Cross-stream migration of drops suspended in Poiseuille flow in the presence of an electric field

The present study focuses on the cross-stream migration of a neutrally buoyant two-dimensional drop in a Poiseuille flow in a channel under the influence of an electric field. In the absence of an electric field, the important nondimensional parameters describing this problem are the viscosity ratio $\left(\lambda \right)$ between the drop fluid and the surrounding medium, the ratio of drop diameter to channel height $\left(a^{*}\right),$ and the capillary number (Ca). The influence of all these parameters on drop migration is investigated. It is observed that a large drop moves slowly as compared to a smaller drop, but attains a steady shape at the center line of the channel. The increase in value of the capillary number enhances the cross-stream migration rate, while the increase in viscosity ratio reduces the tendency of the drops to move towards the channel center line. The presence of an electric field introduces additional interfacial stresses at the drop interface, which in turn alters the dynamics observed in the absence of an electric field. Extensive computations are carried out to analyze the combined effect of the electric field and the shear flow on the cross-stream migration of the drop. The computational results for a perfect dielectric indicate that the droplet migration enhances in the presence of an electric field. The permittivity ratio $\left(S\right)$ and the electric field strength $\left(E\right)$ play major roles in drop migration and deformation. Computations using the leaky dielectric model also show that for certain combinations of electrical properties the drop undergoes immense elongation along the direction of the electric field. The conductivity ratio $\left(R\right)$ is again a vital parameter in such a system of fluids. It is further observed that for certain conditions the leaky dielectric drops exhibit rotation together with translation.

Figure 1

The schematic representation (not to scale) of the lateral migration of a droplet inside a channel under the influence of external Poiseuille flow and uniform electric field.

Figure 2

Comparison of drop shape for three different Ca: 0.05, 0.10, and 0.16. The other parameters are $a^{*}=0.95$ and $\lambda =0.99$. Left panel: Experimental results [4]; right panel: present simulations.

Figure 3

Comparison of drop deformation computed from the present simulations with Taylor's theory [Eq. (16)] for varying ${\mathrm{Ca}}_E$ and $R$, respectively.

Figure 4

Grid convergence test. (a) Snapshots of adaptive mesh generation using different levels of refinement. The refinement level of 7 is selected for the present study. (b) Comparison of $y_c$ for different levels of grid refinement. The different parameters considered are $a^{*}=0.6, \mathrm{Ca}=0.7$, and $\lambda =1.0$.

Figure 5

A drop ($a^{*}=0.6$) initially placed at different off-center positions gradually migrates towards the center line of the channel. The other parameters are Ca $=0.7$ and $\lambda =1$. (a) The lateral position of the drop versus time. (b) The streamlines around the drop placed at $y_c=0,-0.1$, and $-0.15$, respectively. The flow field is presented in the frame of reference of the drop.

Figure 6

Migration of different sized droplets starting from an initial off-center position ($y_c=-0.15$) towards the channel center line. (a) Drop velocity versus time. (b) Lateral position versus time. The other parameters are Ca $=0.7$ and $\lambda =1$.

Figure 7

The effect of varying Ca (Ca $=0.3$ to 1.3) on the migration of a drop initially placed at $y_c=-0.15$. (a) Drop velocity versus time. (b) Lateral position of the drop versus time. The other parameters are $a^{*}=0.4$ and $\lambda =1$.

Figure 8

The effect of varying viscosity ratio ($\lambda =0.01$ to 2.0) on the migration of a drop initially placed at $y_c=-0.15$. (a) Drop velocity versus time. (b) Lateral position of the drop versus time. The other parameters are $a^{*}=0.4$ and Ca $=0.7$.

Figure 9

Effect of varying permittivity ratio ($S=0.2$ to 5) on the deformation and migration of a drop initially placed at $y_c=-0.15$. (a) The shape of a drop at $t^{*}=30$ subjected to different permittivity ratio $S$. (b) The lateral position of the drop versus time. Other parameters are $a^{*}=0.4$, Ca $=0.7, \lambda =1$, and $E=5$.

Figure 10

Effect of elongation of the drop. (a) The normalized pressure drop in the channel for different permittivity ratio compared to the pressure drop in the channel in the absence of any drop (dashed line). (b) The streamlines around a drop subjected to varying permittivity ratio ($S=0.2$ and 5) at $t^{*}=30$. The flow field is presented in the frame of reference of the drop. Other parameters are $a^{*}=0.4$, Ca $=0.7, \lambda =1$, and $E=5$.

Figure 11

Variations of shape a drop ($a^{*}=0.4$) initially placed at $y_c=-0.15$ inside the channel and subjected to electric field from $t^{*}=0$ to 5.2. The other parameters are Ca $=0.7, \lambda =1, E=5$, and $S=5$.

Figure 12

Effect of varying electric field strength ($E=5$ to 20) on the deformation and migration of a drop initially placed at $y_c=-0.15$. (a) The shape of a drop at $t^{*}=30$ subjected to different $E$. (b) The lateral position of the drop versus time. Other parameters are $a^{*}=0.4$, Ca $=0.7, \lambda =1$, and $S=0.2$.

Figure 13

Effect of elongation of the drop. (a) The normalized pressure drop in the channel for different electric field strength compared to the pressure drop in the channel in the absence of any drop (dashed line). (b) The streamlines around a drop subjected to varying electric field strength ($E$ = 5, 10 and 20) at $t^{*}=30$. The flow field is presented in the frame of reference of the drop. Other parameters are $a^{*}=0.4,$ Ca $=0.7, \lambda =1$, and $S=0.2$.

Figure 14

Shapes of the drop at $t^{*}=30$ while migrating from an initial off-center position of $y_c=-0.15$ and subjected to different permittivity ratio $S$, conductivity ratio $R$, and electric field strength $E$. (a) Drops suspended in a system of $S=0.5$ and $R=2$ and subjected to varying $E$. (b) Drops suspended in a system of $S=2$ and $R=0.5$ and subjected to varying $E$. $E$ is varied in the range of 1 to 5. Other parameters are $a^{*}=0.4,$ Ca $=0.7$, and $\lambda =1$.

Figure 15

The migration of a drop initially placed at $y_c=-0.15$ towards the center line of the channel influenced by different electric field strength $E$, permittivity ratio $S$, and conductivity ratio $R$. Dotted and solid lines represent the position of the drop suspended in systems A and B, respectively. Other parameters are $a^{*}=0.4,$ Ca $=0.7$, and $\lambda =1$.

Figure 16

The transient evolution of a drop ($a^{*}=0.4$) initially placed at $y_c=-0.15$ inside the channel. For system A (a) and B (b). The other parameters are Ca $=0.7, \lambda =1$, and $E=5$.

Figure 17

The charge density at the interface of a drop ($a^{*}=0.4$) initially placed at $y_c=-0.15$ inside the channel subjected to electric field. (a) Charge distribution at the interface of a nonrotating drop suspended in system A. (b) Charge distribution at the interface of a rotating drop suspended in system B. The other parameters are Ca $=0.7, \lambda =1$, and $E=5$.

Figure 18

The rotation of a drop ($a^{*}=0.4$) initially placed at $y_c=-0.15$ inside the channel subjected to electric field. (a) Profile of the rotating drop for $t^{*}=0$ to 5.6. (b) The vector field in and around the drop for $t^{*}=1.2,2.3,3.3$, 4.3, 15, and 20. (c) The trajectory of a marker point (marked as red dot) near the interface $(x^{*}=1, y^{*}=0.025)$. The other parameters are Ca $=0.7, \lambda =1, E=5, S=2$, and $R=0.5$. The flow field is presented in the frame of reference of the drop.