Simulations of a heavy ball falling through a sheared suspension

As we shall later examine a sphere sedimenting in a sheared monolayer, a reasonable approximation when gravity acts in the plane of shear (i.e., when gravity acts in the negative x- and y-directions in Table 2), we start by letting a sphere fall through a monolayer of randomly assembled smaller beads. We plot in Fig. 8 the transient fall speed of the large sphere (normalised by its Stokes velocity in pure solvent) as it sediments through this suspension. We find that for three well-differentiated repulsion strengths, the falling ball decelerates before achieving a mean falling velocity which has only a weak dependence on the repulsion magnitude, k. In contrast to Fig. 4, here we have given the suspension some time to equilibrate after applying the repulsion force, before activating gravity on the large ball.

The deceleration is caused by the build-up of concentration beneath the ball, which is also quantified in Fig. 8. With the local area concentration starting at \(c\approx 0.55\), we see that below the particle (upper line), the concentration increases until finding an equilibrium level: from \(c \approx 0.63\) at the highest repulsion to \(c \approx 0.72\) at the lowest. In the lowest repulsion sample, the area concentration is still comfortably below the maximum circle packing fraction of 0.91 with hexagonal packing, but is approximately the square packing fraction of \(\pi /4 \approx 0.79\). The concentration behind (lower line) initially decreases, as the falling ball creates a wake, before increasing again as the wake enters the more concentrated zone created by the falling ball. The wake is unexpected normally in reversible Stokes flow, but occurs due to the repulsive force, which is inherently irreversible.

We perform the same test on a cube of small particles containing the sedimenting sphere. In contrast to the monolayer simulation, the falling velocity and concentration in Fig. 9 stabilise much more quickly. We see a quick deceleration for all three repulsion strengths before reaching a fairly constant velocity. The mean falling velocity is shown to have a stronger dependence on the repulsion strength, k, than in the monolayer, and in fact suggests the opposite dependence: weaker strengths lead to faster mean velocities. In full 3D, the number of neighbouring particles—the ‘kissing number’—is much higher than in a monolayer, and the slower falling speeds with a higher repulsion force can be explained by higher (upwards) repulsion from the large number of smaller particles beneath the falling ball.

The concentration build-up beneath the ball is also shown in the same figure. As expected, behind the ball (lower line), we see a reduction from the starting volume concentration of \(\phi \approx 0.33\) to \(\phi \approx 0.30\). Unlike in the monolayer simulations, however, the concentration in front of the ball (upper line) shows very little change. This can again be rationalised by the larger number of neighbours beneath the ball in full 3D giving stronger resistance to compression, as well as the extra available space for them to move into in full 3D.

We perform one 3D preshear simulation for each of the three gravity directions shown in Table 3. In each, the initial position of the large sphere is set to be towards the ‘top’ of the cube of smaller particles, according to what will be the fall direction during sedimentation. The large sphere, ready for release, is for now made neutrally buoyant. The velocity of the sphere is then measured in what will be the gravity direction, so that we may compare this shear-induced motion to the motion under sedimentation.

Applying two cycles of shear in each simulation, we find that the oscillatory preshear creates structure in the suspension bulk. Figure 10 shows the particle density functions of slices in each plane before and after an oscillatory shear (which, recall, is always in the xy-plane). We see that the preshear slightly emphasises the circular structure near the centre of the plots in the neutral planes, whereas in the shear plane, we see distinctive alignment of particles parallel and perpendicular to the shear direction: square packing.

The observation of square packing agrees with the hypothesis in Blanc et al. [7]: that particles align in the vertical direction under oscillatory shear. However, as we shall see, in our simulations, this does not lead to faster sedimentation.

We now look at the velocity of the large, neutrally buoyant sphere, in a cube of smaller particles, in our three gravity directions. The variations of the velocity as a function of phase during a shear cycle—from now on, ‘velocity plots’—are shown in Fig. 11. In each, the velocity of the large sphere is plotted against the system shear for the second oscillation, and the horizontal dashed line marks the mean velocity. The oscillation begins with motion to the right, and we plot falling velocity as positive. We find that in all cases, the mean velocity over a shear cycle is negative. That is to say, the ball, when neutrally buoyant, travels upwards over the course of the shear. This is purely an effect of our asymmetric geometry: the large ball is placed near the top of the region occupied by particles, so it is driven upwards by the repulsion from the larger number of particles below it than above it. Furthermore, in all cases, the ball translates vertically, mostly in a figure-of-eight shape, although in the flow gradient direction, where the variation in velocity is smaller, this is less clear. Interestingly, we observe that velocity plot in the vorticity direction traverses a figure of eight in the opposite direction to the velocity in the flow direction. We will discuss this observation further in Sect. 8. We also find that the velocity changes are of similar magnitude for sedimentation in the flow and vorticity directions, but are opposite in sign.

