Discontinuous Transition in a Laminar Fluid Flow: A Change of Flow Topology inside a Droplet Moving in a Micron-Size Channel

Even at moderate values of Reynolds number [e.g., $\mathrm{Re}=O(1)$] a curved interface between liquids can induce an abrupt transition between topologically different configurations of laminar flow. Here we show for the first time direct evidence of a sharp transition in the speed of flow of a droplet upon a small increase of the value of the capillary number above a threshold and the associated change of topology of flow. The quantitative results on the dependence of the threshold capillary number on the contrast of viscosities and on the direction of transition cannot be explained by any of the existing theories and call for a new description.

Figure 1

(Top) Droplet of length $l$ in a microchannel of width $w$. Neither the speed of the droplet, nor the topology of the streamlines are a priori known (dashed lines mark the unknown streamlines outside the droplet). The continuous liquid can bypass the droplet ($\beta =u_d/u_c<1$) or the droplet can outrun the average speed of the continuous phase ($\beta >1$). (Bottom) The experimental setup: single droplets were formed on demand [22] and their speed was monitored with the use of a stereoscope and a linear camera.

Figure 2

$\beta$ plotted against $l$ for $\mathrm{Ca}<{\mathrm{Ca}}_{\mathrm{TR}}$ (top) and for $\mathrm{Ca}>{\mathrm{Ca}}_{\mathrm{TR}}$ (bottom). The values of the capillary numbers are given in the graphs. The contrast of viscosities is $\lambda =0.33$.

Figure 3

(a) Crossover in the speed of the droplets upon an increase of the capillary number above the threshold: the value of ${\beta }_{\min }$ exhibits a discontinuity at the transition for $\lambda <1$. (b) A diagram illustrating the transition from the simple to more complicated pattern of flow in droplets in square capillaries. (c) The relation between the threshold values $\lambda$ and Ca as well as the orientation of the transition is different than in the case of circular capillaries [9].

Figure 4

Diagrams of the flow filed in the droplet: (top) $\mathrm{Ca}=5\times {10}^{-4}<{\mathrm{Ca}}_{\mathrm{TR}}$ and (bottom) $\mathrm{Ca}=2\times {10}^{-3}>{\mathrm{Ca}}_{\mathrm{TR}}$. Left panels show the cross section through the center of the droplet with the dots marking the flow coinciding with the orientation of $u_d$ and crosses the counterflow. Center and right panels show the velocity fields obtained at the middle of the channel ($180\mu \mathrm{m}$ from the top wall) and the corresponding schematic representations. The bar codes the speed in $\mathrm{mm}/\mathrm{s}$ in a frame moving with the droplet.