Dynamic jamming of dense suspensions under tilted impact

Dense particulate suspensions can not only increase their viscosity and shear thicken under external forcing, but also jam into a solidlike state that is fully reversible when the force is removed. An impact on the surface of a dense suspension can trigger this jamming process by generating a shear front that propagates into the bulk of the system. Tracking and visualizing such a front are difficult because suspensions are optically opaque and the front can propagate as fast as several meters per second. Recently, high-speed ultrasound imaging has been used to overcome this problem and extract two-dimensional sections of the flow field associated with jamming front propagation. Here we extend this method to reconstruct the three-dimensional flow field. This enables us to investigate the evolution of jamming fronts for which axisymmetry cannot be assumed, such as impact at angles tilted away from the normal to the free surface of the suspension. We find that sufficiently far from solid boundaries, the resulting flow field is approximately identical to that generated by normal impact, but rotated and aligned with the angle of impact. However, once the front approaches the solid boundary at the bottom of the container, it generates a squeeze flow that deforms the front profile and causes jamming to proceed in a nonaxisymmetric manner.

Figure 1

Experimental setup. (a) The suspension sample is placed in a cylindrical container. An impactor motivated by an actuator is placed above the suspension. An ultrasound transducer is placed under the container, held by an optical stage. (b) Examples of the $B$-mode images obtained at different $y$ positions by moving the ultrasound transducer. Relevant dimensions of the setup are labeled in the figure.

Figure 2

Velocity isosurfaces under impact from ${\theta }_{\text{I}}={10}^{\circ }$. The orange, green, blue, and red surfaces are the isosurfaces of $v_{\text{L}}=0.4U_{\text{p}}$ at four consecutive time steps. The time difference between adjacent isosurfaces is 3.7 ms. The surface of suspension is at $z=0.03$ m. Curves at the bottom outline the projections of these isosurfaces in the $x\text{−}y$ plane.

Figure 3

Velocity field in the $x\text{−}z$ plane and isocontours for $v_{\text{L}}$ during impact at time (a), (d), and (g) $t=6.25\mathrm{ms}$, (b), (e), and (h) $t=9.5$ ms, and (c), (f), and (i) $t=12.75$ ms. The surface of the suspension and the bottom of the container are at $z=0.03$ m and 0 m, respectively. The green arrows label local velocities in the $x\text{−}z$ plane. The colors show $v_{\text{L}}$, with the upper limit (white) corresponding to $U_{\text{p}}$. The impactor hits the suspension at incident angles (a)–(c) ${\theta }_{\text{I}}=0^{\circ }$, (d)–(f) ${\theta }_{\text{I}}={20}^{\circ }$, and (g)–(i) ${\theta }_{\text{I}}={40}^{\circ }$. The open black arrows show the directions of impact, with exaggerated penetration depths.

Figure 4

Contours of $v_{\text{L}}=0.4U_{\text{p}}$ for ${\theta }_{\text{I}}$ ranging from $0^{\circ }$ to ${40}^{\circ }$, each rotated clockwise by ${\theta }_I$ accordingly. Each contour is obtained by averaging over three to nine experiments. The impactor hits the surface of the suspension at $x=0$ m and $z=0.03$ m, which is labeled by the cross, for all contours here after rotation. The dashed green line shows the impact direction.

Figure 5

Plots of (a) the $k_{\text{L}}$ and $k_{\text{T}}$ values and (b) the ratio $k_{\text{L}}/k_{\text{T}}$ at different incident angles ${\theta }_{\text{I}}$. Error bars show the standard deviation from multiple measurements at each angle. The horizontal dash-dotted lines in (a) indicate the average values of $k_{\text{L}}$ and $k_{\text{T}}$, which are 8.3 and 4.9, respectively. The horizontal dashed line in (b) indicates $\sqrt{3}$. The overall standard deviations, using the data from all impact angles ${\theta }_{\text{I}}$, are 2.3 for $k_{\text{L}}$, 1.2 for $k_{\text{T}}$, and 0.42 for $k_{\text{L}}/k_{\text{T}}$.

Figure 6

Four components in the shear-rate tensor (3): (a) $\partial v_r/\partial r$, (b) $\partial v_z/\partial z$, (c) $\frac{1}{2}(\partial v_z/\partial r)$, and (d) $\frac{1}{2}(\partial v_r/\partial z)$. Black curves show contours from $v_z=0.1U_{\text{p}}$ to $v_z=0.9U_{\text{p}}$, with increments of $0.1U_{\text{p}}$. The contour at $0.5U_{\text{p}}$ is highlighted by the thick curves.

Figure 7

Distribution of the scalar strain rate $\dot{E}$ at four different time points (a) 2 ms, (b) 4.5 ms, (c) 7 ms, and (d) 9.5 ms after the tip of the impactor reached the surface of the suspension. The contours are defined in the same way as in Fig. 6. Panel (c) and the flow shown in Fig. 6 are at the same time point.

Figure 8

(a) Maximum strain rate ${\dot{E}}_{\text{max}}$ at different polar angles $\theta$ with respect to the initial contact point of the impactor head and the suspension surface. (b) Front width ${\mathrm{\Delta }}_{46}$, which is the distance between the contours of $v_z=0.4U_{\text{p}}$ and $v_z=0.6U_{\text{p}}$.

Figure 9

Velocity directions for ${30}^{\circ }$ impact at (a) $t=9.5$, (b) $t=16.25$, (c) $t=22$, and (d) $t=29$ ms. Grids with a $v_L\ge 0.02$ m/s are colored red, blue, or white according to ${\theta }_v-{\theta }_I$, as shown in the colormap. Grids with $v_L\le 0.02$ m/s are colored gray, which should be distinguished from the white region. The surface and bottom of the suspension are at $z=0.03$ m and 0 m, respectively. Dashed green lines are extension lines of the impact rod. The flow field here is obtained by averaging nine repeated experiments at the same ${\theta }_{\text{I}}$.

Figure 10

Schematic illustration of the flow when the front interacts with the solid boundary. (a), (b), and (c) correspond to Figs. 9, 9, and 9, respectively. The boundary between the solidlike (yellow) and fluidlike (gray) regions is the front. The green and blue arrows represent their motion, respectively. The static region is colored orange and its expansion is shown by the dashed orange curve and arrows. The curved surfaces of the suspension are sketched based on the original ultrasound images.

Figure 11

Demonstration of obtaining ${\dot{E}}_{\text{max}}$ and ${\mathrm{\Delta }}_{46}$ at angle $\theta$. The colormap shows $\dot{E}$. The dashed green line labels the direction at angle $\theta$. The length of the solid green line represents ${\mathrm{\Delta }}_{46}$ at angle $\theta$.