Influence of the viscosity and charge mobility on the shape deformation of critically charged droplets

In this work, we model and simulate the shape evolution of critically charged droplets, from the initial spherical shape to the charge emission and back to the spherical shape. The shape deformation is described using the viscous correction for viscous potential flow model, which is a potential flow approximation of the Navier-Stokes equation for incompressible Newtonian fluids. The simulated shapes are compared to snapshots of experimentally observed drop deformations. We highlight the influence of the dimensionless viscosity and charge carrier mobility of the liquid on the shape evolution of droplets and discuss the observed trends. We give an explanation as to why the observed deformation pathways of positively and negatively charged pure water droplets differ and give a hint as to why negatively charged water droplets emit more charge during charge breakup than positively charged ones.

Figure 1

Comparison between simulated (top) and observed (bottom) snapshots of $\oslash =52\mu \mathrm{m}$ de-ionized water droplets at room temperature. Initially, the red droplets are charged at the Rayleigh limit $X=1$. At charge breakup, 20% of the charge is removed and the fissility for the green droplet is $X=0.64$. Shape labels are as follows: a, lemon; b, peanut; and c, diamond.

Figure 2

Fit parameters $(a/b,n)$ found for various observed shapes using the superellipsoid equation (34). Red contours show the corresponding superellipsoid shape.

Figure 3

Graph $(a/b,n)$ showing the deformation pathway of a critically charged de-ionized water droplet. Starting with the spherical shape (1,2), arrows give the time evolution of the path.

Figure 4

Deformation pathway of a critically charged droplet as a function of the Ohnesorge number Oh. At the pointed tip appearance ($n=1.5$), the fissility is reduced from $X=1$ to $X=0.64$.

Figure 5

Elongation $a/b$ of the droplet at charge breakup (taken at $n=1.5$) as a function of the dimensionless viscosity Oh. Purple squares stand for the VCVPF model and green circles for the VPF model. The horizontal dotted black line is $a/b=3.86$ and gives the elongation of the droplet in the case of the Stokes flow approximation [7, 8].

Figure 6

Deformation pathway of a droplet with $\text{Oh}=0.023$ (typically a $\oslash =50\mu \mathrm{m}$ water droplet at room temperature) for various dimensionless charge carrier mobility $\tilde{\lambda }$, ranging from 0.1 to 10 and for the perfect conductor.

Figure 7

Deformation pathway for positively and negatively charged water droplets having a radius of 26 $\mu \mathrm{m}$. Solid and dashed lines are simulations, black closed circles and green open squares are experimental results of negatively and positively charged de-ionized water droplets, respectively. In the legend, $\lambda$ stands for the dimensionless charge mobility (tilde has been omitted) and $\mathrm{\Delta }Q/Q$ is the relative amount of emitted charge. The red dashed line is added to show the effect on the return path for $\tilde{\lambda }=2$, in the case where 20% of the charge was removed instead of 40%.

Figure 8

(a) Dimensionless velocity of the charges tangent to the surface, ${\tilde{u}}_{\tau }+\tilde{\lambda }{\tilde{E}}_{\tau }$, as a function of the dimensionless distance $r/R$ defined in the inset. Positive values indicate velocities directed towards the tips. (b) Dimensionless surface charge density $\tilde{\sigma }$ as a function of $r/R$. The quantities are calculated for three different charge mobilities $\tilde{\lambda }$. The blue line with pluses gives the equilibrium surface charge density ${\tilde{\sigma }}_{\text{eq}}$ in the case of a perfectly conducting liquid.