Nonspherical liquid droplet falling in air

The dynamics of an initially nonspherical liquid droplet falling in air under the action of gravity is investigated via three-dimensional numerical simulations of the Navier-Stokes and continuity equations in the inertial regime. The surface tension is considered to be high enough so that a droplet does not undergo breakup. Vertically symmetric oscillations which decay with time are observed for low inertia. The amplitude of these oscillations increases for high Gallilei numbers and the shape asymmetry in the vertical direction becomes prominent. The reason for this asymmetry has been attributed to the higher aerodynamic inertia. Moreover, even for large inertia, no path deviations or oscillations are observed.

Figure 1

Initial shapes of a liquid drop falling in air. Here, $a$ and $b$ represent the diameters of the drop along the radial and vertical directions, respectively. The initial shape of the drop is generated as an axisymmetric surface about the vertical axis passing through the centroid of the drop. The aspect ratio $A_r$ of the drop is defined as $a/b$.

Figure 2

Shapes of the drop at $t=11$ (top panel) and $t=12.5$ (bottom panel) obtained using different grids and domain sizes. (a) $L=120$, smallest grid size in the interfacial and the vortical regions are 0.058 and 0.234, (b) $L=160$, smallest grid size in the interfacial and the vortical regions are 0.078 and 0.312, and (c) $L=160$, smallest grid size in the interfacial and the vortical regions are 0.078 and 0.625, respectively.

Figure 3

The temporal variations of the diameters of the drop along the radial and vertical directions $(a-b)$ for different initial oblate shaped droplets. Circles: $A_{r0}=1$; filled triangles: $A_{r0}=1.19$; pluses: $A_{r0}=1.33$; squares: $A_{r0}=1.95$. The rest of the parameters are $\text{Ga}=10,\text{Eo}={10}^{-3},{\mu }_r=55$, and ${\rho }_r=998$. The magenta dotted line shows the aspect ratio of a perfect sphere $(A_r=1)$.

Figure 4

Evolutions of shapes of the drop for (a) $A_{r0}=1.19$ and (b) 1.95. The rest of the parameters are the same as those of Fig. 3.

Figure 5

The temporal variations of the diameters of the drop along the radial and vertical directions $(a-b)$ for different initial prolate shaped droplets. Circles: $A_{r0}=0.857$; filled triangles: $A_{r0}=0.729$; pluses: $A_{r0}=0.614$; squares: $A_{r0}=0.512$. The rest of the parameters are $\text{Ga}=10,\text{Eo}={10}^{-3},{\mu }_r=55$, and ${\rho }_r=998$. The magenta dotted line shows the aspect ratio of a perfect sphere $(A_r=1)$.

Figure 6

Evolutions of shapes of the drop for (a) $A_{r0}=0.729$ and (b) 0.512. The rest of the parameters are the same as those of Fig. 5.

Figure 7

The variations of the oscillation velocity, $V_0=2(a_{\max }-a_{\min })/T_p$, of the drop with $A_{r0}$. The rest of the parameters are the same as those used to generate Figs. 3 and 5.

Figure 8

The temporal variations of the diameters of the drop along the radial and vertical directions, $(a-b)$ for different values of $\text{Ga}$. The initial aspect ratio is $A_{r0}=1.33$ for all values of $\text{Ga}$. The rest of the parameters are $\text{Eo}={10}^{-3},{\mu }_r=55$, and ${\rho }_r=998$. The magenta dotted line shows the aspect ratio of a perfect sphere $(A_r=1)$. The filled red squares and blue circles represent the time instants for which the shapes of the drop are plotted in Fig. 9 for different values of $\text{Ga}$. They correspond to the time instants at which the drop undergo prolate and oblate shapes.

Figure 9

Shapes of the drop for different values of $\text{Ga}$ at the time instants shown by filled red squares and blue circles in Fig. 8. The rest of the parameters are the same as those used to generate Fig. 8.

Figure 10

The variation of the oscillation velocity, $V_0=2(a_{\max }-a_{\min })/T_p$, of the drop with $\text{Ga}$ for $A_{r0}=1.33$ and 1.95. The rest of the parameters are $\text{Eo}={10}^{-3},{\mu }_r=55$, and ${\rho }_r=998$.

Figure 11

Isosurfaces of the vorticity component in the $z$ direction $\left({\omega }_z=\pm 1\right)$: (a) $\text{Ga}=10$ at $t=12.3$, (b) $\text{Ga}=100$ at $t=12.7$, (c) $\text{Ga}=10$ at $t=13.4$, and (d) $\text{Ga}=100$ at $t=13.8$. The rest of the parameters are the same as those used to generate Fig. 8.

Figure 12

Temporal variations of $x$ and $y$ coordinates of the center of gravity of the droplet, i.e., $x_{CG}$ and $y_{CG}$, for different values of $\text{Ga}$. The rest of the parameters are the same as Fig. 8.

Figure 13

Effect of (a) viscosity ratio for ${\rho }_r=998$ (b) density ratio for ${\mu }_r=55$ on the temporal variations of the diameters of the drop along the radial and vertical directions $(a-b)$. The initial aspect ratio is $A_{r0}=1.33$. The magenta dotted lines show the aspect ratio of a perfect sphere $(A_r=1)$. The rest of the parameters are $\text{Eo}={10}^{-3}$ and $\text{Ga}=10$.

Figure 14

(a) Isosurfaces of the vorticity component in the $z$ direction $\left({\omega }_z=\pm 1\right)$ at $t=12$ for ${\mu }_r=100$ (left column) and ${\mu }_r=10$ (right column). (b) Evolutions of shapes of the droplet for ${\mu }_r=100$ (left column) and ${\mu }_r=10$ (right column). The rest of the parameters are the same as those used to generate Fig. 13.

Figure 15

Evolutions of shapes of the droplet for (a) ${\rho }_r=998$ and (b) ${\rho }_r=10$. The rest of the parameters are the same as those used to generate Fig. 13.