Droplet migration characteristics in confined oscillatory microflows

We analyze the migration characteristics of a droplet in an oscillatory flow field in a parallel plate microconfinement. Using phase field formalism, we capture the dynamical evolution of the droplet over a wide range of the frequency of the imposed oscillation in the flow field, drop size relative to the channel gap, and the capillary number. The latter two factors imply the contribution of droplet deformability, commonly considered in the study of droplet migration under steady shear flow conditions. We show that the imposed oscillation brings an additional time complexity in the droplet movement, realized through temporally varying drop shape, flow direction, and the inertial response of the droplet. As a consequence, we observe a spatially complicated pathway of the droplet along the transverse direction, in sharp contrast to the smooth migration under a similar yet steady shear flow condition. Intuitively, the longitudinal component of the droplet movement is in tandem with the flow continuity and evolves with time at the same frequency as that of the imposed oscillation, although with an amplitude decreasing with the frequency. The time complexity of the transverse component of the movement pattern, however, cannot be rationalized through such intuitive arguments. Towards bringing out the underlying physics, we further endeavor in a reciprocal identity based analysis. Following this approach, we unveil the time complexities of the droplet movement, which appear to be sufficient to rationalize the complex movement patterns observed through the comprehensive simulation studies. These results can be of profound importance in designing droplet based microfluidic systems in an oscillatory flow environment.

Figure 1

(a) Model problem setup, showing a Newtonian droplet immersed into another density and viscosity matched immiscible Newtonian carrier fluid, within a parallel plate channel. We consider a rectangular 2D domain of dimensional length $L$ and height $H$. The domain of computation is bounded by periodically connected left and right boundaries $\left(S_L,S_R\right)$, and the top and bottom bounding walls $\left(S_T,S_B\right)$. Droplet of initial radius $a$ is placed at a transverse position $y_d$ form the bottom plate. (b) Typical velocity field at $\frac{1}{4}$ of the cycle of imposed forcing; (c) velocity profiles at different stages of the cycle are shown for the sake of comprehensiveness. The profile is shown for the rightmost boundary $S_R$; however, it remains the same without a droplet at all $x$.

Figure 2

(a) Variations in the order parameter $\varphi$ with $x$ in the absence of imposed flow compared with the $tanh$ profile at steady state for $a/H=0.375, \mathrm{Ca}=0.33, \mathrm{Cn}=0.02$, and $\mathrm{\Delta }x=0.01$. The inset represents the contour of the order parameter. (b) Steady state shape of drop in plane Poiseuille flow for $a/H=0.4375$ and $\mathrm{Ca}=0.286$. (c) Steady state shape of drop in plane Poiseuille flow for $a/H=0.375$ and $\mathrm{Ca}=0.33$. Here we have considered $\mathrm{Re}=1$.

Figure 3

Characteristic shapes of droplets at different $a/H$ and $\mathrm{Ca}$ in plane Poiseuille flow condition, as obtained from the present simulations. For small $a/H$ and $\mathrm{Ca}$, the comparisons are made against the $O\left(\mathrm{Ca}\right)$ theoretical approximation [Eq. (9)]. For larger droplet $\left(a/H=0.4375\right)$ at higher $\mathrm{Ca}\left(=0.286\right)$, the present simulation result is compared against the simulation result of Mortazavi and Tryggvason [15]. Note that here the shapes are shown at 1:1 aspect ratio with axis-tight format. This makes droplets of all sizes to appear as almost the same size, thereby allowing better visualization of the droplets with small $a/H$. However, the actual sizes of the droplets are in accordance with the respective $a/H$.

Figure 4

The distribution of the streamlines, with respect to the droplet reference frame, upon equilibrium settling. The results are shown for $a/H=0.4375$, with $\mathrm{Re}=1$ and $\mathrm{Ca}=0.286$, and the direction of the flow is from left to right, with respect to the figure.