We examine the importance of the shear frequency and repulsion strength on a fully 3D sample. The particle density function plots in the shear plane are shown in Fig. 12. Here we see nine parameter combinations: simulations with different repulsion strengths, k, and different oscillation frequencies, f. Note that since we nondimensionalise the frequencies in terms of the weight of the falling ball in Eq. (4), there is ambiguity when the ball is neutrally buoyant. Although we show all nine parameter combinations for comparison with the later sedimentation phase, we note that here there is only one independent parameter, k / f, which is demonstrable in the diagonal symmetry in the plots.

We see that in the shear plane, the imposed oscillatory shear creates a square-packing structure in the suspension bulk. At low repulsions and high frequencies, there is little difference from the circular structures seen prior to shearing (see Fig. 6), but the effect is clearer for the increasing repulsion strength and the decreasing frequency. We can quantify the degree of square packing in the particle density function plots as a function of angle round the second dark band, \(g(\theta )\). Figure 13 shows two extreme cases. The right-hand plot—the high-frequency, low-repulsion case—shows a mostly constant concentration of particles around the band, but the left-hand plot—the low-frequency, high-repulsion case—shows peaks at \(\theta = 0\), \(\pi /2\) and \(\pi \), corresponding to the observed square packing.

The velocity plot of the falling sphere in the system, sedimenting in each of the three directions, is shown in Fig. 14. In all three gravity directions, the fall velocity of the ball oscillates over a shear cycle. Note that sedimentation in the flow gradient and flow directions leads to figures of eight which are traversed in the opposite sense to that in which it is traversed when the ball sediments in the vorticity direction: in the first two cases, the ball slows down during the central half of the shear cycle (between \(-0.5\) and 0.5), whereas in the latter case, it speeds up. We describe a mechanism for this behaviour in Sect. 8. The third figure in Fig. 14 therefore reproduces the figure-of-eight shape observed in experiment, but traversed in the opposite direction. The ‘steps’ in the first two figures are a sensitivity to a detail of the Stokesian Dynamics method—specifically, how often we calculate the long-range mobility matrix. Increasing how often we calculate this gives smoother plots, as in Fig. 11, but comes with the considerable time expense of simulation.

The velocity plots appear to be a sum of the unsheared falling speed, Fig. 9, and the oscillatory behaviour from the shear, Fig. 11. We do not observe significant changes in the mean velocity from the unsheared case.

We now look at the effect of changing the frequency and repulsion strength. Some elucidation of the effects at work here is provided by examining the repulsive energy in the system over time. This may be calculated from the magnitudes of the repulsive forces in the system, for the exponential force we are using (Eq. (5)): the repulsive energy, E, for a pair of particles i and j with surface separation \(h_{ij}\) is the work done against the repulsive force in bringing them to that position, starting from far apart:This is then summed over all pairs of particles in the system.

Figure 15 measures the repulsive energy in a monolayer system over the second oscillation of shear as the ball sediments in the velocity gradient direction (see Table 2) for different parameter values. We see broadly three regimes. The first, in the bottom-right of the figure, is a U-shape: the high-frequency, low-repulsion regime. Starting from the centre of the plot and moving to the right, the repulsion energy increases as the system is sheared and compressed, and then reverses mostly symmetrically as the system is unsheared and uncompressed. In this limit, the oscillation is too fast for the repulsion to act: the hydrodynamic forces associated with the shear dominate over the weight and repulsion forces, essentially making the particles passive. At high frequencies and low repulsions, the energy plot is therefore reversible. We begin to see hysteresis above and to the left of this plot, as decreasing the frequency gives the repulsion time to act, and increasing the repulsion forces no longer leaves the shear forces dominant.

These hysteresis loops are emphasised in the central plot in the figure—with intermediate parameter values—where we have a figure of eight. Again starting from the centre of the plot and moving rightwards, the system is compressed as it is sheared, building up structure and repulsion energy. At the end of the shear, though, the repulsion force has time to act, whereas it did not in the previous regime. The particles repel each other, the structure is destroyed, and the system relaxes. Then, as the system is sheared in the opposite direction, the particles are compressed again and the structure and energy increases. This alternating between construction and destruction generates the figure of eight.

Finally, the bottom-left plot of the same figure—low repulsion, low frequency—shows ascending loops. This regime is just a slower version of the aforementioned one: the system has time to relax when the shear direction changes, but as shears take longer with low frequencies, the concentration builds up more beneath the falling ball during the shear cycle and thus the system repulsion energy increases over time.

