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

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.

Figure 1

Sketch of the geometry of a film flow coating a fiber.

Figure 2

The boundaries between the convective and absolute instabilities in the $\text{Bo}-\alpha$ plane with various $E_b$ for (a) $\beta =2.0$, (b) $\beta =2.5$, and (c) $\beta =5.0$.

Figure 3

The profiles of the interface at instant time for $\alpha =0.5, \beta =2.5$, and $E_b=0.2$. (a) $\text{Bo}=0.4, t=50$. (b) $\text{Bo}=0.2, t=50$.

Figure 4

Spatial-temporal diagram for $\alpha =0.5, \beta =2.5$, and $E_b=0.2$. The values of $\left|S\right(z,t)-\alpha |$ are marked by different colors. (a) $\text{Bo}=0.4$. (b) $\text{Bo}=0.2$.

Figure 5

The profiles of the interface at instant time for $\alpha =0.4, \beta =5.0$, and $\text{Bo}=0.2$. (a) $E_b=0.5, t=50$. (b) $E_b=2.0, t=60$.

Figure 6

Flow regimes in the $\text{Bo-}\alpha$ plane. (a) $E_b=0.5$. (b) $E_b=2.0$.

Figure 7

The profiles of the interface at instant time for $\alpha =0.2, \beta =2.5$, and $\text{Bo}=0.2$. (a) $E_b=0.5, t=50$. (b) $E_b=1.0, t=6.90$.

Figure 8

The curve of similarity solution $F\left(\xi \right)$.

Figure 9

The real growth rate ${\omega }_i$ vs the wave number $k$ for $\epsilon =0.2$. Other parameters are (a) $\alpha =0.4, \beta =5.0, E_b=0.2$; (b) $\alpha =0.4, \beta =5.0, E_b=1.0$; (c) $\alpha =0.2, \beta =2.5, E_b=0.2$; and (d) $\alpha =0.2, \beta =2.5, E_b=1.0$.