Emerging Attractor in Wavy Poiseuille Flows Triggers Sorting of Biological Cells

Microflows constitute an important instrument to control particle dynamics. A prominent example is the sorting of biological cells, which relies on the ability of deformable cells to move transversely to flow lines. A classic result is that soft microparticles migrate in flows through straight microchannels to an attractor at their center. Here, we show that flows through wavy channels fundamentally change the overall picture. They lead to the emergence of a second, coexisting attractor for soft particles. Its emergence and off-center location depends on the boundary modulation and the particle properties. The related cross-stream migration of soft particles is explained by analytical considerations, Stokesian dynamics simulations in unbounded flows, and Lattice-Boltzmann simulations in bounded flows. The novel off-center attractor can be used, for instance, in diagnostics, for separating cells of different size and elasticity, which is often an indicator of their health status.

Figure 1

(a) The trajectory of a soft capsule (dashed line) in plane Poiseuille flow approaches the attractor at the channel center at $y=0$: The inset shows the channel cross section with the particle attractor (dash-dotted line). (b) A soft capsule in a wavy channel moves to either the off-center attractor (solid line) or to the center, depending on its initial position. The left inset shows the particle’s wavy motion around the trajectory’s mean (cf. animations in the Supplemental Material [38]). The right inset shows the wavy off-center particle attractor (dotted line) and the attractor at the channel center (dash-dotted line). For parameters, see [39].

Figure 2

(a) Two snapshots of a capsule are shown, resulting from their Stokesian dynamics simulation in an unbounded wavy flow field (black arrows). The flow is symmetric to the dashed vertical line, but the deformed particle is not (see, e.g., colored bars). (b) shows the second derivatives ${\partial }_x^2{u}_y(x)$ and ${\partial }_y^2{u}_y(x)$ of the vertical flow velocity $u_y$ at $y=0.7d$.

Figure 3

The averaged CSM velocity $⟨v_{m,y}{⟩}_{\lambda }$ of capsules and RBCs in unbounded wavy Poiseuille flow as a function of $\lambda /R$. $⟨v_{m,y}{⟩}_{\lambda }$ is given for two amplitudes $\tilde{u}=5$, 11. It is negative for small and rather large values of $\lambda /R$ and the particles migrate to the channel center. $⟨v_{m,y}{⟩}_{\lambda }$ is positive in the intermediate range of $\lambda /R$, where particles migrate away from the channel center. For further parameters, cf. [39].

Figure 4

The averaged migration velocity $⟨v_{m,y}{⟩}_{\lambda }/\tilde{u}$ for capsules in bounded flows obtained by LBM simulations as a function of the particle’s $y$ position and for the modulation amplitudes $ϵ=0.0$, 0.2, 0.3. The capsule migrates toward the center ($y=0$) at small $y$ below the repeller (empty circles) and otherwise to the second attractor (solid circles), confirming the analytical considerations and Stokesian dynamics simulations in unbounded flows. The arrows mark the migration direction, i.e., the sign of $⟨v_{m,y}{⟩}_{\lambda }$. For further parameters, cf. [56].

Figure 5

The $y$ position of the second attractors (solid lines) and the repellers (dashed lines) of capsules as a function of $ϵ$ for $\lambda /R=12$, 16, 20. Particles with an initial position below the repeller migrate to the channel center. The second attractor occurs for $ϵ$ larger than the respective critical values ${ϵ}_c=0.14$, 0.19, 0.23, which increase with increasing wavelength.