Fiber alignment in oscillating confined shearing flows

Fiber orientation in oscillatory shearing flow was studied both experimentally and numerically. Optical measurements were made using a custom flow cell containing rigid, noncolloidal fibers suspended at high concentration in a Newtonian fluid. Simulations that account for hydrodynamic drag and excluded volume predict fiber alignment in the vorticity direction for some conditions, in agreement with the measurements. Vorticity alignment was found to be a complex function of strain amplitude and fiber concentration, confinement, and aspect ratio.

Figure 1

Fibers were manufactured from the poly(methyl methacrylate) core of fiber optic cables of diameter $d$ by cutting them in segments of length $L$. Various aspect ratios of $A=L/d$ were generated: (a) $L=5.2\pm 0.2$ mm, $d=0.46\pm 0.06$ mm, and $A=11\pm 2$; (b) $L=5.2\pm 0.2$ mm, $d=0.23\pm 0.02$ mm, and $A=23\pm 2$; and (c) $L=10.4\pm 0.2$ mm, $d=0.46\pm 0.06$ mm, and $A=23\pm 2$.

Figure 2

(a) Schematic of the shear cell as viewed from above. The suspension is sheared by the transparent acrylic belt (red) which is driven by rotating the cylinders (white) at either end of the cell. A laser sheet (green) of thickness $250\mu \mathrm{m}$, centered in the gap, fluoresces the fluid and enables imaging of the particle orientation distribution. (b) Photograph showing the controller and imaging equipment that surrounds the shear cell. A stepper motor attached to the cylinders drives the shearing flow, which is oscillated using a controller that also actuates the shutter and camera. (c) View of the shear cell from the same direction as the camera. The laser sheet fluoresces the suspension in the flow-vorticity plane.

Figure 3

(a) Example image showing fibers in the flow-vorticity plane, where $\alpha$ is the angle between the projection of the fiber in this plane with the flow direction. (b) Fibers within each image were identified and the orientation of each was determined. Only those particles at least one fiber length from the free surface and bottom plate were included in calculations of the orientation distribution.

Figure 4

(a) The $x$, $y$, and $z$ directions are the flow, gradient, and vorticity directions, respectively. The angle between the fiber projection in the flow-vorticity plane and the flow direction is $\alpha$. (b) Each fiber of aspect ratio $A$ is defined by its center-of-mass position ${\mathit{x}}_i$ and orientation ${\mathit{p}}_i$. A Hertzian repulsive force acts to separate pairs of fibers when their minimum separation distance $h_{ij}$ indicates an overlap [refer to Eq. (3)].

Figure 5

The order parameter $S_{\alpha }$ as a function of the oscillation number $N$ of the simple contact force simulations used in previous calculations [17] compared to the change in the forcing to a Hertzian contact force. The conditions used were $H=1.5L$, $A=11$, $\varphi =0.20$, and ${\gamma }_0=2.5$; both curves were produced using the sine-wave oscillation method and the initial distributions are identical. Averaging is performed over all fibers within the gap and over 12 runs.

Figure 6

Visualization of an initial condition for the fibers and their predicted distribution after $N=400$ and 4000 oscillations. Fibers generally rotated into alignment with the vorticity direction for this set of conditions, which consisted of $H=1.5L$, $A=11$, $\varphi =0.20$, and ${\gamma }_0=2.5$ with a square-wave oscillation. Vorticity alignment is reflected by the increased values of $S_{\alpha }$, which were calculated using all fibers in the gap.

Figure 7

Results from simulations with $H=1.5L$, $A=11$, $\varphi =0.20$, ${\gamma }_0=2.5$, and a square-wave oscillation. (a) Illustrations of slices of the suspension at the center of the gap (left column) and near the wall (right column) evolving from an initial state ($N=0$) through the steady-state structure at $N=4000$. Reported values of $S_{\alpha }$ were calculated only over those fibers intersecting each slice. (b) The order parameter $S_{\alpha }$ as a function of the number of oscillations $N$ for the square-wave model presented in Sec. 2 using every fiber in the gap (circles), fibers intersecting the center (open triangles), and fibers within a diameter of the wall (square symbols). Order parameters measured from experiments are also shown (filled triangles). The inset shows the calculated average number of fibers ${\nu }_S$ present in the wall and center slices as a function of $N$ from the square-wave model. Error bars represent the standard deviation of the mean values calculated from multiple realizations of either the simulations or experiments.

