Dynamics of rigid fibers interacting with triangular obstacles in microchannel flows

Fiber suspensions flowing in structured media are encountered in many biological and industrial systems. Interactions between fibers and the transporting flow as well as fiber contact with obstacles can lead to complex dynamics. In this work, we combine microfluidic experiments and numerical simulations to study the interactions of a rigid fiber with an individual equilateral triangular pillar in a microfluidic channel. Four dominant fiber dynamics are identified: transport above or below the obstacle, pole vaulting, and trapping, in excellent agreement between experiments and modeling. The dynamics are classified as a function of the length, angle, and lateral position of the fibers at the channel entry. We show that the orientation and lateral position close to the obstacle are responsible for the fiber dynamics and we link those to the initial conditions of the fibers at the channel entrance. Direct contact between the fibers and the pillar is required to obtain strong modification of the fiber trajectories, which is associated to irreversible dynamics. Longer fibers are found to be more laterally shifted by the pillar than shorter fibers that rather tend to remain on their initial streamline. Our findings could in the future be used to design and optimize microfluidic sorting devices to sort rigid fibers by length.

Figure 1

Experimental method and geometrical parameters. (a) The rigid SU-8 fiber fabrication process. The right panel is a real image of the SU-8 fibers under the microscope. (b) The length and radius distributions of the SU-8 fibers. (c) A detailed sketch of the experimental setup showing the microfluidic channel, flow control, and data acquisition. (d) Zoom in on the region of interest (ROI) showing the main geometrical parameters of the study (figure not to scale).

Figure 2

Numerical method. (a) Sketch showing the discretization of the fiber by rigid spherical beads connected by springs. (b) Representation of the Eulerian grid where the fiber and the obstacle are immersed. Fluid nodes are shown as blue circles and solid nodes as red squares. (c) Zoom on the contact between the fiber and the obstacle surface. The resulting repulsive force ${\mathbf{F}}^{\mathrm{R}}={\mathbf{F}}_{ij}^{\mathrm{R}}+{\mathbf{F}}_{ik}^{\mathrm{R}}+{\mathbf{F}}_{il}^{\mathrm{R}}$ is not strictly normal to the obstacle surface (the red line) because of the discretization using spherical beads.

Figure 3

Velocity fields in the vicinity of the pillar in the experiments and simulations. (a) Normalized velocity magnitude and streamlines obtained experimentally by $\mu \mathrm{PIV}$ (left) and computed by LBM simulation (right). The thick red line is the flow separatrix. (b) Velocity profiles along the $x$ axis [horizontal lines in panel (a)]. (c) Velocity profiles along the $y$ axis [vertical lines in panel (a)]. $U_{\mathrm{center}}$ is the velocity magnitude at the channel centerline.

Figure 4

The four different fiber dynamics observed experimentally and accurately reproduced by simulations. The first line of each panel shows images from the experiments, the second line represents the chronophotograph of the simulation and the third line compares the trajectories of the fiber center of mass in the experiment (dots) and in the simulation (solid line). (a) The fiber goes below the pillar. Initial conditions are: ${\theta }_0=0^{\circ }, y_0/h_{\mathrm{obs}}=0.197$, and $L/l_{\mathrm{obs}}=0.94$ both in the experiment and simulation. (b) The fiber goes above the pillar. Initial conditions are: ${\theta }_0=0^{\circ }, y_0/h_{\mathrm{obs}}=0.678$, and $L/l_{\mathrm{obs}}=0.64$ both in the experiment and simulation. (c) Pole-vaulting. Vertical black double-arrow indicates the lateral shift that results from the interaction with the obstacle. Initial conditions are: ${\theta }_0=-3.7^{\circ }, y_0/h_{\mathrm{obs}}=0.373$, and $L/l_{\mathrm{obs}}=0.82$ both in the experiment and simulation. (d) Permanent trapping. Initial conditions are: ${\theta }_0=8^{\circ }, y_0/h_{\mathrm{obs}}=0.351$, and $L/l_{\mathrm{obs}}=0.75$ both in the experiment and simulation. The colors in the two last rows of each panel indicate the time with the color-code indicated at the bottom of the figure. All figures share the same scale bar (100 $\mu \mathrm{m}$).