These different regimes loosely match with the velocity plots shown in Fig. 16, measured over the same single oscillation of the shear flow for the same nine different parameter combinations. Again, the oscillation begins with motion to the right, and we plot falling velocity as positive.

Starting from the bottom-left, the low-repulsion, low-frequency case, we see the velocity decrease over time as the concentration builds up beneath the ball, agreeing with the loops in the repulsive energy plot. In the central and right-hand plots we see figures of eight, as the particle slows down under the increased repulsion from the structured, compressed suspension. As the system relaxes and the structure is lost—the small particles ‘space out’ under repulsion—the large particle is able to fall with less hindrance, and its speed increases until the reverse shear increases the local concentration and repulsion forces again.

Looking from left to right across the figure, we can see that increasing the frequency increases the velocity difference over the shear: speeding up the oscillation by a factor of 10 leads to an approximate doubling of the percentage velocity gain. This suggests that increasing the frequency of oscillation further, or reducing the weight of the falling ball, should lead to greater velocity differences, allowing the particle to rise over parts of the shear. This would mimic the behaviour seen experimentally. These results are all for sedimentation in the velocity gradient direction through a monolayer. It is not until we shear a fully 3D cell, simulating the experiments, that we see the ball travel upwards.

The plots shown are averages of two simulations with different initial suspension distributions. It is worth noting that although on average, the plots give figures of eight, individual runs can be sensitive to initial configurations, producing anomalous plots. We understand that this is also an occasional feature of the experiments. We find that the bottom-right parameters in Fig. 16 (low repulsion and high frequency) is the most likely to produce such results, and an example is compared with the average in Fig. 17.

Here, we see behaviour that appears counterintuitive: when the shear is moving to the right the ball accelerates, but when the shear is moving to the left it decelerates, which would appear to break the intrinsic symmetries of the system. However, with both repulsive forces and the weight of the ball being very small in this limit compared with the hydrodynamic forces associated with the external shear flow, all the particles are essentially passive, and simply move with the shear flow, hindered by each other’s presence in a purely hydrodynamic way. The falling ball is pushed primarily horizontally, with small perturbations in the vertical direction. One of these is caused by the presence of the small particles, which happen (for this simulation) to be arranged in such a way that the perturbation is upwards when the motion is to the left and (by reversibility) downwards when the motion is to the right. The other is caused by the particle’s weight, which always drives it downwards. Because we have made velocities dimensionless using the Stokes velocity of the large sphere in the absence of small particles, these velocities both appear order 1 in our plots.

Figures 18 and 19 show the repulsive energy and velocity plots for the falling sphere in a cube of smaller particles, undergoing oscillatory shear. The repulsive energy is of a similar magnitude to in the monolayer case, but the loops retain a stronger U-shape in the moderate parameter values. The velocity plots produce more consistent figures of eight in the central and right-hand column, allowing for the appearance of steps which are an artefact of the method. This is not surprising, given that the increased number of neighbours in fully 3D suspensions offers more consistent resistance to the movement of the large sphere. As mentioned when looking at preshear, the velocity plots are traversed in the opposite sense to which it is traversed in the monolayer simulations. In this geometry, the build-up of structure corresponds to the ball speeding up, which is lost when the structure is destroyed. The mechanism is discussed fully in Sect. 8. The left-hand plots are mostly unaffected by the slow shear, showing the velocity slowly increasing during the shear: this simply reflects the large particle getting closer to the bottom of the simulation box over the shear period, thus having fewer particles in its way.

Some observations are the same as in the monolayer: increasing the frequency by a factor of 10 leads to a similar proportional increase in the velocity difference over the shear cycle. In our high frequency cases, this leads to the ball traversing upwards over the course of the cycle: the same behaviour seen in the experiments, although with the plot again traversed in the opposite direction. Changing the repulsion strength does not seem to lead to significant changes in the plots.

The concentration of the suspension is varied in Fig. 20 to \(\phi =0.1\) and 0.2, as well as the existing \(\phi =0.4\) which we use for the rest of the study. Here the ball is falling in the vorticity, or negative z-, direction (see Table 2), with the moderate frequency and repulsion parameters \(f=1\), \(k=2\). We see that, as expected, decreasing the concentration increases the mean fall velocity. However, it does not appear to dramatically change the variation in velocity over the shear, which remains at approximately \(\pm 0.05\).

The amplitude of the shear is varied for the same moderate parameters in Fig. 21. We see that the mean velocity of the falling sphere over the shear cycle is mostly unchanged from the unsheared suspension, but the variation in velocity across the shear is increased by increasing the shear amplitude proportionally. This makes sense: increasing the amplitude increases the energy put into the system.