Figure 5

The time course of the migration characteristics of a droplet in a plane Poiseuille flow condition, for $a/H=0.4375$ and $0.375$ (indicated by the open markers), with $\mathrm{Re}=1$ and 4, respectively, and $\mathrm{Ca}=0.286$. The parametric setup is in tune with the setup of Mortazavi and Tryggvason [15]; their results are indicated by the corresponding solid lines. The adopted numerical setup consists of a 2D rectangular domain of dimensionless size $\mathrm{length}\times \mathrm{height}=3\times 1$ with grid size $\mathrm{\Delta }x=0.01$. The dimensionless parameters related to the phase field method are taken as $\mathrm{Cn}=0.02$ and $\mathrm{Pe}=10^6$.

Figure 6

Trajectory of a confined droplet ($a/H=0.4375$ and $\mathrm{Ca}=0.286$) in an oscillating flow field, for different $\mathrm{St}$. The numerical setup corresponds to $\mathrm{Cn}=0.02, \mathrm{\Delta }x=0.01$, and $\mathrm{Pe}={10}^6$, taking accuracy and computational expenses into combined consideration. Please refer to the movie in the Supplemental Material [54] for visual appreciation of the pathway that the droplet follows.

Figure 7

Temporal and spatial variations in the net force acting on the droplet at two different $\mathrm{St}$. Here, we present the characteristics with respect to $\theta =t\mathrm{St}$, to have a common basis for analysis for different $\mathrm{St}$. Panels (a,b) show the variations in the $x$ and $y$ components, respectively, $F_x$ and $F_y$ of the net force acting on the droplet. Panel (c) describes the force magnitudes at different locations ($x_d, y_d$) of the droplet. The inset in (b) magnifies in the variations in $F_y$ at $\mathrm{St}=0.5$ for better visualization. The results are shown for the same setup as in Fig. 6.

Figure 8

Characteristics of the pathway of a droplet with $a/H=0.4375, \mathrm{Re}=1$, and $\mathrm{Ca}=0.286$, in an oscillating flow field at different $\mathrm{St}$. Corresponding grid independent and $\mathrm{Cn}$ independent studies are presented in the Appendix. The inset in $u_{dy}\left(\theta \right)$ magnifies the domain $\theta =10\mathrm{to}25$, for clarity.

Figure 9

Variations in drop velocities $\left(u_{dx},u_{dy}\right)$ with the imposed forcing $\left(\psi \right)$ for (a) $\mathrm{St}=0.75$, (b) $\mathrm{St}=1$, and (c) $\mathrm{St}=3$. Other parameters have the following values: $a/H=0.4375, \mathrm{Ca}=0.286, \mathrm{Re}=1$, and $y_d=0.525$.

Figure 10

The migration characteristics of a droplet in an oscillating flow field, for $\mathrm{St}=1$ at different (a) $\mathrm{Ca}$ (keeping $a/H=0.4375$) and (b) $a/H$ (keeping $\mathrm{Ca}=0.286$), with $\mathrm{Re}=1$ for both cases.

Figure 11

(a) Variations in drop perimeter and drop shape with time for $\mathrm{St}=1$. (b) Variation of drop perimeter with time for three different values of $\mathrm{St}=0.5,1$, and 2. Other parameters have the following values: $a/H=0.4375, \mathrm{Ca}=0.286, y_d=0.525$, and $\mathrm{Re}=1$.

Figure 12

Comparison of the present simulation results on the streamwise migration characteristics with the $O\left(1\right)$ theoretical approximation. In the figure, the markers represent the simulation data points while the solid lines represent the theoretical approximations. Note that the light shaded band appearing in the $u_{dx}$ characteristics are the markers representing simulation data points over a wide range of $\mathrm{St}$. Here the simulation data are taken for $a/H=0.4375$ and $\mathrm{Ca}=0.286$.

Figure 13

Comparison of the present scaling estimation of the time complexity of $u_{dy}$ with the simulation data points for different $\mathrm{St}$. Here, the simulation data are taken for $a/H=0.4375$ and $\mathrm{Ca}=0.286$.

Figure 14

Grid and Cn independent test for the migration characteristics of a droplet with $a/H=0.4375, \mathrm{Re}=1$, and $\mathrm{Ca}=0.286$, in an oscillating flow field at $\mathrm{St}=1$. The figure also demonstrates a comparison between the different numerical setups, as decided by the combinations of $\mathrm{Cn}$ and $\mathrm{\Delta }x$.