlinear evolution study showed that the electric field enhanced the instability which may cause the interface to become singular. When $\beta >e$, the nonlinear evolution studies showed that the influence of the electric field on the nonlinear behaviour of the interface was complex. The electric field either enhanced or impeded the interfacial instability. In addition, an interesting phenomenon was observed by the nonlinear evolution study that the electric field may cause an oscillation in the amplitude of permanent waves when $\beta \ge e$. Further study on steady travelling waves was conducted to reveal the influence of electric field on the wave speed. Results showed that the electric field either increased or decreased the wave speed as well as the wave amplitude and flow rate. In some situations, the wave speed may increase/decrease while its amplitude decreased/increased as the strength of the external electric field increased.