Deformation and breakup of a liquid droplet past a solid circular cylinder: A lattice Boltzmann study

In this paper, we present a numerical study on the deformation and breakup behavior of liquid droplet past a solid circular cylinder by using an improved interparticle-potential lattice Boltzmann method. The effects of the eccentric ratio $\beta$, viscosity ratio $\lambda$ between the droplet and the surrounding fluid, surface wettability, and Bond number (Bo) on the dynamic behavior of the liquid droplet are considered. The parameter $\beta$ represents the degree that the solid cylinder deviates from the center line, and Bo is the ratio between the inertial force and capillary force. Numerical results show that there are two typical patterns, i.e., breakup and no breakup, which are greatly influenced by the aforementioned parameters. When $\beta$ increases to a critical value ${\beta }_c$, the droplet can pass the circular cylinder without a breakup, otherwise, the breakup phenomenon occurs. The critical eccentric ratio ${\beta }_c$ increases significantly with increasing Bo for case with $\lambda >1$, while for the case with $\lambda <1$, the viscosity effects on the ${\beta }_c$ is not obvious when Bo is large. For the breakup case, the amount of deposited liquid on the tip of the circular cylinder is almost unaffected by $\beta$. In addition, the results also show that the viscosity ratio and wettability affect the deformation and breakup process of the droplet. For case with $\lambda <1$, the viscosity ratio plays a minor role in the thickness variations of the deposited liquid, which decreases to a nonzero constant eventually; while for $\lambda >1$, the increase of the viscosity ratio significantly accelerates the decrease of the deposited liquid, and finally no fluid deposits on the cylinder. In term of the wettability, there occurs continuous gas phase trapped by the wetting droplet, but this does not happen for nonwetting droplet. Besides, for $\lambda <1$, the time required to pass the cylinder $\left(t_p\right)$ decreases monotonically with decreasing contact angle, while a nonmonotonic decrease appears for $\lambda >1$. It is also found that $t_p$ decreases monotonically with increasing Bo and is less sensitive to $\lambda$ at a large Bo.

Figure 1

Steady state velocity profiles as a function of the channel width for (a) $M=1/150,$ (b) $M=150$.

Figure 2

Schematic illustration of the flow geometry.

Figure 3

Effects of the grid size on the deformation and breakup of liquid droplet past a solid circular cylinder for (left) $\lambda$ = 0.02, (right) $\lambda$ = 50, and grid size with $Lx\times Ly$ (a) $300\times 1800$, (b) $150\times 900$, and (c) $74\times 450$.

Figure 4

Comparisons between the present simulations and previous experiments [5] on the droplet past an obstruction $\text{Ca}=\frac{{\mu }_{d}U}{\sigma }=0.01,\lambda =0.2$; (a) the droplet past a square cylinder (left: numerical, right: experimental), (b) the droplet past a circular cylinder (left: numerical, right: experimental).

Figure 5

Time evolution of the droplet past a circular cylinder at $\theta ={170}^{\circ },\lambda =0.02$ (a) head-to-head impact: $\beta =0$, (b) eccentric impact: $\beta =1/7$, (c) eccentric impact: $\beta =2/7$.

Figure 6

Effect of the eccentric ratio $\beta$ on the size ratio of daughter droplets.

Figure 7

Time evolution of the droplet past a circular cylinder at $\lambda$ = 50, $\theta ={170}^{\circ }$.

Figure 8

Time evolution of the normalized thickness of the deposited film with different $\lambda$.

Figure 9

Time evolution of the droplet past a circular cylinder at $\theta ={70}^{\circ },\lambda$ = 0.02.

Figure 10

Time evolution of the droplet past a circular cylinder at $\theta ={70}^{\circ },\lambda$ = 50.

Figure 11

Effect of the surface wettability on the passing time $t_p$: (a) $\lambda <1$, (b) $\lambda >1$.

Figure 12

Effect of Bo on the passing time $t_p$.

Figure 13

Effect of the Bo on the critical eccentric ratio ${\beta }_c$.