The motion of long drops in rectangular microchannels at low capillary numbers

Abstract

Drop flow in rectangular microchannels has been utilized extensively in microfluidics. However, the pressure-gradient versus flow-rate relation is still not well understood. We study the motion of a long drop in a rectangular microchannel in the limit the capillary number $Ca\rightarrow 0$ ($Ca=\unicode[STIX]{x1D707}U/\unicode[STIX]{x1D70E}$, where $U$ is the constant drop velocity, $\unicode[STIX]{x1D707}$ is the viscosity of the carrier liquid and $\unicode[STIX]{x1D70E}$ is the interfacial tension). In this limit, the moving drop looks like the static drop and has two end caps connected by a long column, which is surrounded by thin films on the microchannel wall and by menisci along the microchannel corners. Integral axial force balances on the drop fluid and on the carrier liquid surrounding the drop relate the carrier-liquid pressure gradient to the drop-fluid pressure gradient and the contact-line drag. The contact-line drag is argued to be the same as that for a long bubble (which has been determined by Wong et al. (J. Fluid Mech., vol. 292, 1995b, pp. 95–110)) if the viscosity ratio $\unicode[STIX]{x1D706}\ll Ca^{-1/3}$ and $\unicode[STIX]{x1D706}\ll L$, where $\unicode[STIX]{x1D706}=\bar{\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D707}$, $\bar{\unicode[STIX]{x1D707}}$ is the drop viscosity and $L~(\gg 1)$ is the dimensionless drop length. Thus, the force balances yield one equation relating the two pressure gradients. The two pressure gradients also drive unidirectional flows in the drop and in the corner channels along the long middle column. These coupled flows are solved by a finite-element method to yield another equation relating the two pressure gradients. From the two equations, we determine the pressure gradients and thus the unidirectional velocity fields inside and outside the drop for $\unicode[STIX]{x1D706}=0$–100 and various microchannel aspect ratios. We find that in the limit $LCa^{1/3}\rightarrow 0$, the contact-line drag dominates and the carrier liquid bypasses the drop through the corner channels alongside the drop. For $LCa^{1/3}\gg 1$, the contact-line drag is negligible and the corner fluid is stationary. Thus, the drop moves as a leaky piston. We extend our model to a train of long drops, and compare our model predictions with published experiments.