Effect of confinement on the deformation of microfluidic drops

We study the deformation of drops squeezed between the floor and ceiling of a microchannel and subjected to a hyperbolic flow. We observe that the maximum deformation of drops depends on both the drop size and the rate of strain of the external flow and can be described with power laws with exponents $2.59\pm 0.28$ and $0.91\pm 0.05$, respectively. We develop a theoretical model to describe the deformation of squeezed drops based on the Darcy approximation for shallow geometries and the use of complex potentials. The model describes the steady-state deformation of the drops as a function of a nondimensional parameter $\text{Ca}{\delta }^2$, where $\text{Ca}$ is the capillary number (proportional to the strain rate and the drop size) and $\delta$ is a confinement parameter equal to the drop size divided by the channel height. For small deformations, the theoretical model predicts a linear relationship between the deformation of drops and this parameter, in good agreement with the experimental observations.

Figure 1

(a) Schematic of the microchannel geometry (not to scale), showing the section for drop formation (1), the second oil inlet for the control of the flow velocity (2), and the test section in which drops deform (3). Thick arrows represent the direction of flow. The width at the test section ($w$) is indicated and the height $h$ of the channel is in the direction perpendicular to the figure. (b) Micrography of the channel, with a drop being accelerated by the injection of $Q_o^{\left(2\right)}$ and another one being deformed in the test section. The flow in the test section is visualized with fluorescent beads in the absence of drops and several photographs are superimposed to represent the oil streamlines.

Figure 2

Velocity field (arrows) and strain rate (gray scale) in the test region. Solid lines are drawn at the positions of the channel walls.

Figure 3

(a) Images of deformed drops. The white thick curve shows the digitally obtained contour of the drop. The red thin dashed curve corresponds to a fitted ellipse. (b) Deformation of a drop as it travels through the microchannel. $D$ is plotted against the normalized distance between the drop and the test section centers when the drop flows into (blue circles) and out of (red crosses) the test section. This case corresponds to data from channel I, with $R_o=58\mu$m and $G=9.1$ s${}^{-1}$.

Figure 4

(a) Maximum deformation of drops $D_{\max }$ as a function of drop size for constant strain rate $G=12.6$ s${}^{-1}$ in channel II. (b) $D_{\max }$ as a function of strain rate for fixed drop size $R_o=60\mu$m in channel I. In (a) and (b) the symbols correspond to experimental data and the solid lines correspond to power law fits. Insets: (a) Dependence of the fit coefficient $\alpha$ on the strain rate $G$. (b) Dependence of the fit coefficient $\beta$ on the drop size $R_o$. The symbols represent fit data and the solid line shows the average.

Figure 5

A quasi-two-dimensional drop subjected to a hyperbolic flow.

Figure 6

Solution of Eq. (12). The lower branch (solid blue line) corresponds to the physical solution, while the upper branch (dashed red line) is unphysical. The linear relationship $D_t=P$ is plotted as a black dash-dotted line for comparison. The vertical black dotted line shows the position of $P_{\mathrm{cr}}$.

Figure 7

$D_{\max }$ as a function of $P=\text{Ca}{\delta }^2$ for different drop sizes and strain rates from data taken in the two channels used in the study (see Sec. 2a). The symbols correspond to experimental data, and the black solid line and the red dashed line correspond to linear fits of the data from channel I ($D_{\max }=3.9P$) and channel II ($D_{\max }=5P$), respectively.