Absolute and convective instabilities of a film flow down a vertical fiber subjected to a radial electric field

Abstract

We consider the motion of a gravity-driven flow down a vertical fiber subjected to a radial electric field. This flow exhibits rich dynamics including the formation of droplets, or beads, driven by a Rayleigh-Plateau mechanism modified by the presence of gravity as well as the Maxwell stress at the interface. A spatiotemporal stability analysis is performed to investigate the effect of electric field on the absolute-convective instability (AI-CI) characteristics. We performed a numerical simulation on the nonlinear evolution of the film to examine the transition from CI to AI regime. The numerical results are in excellent agreement with the spatiotemporal stability analysis. The blowup behavior of nonlinear simulation predicts the formation of touchdown singularity of the interface due to the effect of electric field. We try to connect the blowup behavior with the AI-CI characteristics. It is found that the singularities mainly occur in the AI regime. The results indicate that the film may have a tendency to form very sharp tips due to the enhancement of the absolute instability induced by the electric field. We perform a theoretical analysis to study the behaviors of the singularities. The results show that there exists a self-similarity between the temporal and spatial distances from the singularities.

Relative periodic solutions in conducting liquid films flowing down vertical fibres

“…In the following discussion, we fix the small parameter ǫ, and the radius of the fibre α at (ǫ, α) = (0.2, 0.25) following previous studies, e.g. ; Liu, Chen & Ding (2018). To understand the oscillation dynamics between big droplets, we consider a periodic computational domain that is large enough for two big droplets.…”

“…using an electric field to control pattern formation in a liquid film on a flat plane (Schäffer et al 2000;Craster & Matar 2005) and droplet production from liquid jets (Wang et al 2011;Conroy et al 2011). The idea of using an electric field was also transplanted into the control of instability of liquid film flows on vertical fibres (Wray et al 2012(Wray et al , 2013aDing et al 2014;Liu, Chen & Ding 2018). The asymptotic model was used to investigate the dynamics of the electrified films (Wray et al 2012(Wray et al , 2013aDing et al 2014;Liu, Chen & Ding 2018).…”

“…The idea of using an electric field was also transplanted into the control of instability of liquid film flows on vertical fibres (Wray et al 2012(Wray et al , 2013aDing et al 2014;Liu, Chen & Ding 2018). The asymptotic model was used to investigate the dynamics of the electrified films (Wray et al 2012(Wray et al , 2013aDing et al 2014;Liu, Chen & Ding 2018). Recently, other control methods, e.g., varying the nozzle size or using a countercurrent gas flow (Zeng et al 2017), have been used to modulate the pattern formation in liquid films on vertical fibres.…”

“…When the liquid is a leaky dielectric, the electric field can suppress the Plateau-Rayleigh instability or attract the liquid to the electrode, leading to the 'touch-down' structure (Wray et al 2012(Wray et al , 2013a, where the liquid interface is attracted to the fibre (the inner electrode). When the liquid is perfectly conducting, instead, a spike-like structure is observed (Ding et al 2014;Liu, Chen & Ding 2018), where the liquid interface touches the outer electrode. Self-similarity analysis has shown that the minimal film thickness in touch-down state scales as (∆t) 1/3 (Wray et al 2013a), and the spike-like structure develops as (∆t) 1/6 (∆t = t s − t with singularity formation time t s ) (Liu, Chen & Ding 2018).…”

“…When the liquid is perfectly conducting, instead, a spike-like structure is observed (Ding et al 2014;Liu, Chen & Ding 2018), where the liquid interface touches the outer electrode. Self-similarity analysis has shown that the minimal film thickness in touch-down state scales as (∆t) 1/3 (Wray et al 2013a), and the spike-like structure develops as (∆t) 1/6 (∆t = t s − t with singularity formation time t s ) (Liu, Chen & Ding 2018). The interested reader may consult the review paper by Saville (1997) for more details on the dielectric models of liquids.…”