Shape oscillations of a nonspherical water droplet

The dynamics of a nonspherical droplet falling in air is investigated via three-dimensional numerical simulations of the Navier-Stokes and continuity equations. The main focus of this work is to study the effects of the relative influence of surface tension force over the inertial force and the orientation of the droplet. The values of the dimensionless parameters considered in this study are similar to those of a typical falling raindrop. Our results obtained from the numerical simulations for low values of Galilei number agree well with the theoretical predictions in the microgravity condition and in the limit of creeping flow. However, in the inertia-dominated region and for high initial aspect ratio, the droplet undergoes predominant nonlinear oscillations and subsequently breaks up. To the best of our knowledge, none of the previous studies have investigated dynamics of a tilted drop and observed the behavior reported in the present study.

Figure 1

(a) Schematic diagram of a nonspherical water droplet falling in air. A cubic computational domain of size $H\times H\times H$ is used and the bubble is placed at $(0,0,z_i)$. The values of $H=75R_{\mathrm{eq}}$ and $z_i=70R_{\mathrm{eq}}$. The acceleration due to gravity $\left(g\right)$ acts in the negative $z$ direction. (b) The initial shape of a typical tilted droplet. Here $a$ and $b$ represent the diameters of the droplet along its initial major and minor axes, respectively. $\theta$ is the angle of the initial major axis of the droplet to the negative $y$ axis.

Figure 2

(a) The temporal variations of $(a-b)$ obtained using $\mathrm{\Delta }=0.009$ (solid line), $\mathrm{\Delta }=0.012$ (circles), and $\mathrm{\Delta }=0.015$ (squares). (b) Shapes of the droplet ($y\text{−}z$ view) at different time instants (which are indistinguishable for different grids considered). The remaining parameter values are $\mathrm{Ga}=100$, $\mathrm{Eo}={10}^{-3}$, $\theta =0$, and $A_{r0}=1.331$. The red dotted line shows the aspect ratio of a perfect sphere.

Figure 3

(a) The temporal variations of $(a-b)$ for $A_{r0}=1$ (circles), $A_{r0}=1.331$ (filled triangles), and $A_{r0}=1.953$ (squares). The red dotted line, which coincides with the result for $A_{r0}=1$, shows the aspect ratio of a perfect sphere. (b) The evolutions of the shape of the droplet ($y\text{−}z$ view) obtained for $A_{r0}=1.331$ (left column) and 1.953 (right column). The remaining parameter values are $\mathrm{Ga}=48.4$, $\mathrm{Eo}=1.13\times {10}^{-3}$, and $\theta =0$.

Figure 4

(a) The temporal variations of $(a-b)$ for $A_{r0}=2.46$ and 2.74. (b) The evolutions of the shape of the droplet obtained for $A_{r0}=2.46$ (left column) and 2.74 (right column). The remaining parameter values are $\mathrm{Ga}=48.4$, $\mathrm{Eo}=1.13\times {10}^{-3}$, and $\theta =0$.

Figure 5

Temporal variations of $(a-b)$ of the droplet for different values of initial tilt angle, $\theta$. The remaining parameter values are $\mathrm{Ga}=100$ and $\mathrm{Eo}={10}^{-3}$. The red dotted line shows the aspect ratio of a perfect sphere.

Figure 6

Evolutions of the shape of the droplet ($y\text{−}z$ view) for different tilt angles: (a) $\theta =0^{\circ }$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={60}^{\circ }$, and (d) $\theta ={90}^{\circ }$. The remaining parameter values are $\mathrm{Ga}=100$, $\mathrm{Eo}={10}^{-3}$, and $A_{r0}=1.331$.

Figure 7

Evolutions of the $y$ vorticity contours (${\omega }_y=\pm 0.1$) for different values of the tilt angle, $\theta$ (left to right for each value of $\theta$: $t=4.6$, 5, 5.8, 6.2, and 7). The remaining parameter values are $\mathrm{Ga}=100$, $\mathrm{Eo}={10}^{-3}$, and $A_{r0}=1.331$.

Figure 8

The temporal variations of $(a-b)$ for $A_{r0}=1.331$ ($\mathrm{Ga}=100$ and $\theta ={45}^{\circ }$) for $\mathrm{Eo}=0.001$ (circles), 0.004 (filled triangles), and 0.005 (squares). The red dotted line shows the aspect ratio of a perfect sphere.

Figure 9

The dimensionless time period of the oscillations, $T_p$ versus $\mathrm{Eo}$. The results from the numerical simulations and theoretical analysis [obtained from Eq. (11)] are shown by symbols and a solid line, respectively. The rest of the parameter values are the same as those used to generate Fig. 8.

Figure 10

The temporal variations of $(a-b)$ for $A_{r0}=1.331$ ($\mathrm{Eo}={10}^{-3}$ and $\theta ={45}^{\circ }$) for different values of $\mathrm{Ga}$.

Figure 11

Evolutions of the shape of the droplet ($y\text{−}z$ view) for $\theta ={45}^{\circ }$: (a) $\mathrm{Ga}=300$ and (b) $\mathrm{Ga}=500$. The remaining parameter values are $\mathrm{Eo}={10}^{-3}$ and $A_{r0}=1.331$.

Figure 12

The dimensionless decay rate of the oscillations, ${\alpha }_n$, obtained from the theoretical analysis [Eq. (12)] versus $\mathrm{Ga}$ (solid line). The results from the numerical simulations are shown by symbols. The rest of the parameter values are the same as those used to generate Fig. 10.