Velocity oscillations and stop-go cycles: The trajectory of an object settling in a cornstarch suspension

We present results for objects settling in a cornstarch suspension. Two surprising phenomena can be found in concentrated suspensions. First, the settling object does not attain a terminal velocity but exhibits oscillations around a terminal velocity when traveling through the bulk of the liquid. Second, close to the bottom, the object comes to a full stop but then reaccelerates before coming to another stop. This cycle can be repeated up to 6 or 7 times before the object reaches the bottom to come to a final stop. For the bulk, we show that shear-thickening models are insufficient to account for the observed oscillations and that the history of the suspension needs to be taken into account. A hysteretic model, that goes beyond the traditional viscoelastic ones, describes the experiments quite well but still misses some details. The behavior at the bottom can be modeled with a minimal jamming model.

Figure 1

(a) Schematic view of the setup with, from left to right, a light source and diffusing plate, the container filled with suspension, above that the object with tracers attached, and a high-speed camera. (b) Microscopic picture of the cornstarch, used in the experiments.

Figure 2

The velocity $\dot{x}$ of the settling object versus time $t$, determined by two different methods, namely (i) using a local quadratic fit and (ii) employing a first-order difference. The two methods are shown for different time intervals of the fitting procedures, namely $4.0$, 12, and 20 ms, to illustrate the trade-off when choosing between higher spatial or temporal resolution.

Figure 3

The settling velocities $\dot{x}$ of a steel sphere with a diameter of 0.5 cm in glycerine [blue (gray) line] and a steel sphere of diameter 1.6 cm in a cornstarch suspension with $\varphi =0.41$ (black line) as a function of time $t$. The inset shows the last part of the actual trajectory, clearly showing the stop-go cycles near the bottom in cornstarch in the position versus time curve.

Figure 4

Settling velocity $\dot{x}$ of a stainless steel sphere (diameter $1.6$ cm) in a cornstarch suspension as a function of time $t$ and for different cornstarch packing fractions $\varphi$ varying from $0.35$ to $0.41$.

Figure 5

The settling velocity $\dot{x}$ versus time $t$ in a $12\times 12$ cm${}^2$ square container and a cylindrical container with a diameter of $5.0$ cm for a ball of $1.6$ cm diameter impacting with a velocity of $1.5$ m/s on a cornstarch suspension with a concentration of $\varphi =0.42$. For comparison, the result of a cylindrical disk settling in a quasi-2D setup is added.

Figure 6

Time evolution of the velocity $\dot{x}$ of a hollow ping-pong ball filled with different masses settling in a cornstarch suspension with $\varphi =0.44$. The buoyancy-corrected mass varies from $\mu =10$ to $\mu =132$ g. Also added is an experiment with a steel sphere of $\mu =217$ g, with the same diameter ($4.0$ cm) as the ping-pong ball. The inset shows the frequency of the bulk oscillations for the ping-pong ball.

Figure 7

Time evolution of the velocity $\dot{x}$ of a settling cylinder in a cornstarch suspension ($\varphi =0.44$) for different cylinder masses $m_{\mathrm{cyl}}$, varying from 40 to 120 g. A buoyancy-corrected mass cannot be used here since it changes along the trajectory of the cylinder.

Figure 8

The drag $D=m\ddot{x}-\mu g$ on a ping-pong ball with buoyancy-corrected mass $\mu =122$ g versus its velocity $\dot{x}$, calculated from the object's trajectory $x\left(t\right)$ during a settling experiment in a cornstarch suspension with $\varphi =0.44$.

Figure 9

(a) Schematic of the drag force $D$ defined by Eq. (7) as a function of the velocity $\dot{x}$, with the hysteresis loop between $\dot{x}=u_1$ and $\dot{x}=u_2$. (b) Schematic of the oscillatory solution of Eq. (1) using the drag force of (a): For suitable values of $B_1$, $B_2$, $u_1$, and $u_2$ the system alternately switches from the low to the high branch in the hysteresis loop and back. In these schematics, all quantities are in arbitrary units.

Figure 10

Comparison of the experimental results to the hysteresis model. (a) In the region where bulk oscillations are observed, the velocity of the sphere is plotted versus time for four different buoyancy-corrected masses ($\mu =$ 62, 82, 102, and 132 g) for both the experiment (colored symbols) and the model (black lines), where the values for $B_1$ and $B_2$ have been obtained from the experiment with the highest mass. (b) The corresponding drag $D=m\ddot{x}-\mu g$ versus velocity $\dot{x}$ plots also for both the experiment (colored symbols) and the model (black lines).

Figure 11

The drag coefficients $B_1$ and $B_2$ as a function of buoyancy-corrected mass $\mu$, now calculated from the experiment using Eq. (10) separately for each value of $\mu$. Below $\mu =$ 62 g no bulk oscillations could be discerned. The different symbols correspond to two different series of experiments.

Figure 12

Comparison of the experimental results to the hysteresis model for a cylindrical object in the region where bulk oscillations are observed. The velocity of the cylinder is plotted versus time for three different masses ($m_{\mathrm{cyl}}=40$, 80, and 120 g) for both the experiment (colored symbols) and the model (black lines), where the values for $B_1$ and $B_2$ have been obtained from the experiment with the highest mass.

Figure 13

Time evolution of the velocity during the stop-go cycles for the settling ping-pong ball for three different (buoyancy-corrected) masses, $\mu =17$, 77, and 132 g, and for the settling cylinder, also for three different buoyancy-corrected masses, $\mu =97$, 57, and 17 g. The noisy lines represent the experimental results and the dashed blue lines correspond to the model of Eq. (11).

Figure 14

Three quantities that characterize the first stop-go cycle as a function of buoyancy-corrected mass for the cylinder (blue crosses) and pingpong ball (red circles): (a) Distance $x_0$ to the bottom at which the first stop takes place. (b) The maximum velocity reached in the relaxation period after the first stop. (c) The time needed to reach the maximum velocity after the first stop. The solid lines in (b) and (c) represent the results obtained with the model of Eq. (11).

Figure 15

The duration $\Delta \tilde{t}$ (dashed blue curve) of and the traveled distance $\Delta \tilde{x}$ (solid red curve) during a stop-go cycle as a function of the logarithm of the relaxation parameter $\tilde{\kappa }$. Modeled with Eq. (14), using $\tilde{B}\ll 1$ and $c=1$.

Figure 16

The maximum velocities reached during the stop-go cycles in the experiments (circles) and the model (lines) per jump, normalized by the mean bulk velocity, positioned at cycle 0.

Figure 17

Time evolution of the velocity of a steel sphere with a diameter of 1.6 cm in three different suspensions: quartz flower, polystyrene beads, glass beads, and cornstarch. Clearly, both the bulk oscillations and the stop-go cycles are only observable in the cornstarch suspension.

Figure 18

Schematics of the three linear viscoelastic models discussed in this Appendix: (a) Maxwell fluid, (b) extended Maxwell fluid, and (c) modified Kelvin-Voigt solid.