Figure 5

Effect of the fiber initial configuration on the resulting dynamics. (a) Comparison between experiments (open symbols) and simulations (closed symbols). The fiber lengths are $0.5l_{\mathrm{obs}}\le L\le 1.5l_{\mathrm{obs}}$ in the experiments and $L=0.8l_{\mathrm{obs}}$ in the simulations, which is the average length of the fibers in the experiments. The dashed line represents the position of the flow separatrix at the entrance of the channel. (b) Simulated dynamics for different fiber lengths ranging in $0.5l_{\mathrm{obs}}<L<1.4l_{\mathrm{obs}}$. Red circles: the fiber goes below the pillar; green squares: the fiber goes above the pillar; blue triangles: pole-vaulting; yellow diamonds: permanent trapping.

Figure 6

Effect of the fiber contact configuration on the dynamics. (a) Definitions of the contact angle ${\theta }_{\mathrm{c}}$ and the contact position $y_{\mathrm{c}}$. (b) Phase diagram showing the dynamics of the fiber as a function of ${\theta }_{\mathrm{c}}$ and $y_{\mathrm{c}}$ in simulations where the fiber is initially in contact with the pillar at a well-controlled position $({\theta }_{\mathrm{c}}={\theta }_0,y_{\mathrm{c}}=y_0)$. The fiber length is $L/l_{\mathrm{obs}}=1$. Red circles: the fiber goes below the pillar; green squares: the fiber goes above the pillar; blue triangles: pole-vaulting; yellow diamonds: permanent trapping. (c) Sketch showing the motion of fibers starting at a contact position $y_{\mathrm{c}}/h_{\mathrm{obs}}=0.4$ and ${\theta }_{\mathrm{c}}=-{45}^{\circ },-{15}^{\circ },{15}^{\circ }$, and ${45}^{\circ }$. Same color code as in panel (b) is used to represent the fiber dynamics, light colors correspond to the contact position while darker colors to subsequent moment. Black streamline: flow separatrix.

Figure 7

Influence of the fiber initial position on the contact configuration. (a) Phase diagram representing the dynamics of fibers initially located far away from the pillar as a function of their contact configuration ${\theta }_{\mathrm{c}}$ and $y_{\mathrm{c}}$. Open symbols: experiments; closed symbols: simulations at $L/l_{\mathrm{obs}}=1$. (b) Typical examples of chronophotographs illustrating the high sensitivity of the contact configuration on $y_0$ (top), ${\theta }_0$ (middle), and $L$ (bottom). Red line: flow separatrix.

Figure 8

Effect of the fiber initial configuration and length on the lateral displacement $\delta$. (a) Comparison between experiments (thin edges) and simulations (thick edges). The fiber lengths are $0.5l_{\mathrm{obs}}\le L\le 1.5l_{\mathrm{obs}}$ in the experiments and $L=0.8l_{\mathrm{obs}}$ in the simulations, which is the average fiber length in the experiments. The dashed line represents the position of the flow separatrix at the entrance of the channel. (b) Lateral displacement computed by numerical simulations while varying the initial angle ${\theta }_0$, the initial lateral position $y_0$ and the fiber length $L$. (c, d) Typical examples of chronophotographs and trajectories showing the influence of the fiber length and initial configuration on the lateral displacement.The black streamline is the flow separatrix.

Figure 9

Influence of contact on the fiber trajectory. (a) Phase diagram showing a close link between direct contact and high lateral deviation. The dashed line represents the position of the flow separatrix at the entrance of the channel. (b) Probability of $\delta$ with and without contact in the simulations. Inset shows the experimental probabilities. Initial conditions are: $-{10}^{\circ }<{\theta }_0<{10}^{\circ }, 0<y_0/h_{\mathrm{obs}}<1$, and $0.5<L/l_{\mathrm{obs}}<1.5$ both in the experiments and simulations. (c) Typical example of chronophotographs and trajectories in the absence of contact (blue: flow from left to right, yellow: flow from right to left). Reversibility of the fiber motion is shown by the superimposition of blue and yellow trajectories. Small lateral deviation, $\delta$, is obtained (this case corresponds to the highest observed value of $\delta$ in the absence of contact). (d) Typical example of chronophotographs and trajectories in case of contact, showing irreversibility of the fiber motion (blue and yellow trajectories do not superimpose), and larger lateral displacement is observed.

