(160d) Motion of a Deformable Drop in Microchannels of Complex Shape

A theoretical model and a robust algorithm have been developed to simulate tight low-Reynolds number motion of a three-dimensional, freely suspended drop in plane microchannels of complex shape (assuming the channel depth but not necessarily the width to be large compared to the drop size). The channel profile may consist of an arbitrary number of straight line segments with sharp corners in an arbitrary configuration. This geometry provides a suitable model for drop transport in many microfluidic devices with multiple branch bifurcations. One example is the pinched-flow fractionation (PFF) device [1] consisting of two inlet channels at an angle with different flow rates, followed by a plane-parallel and widening sections, and the outlet. The drops fed into one of the input branches (at dilute concentration) are pressed by a buffer fluid (entering the second branch) to the wall of the pinched area, and eventually separate into the outlet according to droplet size. Another, simpler example is drop motion in a T-bifurcation with sharp corners. Here, the fluid flux Q=Q₁ + Q₂ through the upstream branch is split into the fluxes through the side (Q₁ ) and the downstream (Q₂) branches, with a specified partition ratio Q₁/Q₂; of primary interest is to determine the corresponding partition of the drop phase fluxes. Solid sphere transport through a PFF channel and a T-junction has been previously simulated [2]. The main goal of the present study is to determine how different these phenomena are for strongly deformable drops, and explore the effects of the capillary number and drop-to-medium viscosity ratio on the drop transport. In particular, for a T-junction and a large enough solid sphere (still well fitting the junction branches), it was demonstrated [2] that the particle-corner interactions (when the particle rolls around the corner before separating) can make the side branch totally inaccessible to particles, regardless of their initial location in the inlet, even for relatively strong fluid suction through the side branch. The case of a sufficiently deformable drop is much different in that the drop does not hit a corner but wraps around it with some clearance. The occurrence of particle-corner contacts did not allow us [2] to simulate the true temporal dynamics of particle motion (only the particle trajectories could be simulated). In contrast, for a sufficiently deformable drop capable of staying away from sharp corners, both the drop trajectories and temporal dynamics are captured by the present simulations.

The drop size is comparable (or close) to the width of the narrowest branch sections, but is generally much smaller than the overall channel domain size. This difference in the scales makes it difficult to address the problem by a standard boundary-integral (BI) algorithm (or other approaches) for the entire domain due to necessary resolution on the drop and the local vicinity as the drop proceeds through the channel, resulting in a very large total number of BI elements. Instead, we adapt the idea of the “moving frame” BI method [2]. Namely, the drop is embedded in a dynamically constructed computational cell (moving frame, MF) and the 3D BI problem is solved in the cell only, with the outer boundary conditions for the fluid velocity provided by the 2D flow that would exist in the channel without the drop. The MF cell is obtained by intersecting a cubic box around the drop with the entire channel domain, and it can acquire quite complex shapes as the drop proceeds. Parts of the MF boundary may be inside the flow domain, while other parts may belong to the channel walls, so our method captures strong drop-wall hydrodynamical interactions. The cutting box has an intermediate size, several times larger than the non-deformed drop but typically much smaller than the overall channel domain, thus resulting in many-fold computational savings compared to the standard approach. Robustness of the proposed algorithm is further demonstrated by an example of tight drop motion through a multiple bifurcation system (akin to those in experiments [3]).

[1] Nakashima M., Yamada M. and Seki M. 2004 “Pinched flow fractionation (PFF) for continuous particle separation in a microfluidic device”, in proceedings of 17^thIEEE International Conference on Micro Electro Mechanical Systems, Maastricht, The Netherlands (Piscataway, NJ), pp. 33-36.

[2] Zinchenko A.Z., Ashley J.F. and Davis R.H. 2012 “A moving-frame boundary-integral method for particle transport in microchannels of complex shape”. Phys. Fluids, vol. 24, 043302.

[3] Roberts B.W. and Olbricht W.L. 2006 “The distribution of freely suspended particles at microfluidic bifurcations”. AIChE J., vol. 52, pp.199-206.