(519g) On the Mixing Induced By Tightly Confined Spheres in Bounded Shear Flows

In the past two decades microfluidics based technologies have made significant progress towards the development of lab-on-a-chip systems, especially for biomedical applications. Rapid and efficient mixing, however, still remains one of the toughest challenges due to miniaturization [1]. A microfluidic system is often dominated by viscous forces leading to low Reynolds number laminar flows. As a direct result, there is no turbulent mixing in the system and the time scales associated with diffusive mixing are large. Recent work by Zurita-Gotor et al. [2] has shown that in the presence of spherical particles under creeping flow conditions, the proximity of bounding walls gives rise to open loop flipping trajectories. These trajectories can cause a very large dispersion of both the particles themselves and fluid tracers. The large shear induced self diffusivity of dilute suspensions measured by Zarraga & Leighton [3] and Beimfohr et al. [4], for example, was shown to be a direct consequence of these flipping trajectories. Analytical models show that the dispersion grows as ¼ ϕϒa³/h where ϕ is the concentration, ϒ is the shear rate, a is the radius of the particle and h is the distance from the wall. This suggests that dispersion will be greatest close to the boundary walls (i.e. h = a). In actuality, however, as the particle approaches a wall, the wall retards the translational motion of the sphere causing the open loop trajectories (and corresponding dispersion) to vanish entirely. A single sphere rolling along a wall in a simple shear flow, for example, produces no dispersion via this mechanism.

In this work we explore the related situation where a sphere is tightly confined between two walls in simple shear flow. Because of the symmetry of the geometry, the sphere moves with the average velocity of the flow evaluated at the streamline corresponding to the sphere center, reopening the flipping trajectories. We demonstrate experimentally that this gives rise to rapid mixing across the central plane of a parallel plate system. The effect is examined by measuring the distributions of much smaller particles whose equilibrium positions in the gap are governed by a balance between inertial lift and sedimentation due to density mismatch. In the absence of tightly confined spacer particles, the measured positions of the tracers are in excellent agreement with that predicted from the lift model of Ho & Leal [5] and Stokes sedimentation. However, in the presence of extremely dilute concentrations (ϕ ~ 3x10^-6) of gap spanning spacer particles, the distributions become well mixed. This mixing effect, rather than being chaotic over the whole system, is confined to the bounds of the flipping envelope. Outside of the envelope, there is no effect of the presence of the spacer particles for such dilute systems. Thus, as the shear rate (and inertial lift) is increased, tracer trajectories initially follow the behavior expected in the absence of mixing (an equilibrium position close to the lower wall), but then abruptly transition to a distribution focused at the center.

References:

[1] C.Y. Lee, C.L. Chang, Y.N. Wang and L.M. Fu. Microfluidic Mixing: A Review. Int J Mol Sci. 2011; 12(5): 3263–3287.                                                                       

[2] M. Zurita-Gotor, J. Blawzdziewicz and E. Wajnryb. Swapping trajectories: a new wall induced cross-streamline particle migration mechanism in a dilute suspension of spheres. J. Fluid Mech., 592:447–469, 2007.                  

[3] I.E. Zarraga and D.T. Leighton. Measurement of an unexpectedly large shear-induced self diffusivity in a dilute suspension of spheres. Phys. Fluids., 14:2194–2201, 2002.                     

[4] S. Beimfohr, T. Looby and D.T. Leighton. Measurement of the shear-induced coefficient of self-diffusion in dilute suspensions. Proceedings of the DOE/NSF Workshop on Flow of Particles and Fluids, Ithaca, NY, 1993.                                                  

[5] B. P. Ho and L. G. Leal. Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech., 65(2):365-400, (1974).