Figure 10

Effect of fiber-obstacle HI on fiber motion. (a) Chronophotographs and comparison of the trajectories of the fiber COM between the simulations with and without fiber-obstacle HI. Fiber velocity (b) and orientation (c) as a function of the horizontal distance from the obstacle $x/l_{\mathrm{obs}}$. Blue line: with fiber-obstacle HI. Red dashed line: without fiber-obstacle HI.

Figure 11

Velocity fields and profiles around the triangular pillar computed by LBM for different channel heights: (a) $H_{\mathrm{ch}}=40\mu \mathrm{m}$, (b) $H_{\mathrm{ch}}=80\mu \mathrm{m}$, (c) $H_{\mathrm{ch}}=120\mu \mathrm{m}$. Panels (d) and (e), respectively, show the velocity profiles along the $x$ and $y$ axes. $U_{\mathrm{center}}$ is the velocity magnitude at the channel centerline.

Figure 12

Hydrodynamic forces along three permanently trapped fibers. Blue and red arrows, respectively, show the hydrodynamic forces above and below the contact point (yellow dot). Their moment $M_{\mathrm{u}}$ and $M_{\mathrm{l}}$ on both sides of the contact point counterbalances, which prevents the rotation of the fiber. The fiber length is $L/l_{\mathrm{obs}}=1$ and the initial conditions are: (a) $y_0/h_{\mathrm{obs}}=0.35, {\theta }_0=2.5^{\circ }$, (b) $y_0/h_{\mathrm{obs}}=0.325, {\theta }_0=5^{\circ }$, (c) $y_0/h_{\mathrm{obs}}=0.375, {\theta }_0=-{10}^{\circ }$.

Figure 13

Influence of the fiber initial position, initial orientation and length on the contact configuration. (a) Angle and lateral position differences between the initial and first contact configurations as a function of (${\theta }_0,y_0$) for a fiber length $L/l_{\mathrm{obs}}=1$. The colors and angle of the lines, respectively, indicate $(y_{\mathrm{c}}-y_0)/h_{\mathrm{obs}}$ and ${\theta }_{\mathrm{c}}-{\theta }_0$. (b) Influence of the fiber length $L$ and initial condition (${\theta }_0,y_0$) on the contact angle ${\theta }_{\mathrm{c}}$. (c) Fiber orientation difference with respect to the initial angle, $\mathrm{\Delta }\theta =\theta \left(x\right)-{\theta }_0$, as a function of the $x$ position of the fiber's center of mass for $y_0/h_{\mathrm{obs}}=0.425$ and ${\theta }_0=-7.5^{\circ }$. The triangular symbols and dashed lines are the positions of the first contact with obstacle surface. The inset shows snapshots of two fibers of length $L/l_{\mathrm{obs}}=0.6$ and $L/l_{\mathrm{obs}}=1.4$ approaching the obstacle. When the longer fiber hits the obstacle at $t=t_1+0.16\text{s}$, the shorter fiber is still transported and rotated by the flow until it first touches the pillar at $t=t_1+0.28\text{s}$, leading to very different contact angles between the two fibers.

Figure 14

Effect of the pillar roughness on the fiber dynamics. (a) Discretization of the pillar using beads (top) and six quadratic Bézier curves (bottom). (b) Phase diagram showing the fiber dynamics computed with the rough pillar discretized using spherical beads. (c) Phase diagram showing the fiber dynamics computed with the perfectly smooth pillar described by Bézier curves. The fiber length in the simulations is $L=0.8l_{\mathrm{obs}}$. Data are only shown in the range $0.3\le y_0/h_{\mathrm{obs}}\le 0.5$ for which direct fiber-pillar contact occurs. The dashed line in (b) and (c) represents the position of the flow separatrix at the channel entrance. Circles: below; squares: above; triangles: pole-vaulting; diamonds: trapping.

Figure 15

Sketch showing the computation of the repulsive force between a fiber bead and a quadratic Bézier curve.