Migration of a droplet in a cylindrical tube in the creeping flow regime

The migration of a neutrally buoyant droplet in a tube containing another immiscible liquid is investigated numerically in the creeping flow regime. A fully developed velocity profile is imposed at the inlet of the tube. The interface between the two immiscible fluids is captured using a coupled level-set and volume-of-fluid approach. The deformation and breakup dynamics of the droplet are investigated in terms of three dimensionless parameters, namely, the ratio between the radius of the undeformed droplet and the radius of the capillary tube, the viscosity ratio between the dispersed and the continuous phases, and the capillary number that measures the relative importance of the viscous force over the surface tension force. It has been observed that the droplet, while traversing through the tube, either approaches a steady bulletlike shape or develops a prominent reentrant cavity at its rear. Depending on the initial droplet size, there exists a critical capillary number for every flow configuration beyond which the drop fails to maintain a steady shape and breaks into fragments. The deformation and breakup phenomena depend primarily on the droplet size, the viscosity ratio, and the capillary number. Special attention has been given to the case where the drop diameter is comparable with the tube diameter. A thorough computational study has been conducted to find the critical capillary number for a range of droplets of varied sizes suspended in systems having different viscosity ratios.

Figure 1

Schematic diagram (not to scale) showing the initial flow configuration of a droplet of radius $a_o$ located at $(r_o,z_0)$ in a cylindrical tube of radius $R_0$. A fully developed flow is imposed at the inlet of the tube. The length $L$ of the tube is taken to be equal to $40R_0$.

Figure 2

Comparison of the velocity of the drop, $V_d$, for different grid meshes ($1000\times 25, 1600\times 40, 2000\times 40$, and $3200\times 80$). The parameters are $\text{Ca}=1.0, \lambda =0.1$, and $a=0.7$. $N_z$ and $N_r$ represent the number of grids in the $z$ and $r$ directions of the computational domain, respectively.

Figure 3

Comparison of the steady shapes of a droplet of size $a=0.95$ migrating in a capillary tube for $\lambda =0.99$ for $\text{Ca}=0.05$, 0.10, and 0.16: (a) experimental results [17] and (b) present study.

Figure 4

Comparison of the evolution of droplet shape: (a) present study and (b) the results of Tsai and Miksis [20]. (c) The shapes of the droplet obtained by Olbritch [15] in the same configuration. The parameters used are $\text{Ca}=1, \lambda =0.1$, and $a=0.9$.

Figure 5

The effect of viscosity ratio on the droplet velocity $V_d$ for (a) $a=0.726$ and (b) $a=0.914$. Here, $\text{Ca}=0.1$. The results obtained from the present computation are compared with those presented by Ho and Leal [16] and Martinez and Udell [19].

Figure 6

The effect of the variation of the aspect ratio $a$ (i.e., initial droplet size) on the droplet velocity $V_d$. The steady shapes of the droplet for $a=0.3$, 0.5, 0.7, and 0.9 are inserted in the plot. The rest of the parameters are $\text{Ca}=0.1$ and $\lambda =0.1$.

Figure 7

The steady shapes attained by a droplet for $a=0.7$ migrating in a tube for different values of $\text{Ca}$. The streamlines (in the relative reference frames moving with the droplets) are also shown in this figure for different values of $\text{Ca}$. The viscosity ratio is $\lambda =0.1$.

Figure 8

The temporal variation of droplet velocity $V_d$ as it is migrating inside the tube for different values of $\text{Ca}$. The other parameters are $a=0.7$ and $\lambda =0.1$.

Figure 9

(a) Variation of droplet velocity $V_d$ with viscosity ratio $\lambda$ for different values of $\text{Ca}$. Variations of dimensionless (b) length of the droplet, $l_d$, (c) width $w_d$ of the droplet, and (d) film thickness $f_t$ with Ca for different values of the viscosity ratio $\lambda$. The aspect ratio $a$ is fixed at 0.7.

Figure 10

The shape evolution of a droplet for $\text{Ca}=1$ and $\lambda =0.1$. The aspect ratio is $a=0.7$.

Figure 11

The shape evolution of a droplet for $\text{Ca}=1$ and $\lambda =0.1$. The aspect ratio is $a=0.8$.

Figure 12

The shape evolution of a droplet for $\text{Ca}=1$ and $\lambda =0.1$. The aspect ratio is $a=0.9$.

Figure 13

The shape evolution of a droplet for $\lambda =1\times {10}^{-3}$ and $\text{Ca}=1$. The aspect ratio is $a=0.9$.

Figure 14

The shape evolution of a droplet for $\lambda =0.58$ and $\text{Ca}=1$. The aspect ratio is $a=0.9$.

Figure 15

The shape evolution of a droplet for $\lambda =2.04$ and $\text{Ca}=1$. The aspect ratio is $a=0.9$.

Figure 16

Variation of critical capillary number ${\text{Ca}}_{cr}$ with viscosity ratio $\lambda$ for different values of $a$.