Figure 8

Experimental and simulation results for $H=1.5L$, $A=11$, and $\varphi =0.20$. (a) The bulk order parameter $S_{\alpha }$ measured in previous experiments [15] is compared to simulation results, including those of Snook et al. [17]. The inset indicates the oscillation number $N_R$ at which $S_{\alpha }$ was determined for each ${\gamma }_0$. (b) The order parameter $S_{\alpha }$ measured from experiments are compared to simulations using square-wave oscillations. Simulations show $S_{\alpha }$ as calculated using all fibers in the gap, though the open triangles and dashed line used only those fibers intersecting the center plane of the gap when calculating $S_{\alpha }$.

Figure 9

Bulk values of $S_{\alpha }$ at $H=1.5L$, $A=11$, $\varphi =0.20$, and ${\gamma }_0=2.5$ from simulations. (a) The order parameter $S_{\alpha }$ as a function of the number of oscillations $N$ is different for the sine-wave and square-wave patterns of shear and measurement, and the inset shows the first nine cycles. (b) The order parameter as a function of $N$ when using the square-wave and recording the orientation distribution at end and half point of each cycle.

Figure 10

Calculations and measurements performed with $A=11$ and $\varphi =0.20$. (a) The order parameter $S_{\alpha }$ was evaluated for fibers intersecting the center of the gap for simulations and experiments at different $H$ and ${\gamma }_0$; here, $S_{\alpha }$ was measured at $N=1283$, the largest common number of oscillations across every experiment and simulation. (b) The bulk, center, and wall-adjacent values of $S_{\alpha }$ as a function of $N$ for $H/L=3.0$ and ${\gamma }_0=1.5$. The inset shows the calculated average number of fibers ${\nu }_S$ that intersect the center and wall plane as a function of $N$.

Figure 11

Simulation results for $A=11$ and $\varphi =0.20$. (a) The bulk order parameter $S_{\alpha }$ was evaluated for multiple gap sizes $H/L$ at $N=4000$ across multiple ${\gamma }_0$. The inset shows the maximum $S_{\alpha }$ for each gap size $H$ (occurs at ${\gamma }_0=2.5$) as a function of $H/L$. (b) Spatial variation of $S_{\alpha }$ for various $H$ when ${\gamma }_0=2.5$, where $h/L$ is the distance from the wall [see Fig. 2]. The inset shows the calculated average number of fibers ${\nu }_S$ as a function of $h/L$.

Figure 12

Measurements and simulations with $A=23$, $H/L=1.5$, and $\varphi =0.10$. The order parameter $S_{\alpha }$ from experiments (solid symbols) is compared to simulation results where $S_{\alpha }$ was evaluated for fibers intersecting the center of the gap (open symbols). The results are reported at the common terminal oscillation $N=1395$ of the experiments. The bulk order parameter $S_{\alpha }$ evaluated from simulation results is also shown.

Figure 13

(a) Bulk values of $S_{\alpha }$ from simulations for $N=2340$, $H=1.5L$, and $A=11$ and 23 for a range of concentrations. Some experimental results [16] are also shown for comparison; the inset indicates the oscillation number at which the experimental data were reported. (b) Simulations results for the bulk values of $S_{\alpha }$ are replotted by scaling the strain amplitude by $\varphi /{\mathrm{\Phi }}_c$, where ${\mathrm{\Phi }}_c=0.4$ as found by Franceschini et al. [16] from rheological measurements.

Figure 14

Average number of contacts (${\nu }^{\left(c\right)}$) extracted from simulations with $H=1.5L$ and $A=11$ and 23. (a) The total number of fiber contacts with other fibers and bounding walls as a function of the oscillation number $N$. Results below, near, and above the critical rescaled strain amplitude of ${\gamma }_0\varphi /{\mathrm{\Phi }}_c=1$ are shown for volume fractions $\varphi =0.10$ and 0.20. (b) The total number of contacts are compared at $N=4000$, after having subtracted the minimum value of ${\nu }^{\left(c\right)}$ at $N=4000$ for each volume fraction. The number of contacts markedly increases at ${\gamma }_0\varphi /{\mathrm{\Phi }}_c\approx 1$. The inset shows the spatial distribution of contacts for $A=11$ and $\varphi =0.2$, where $h/L$ is the distance from the bounding wall.