4.1 Comparison with baseline

We reproduce the results of the baseline cases of Gazzola et al. (Reference Gazzola, Mimeau, Tchieu and Koumoutsakos2012a ) in figure 3(a) for clarity. As we increase $Re$ , we observe two distinct, sharp transitions (at $Re\approx 17$ and $Re\approx 82$ ), between which the slave gets transported (that is, carried along) by the master due to its linear motion (and associated wake) alone. The extent of this transport range is determined by the presence of a maximum in the favourable pressure gradient (consequence of the low-pressure region generated by the wake behind the master) around $Re\sim 50$ , and its decay in the lower and higher $Re$ limits (supplementary material, § 3.5). The above transport behaviour is then quantified in figure 3(b), where we plot the normalized master–slave separation distance $s_{x}/D_{m}$ against the non-dimensional time $T=2U_{l}t/D_{m}$ . We note that the non-dimensional time $T$ is representative of the number of body lengths ( $D_{m}$ ) travelled by the master. In this plot, a plateauing $s_{x}$ or $s_{x}\rightarrow 0$ (ergo, $\text{d}s_{x}/\text{d}T\leqslant 0$ ) indicates transport/trapping and is characteristic of $17\lessapprox Re\lessapprox 82$ . Throughout the paper, we will use the terms transport and trapping interchangeably based on the above mathematical condition, the calculation details of which can be found in the supplementary material, § 4.2. For cases with $Re\lessapprox 17$ and $Re\gtrapprox 82$ , $s_{x}$ increases with time indicating that the slave is left behind as the master moves forward and hence it is not transported. At $Re\approx 17$ and $Re\approx 82$ , $s_{x}$ plateaus with increasing $T$ , indicating that the slave is transported while maintaining a constant distance from the master (and thus travels with the same speed). These two $Re$ then denote the boundaries of transition between trapping and non-trapping regimes. Moreover, at $Re\approx 82$ , this transition is sharp, as small changes in $Re$ (from $80$ to $82$ to $90$ ) lead to large changes in the transport characteristics. Drawing from prior works (Lin et al. Reference Lin, Liang and Zhao2018), we expect the system to be sensitive to changes in the initial master–slave separation $s_{x}(0)$ . We then use this as a proxy to quantify the system sensitivity by perturbing $s_{x}(0)/D_{m}$ by $\pm 2\,\%$ and observing the resulting transport characteristics. We depict this for a few key $Re$ in figure 3(c) for clarity, wherein the shaded regions highlight deviations associated with the perturbation. As expected, the system response is seen to be very sensitive at the transitional $Re\approx 82$ , relative to larger and smaller $Re$ . We consider this as an opportunity to enhance transport. Indeed a carefully constructed system perturbation can ‘pull’ the system response from the non-transport regime into the sensitive region and make it ‘jump over’, enabling transport. We choose to control the system by perturbing the flow via viscous streaming, by oscillating the master at different levels of intensity $\unicode[STIX]{x1D701}=R_{o}/Re$ , and report our observations in figure 3(d–g). For $Re=90>82$ (above transition, no baseline transport) shown in figure 3(d), mild oscillations ( $\unicode[STIX]{x1D701}=0.5$ , blue bands) assist transport and bring the system into the sensitive region. Increasing $\unicode[STIX]{x1D701}$ further pushes the system well beyond the transition, enabling (and quickening) slave transport (green and red bands). Figure 3(e–g) shows the oscillation-enabled characteristics for $Re=82,50$ and $17$ , where we consistently observe a similar behaviour.

We recognize that the system response is dependent not only on initial separation distance $s_{x}(0)/D_{m}$ , wake $Re$ and oscillation $R_{o}$ (encoded in $\unicode[STIX]{x1D701}$ ) effects, but also on the slave’s inertia captured through $D_{s}/D_{m}$ . Therefore, we undertake a parametric investigation to further characterize these dependencies by varying $D_{s}/D_{m}$ between 0.125 and 0.375, $Re$ between $10$ and $200$ and $\unicode[STIX]{x1D701}$ between $0$ and $2$ . In figure 4, we report the corresponding $Re$ – $\unicode[STIX]{x1D701}$ phase spaces in which we indicate whether the slave is trapped or not. From these phase spaces, we observe that oscillations remain beneficial for transport (figure 4 a–f) across a range of $Re$ , with smaller slaves more amenable to being trapped. Moreover, as the slave size relative to the master increases, we observe that the trapping phase narrows down and eventually vanishes for large $D_{s}/D_{m}$ ratios, consistent with the results of Gazzola et al. (Reference Gazzola, Mimeau, Tchieu and Koumoutsakos2012a ). A minimal scaling analysis provides intuition on the narrowing of the trapping phase. The propulsive force at a given $Re$ , obtained by integrating the pressure and viscous shear forces along the slave’s surface, scales approximately linearly with $D_{s}$ (since the circumference is $\unicode[STIX]{x03C0}D_{s}$ ). The inertia of the slave, however, is proportional to its mass (i.e. area) and hence scales as $D_{s}^{2}$ . The forces necessary to overcome inertia then grow faster than the forces exerted on the slave’s surface, as a result of which the trapping phase narrows (supplementary material, § 3.4).

Interestingly, in the lower $Re$ regime, a new phase appears in which oscillations introduce an adverse effect on transport. This is quantified by the ratio $s_{x}/s_{b}$ (at $T=10$ ), where $s_{b}$ is the separation distance of the reference baseline ( $\unicode[STIX]{x1D701}=0$ ) at the corresponding $Re$ . Thus, in this region $s_{x}/s_{b}>1$ . We add that in some cases, the adverse phase can coexist with the trapping phase (i.e. they are not mutually exclusive). In such cases, the baseline exhibits transport and adding oscillations does not disrupt this behaviour, and the slave simply trails behind the master at larger distances, when compared to the baseline. We defer the explanation of this adverse effect on transport to § 4.2, since the analysis carried out there naturally extends to this. Here, we instead draw attention to the fact that the system retains its inherent sensitivity to $Re$ regardless of the slave’s inertia, illustrated by the sharp line separating the trapping/non-trapping phases (supplementary material, § 4.4). We conclude that introducing oscillations, modulated by $\unicode[STIX]{x1D701}$ , enhances inertial particle transport across a range of $Re$ as well as across different slave inertias. The causal mechanism, be it viscous streaming or other wake–oscillation interactions, is ascertained in the next sections. In the following sections, we will fix $D_{s}/D_{m}=0.25$ for consistency with previous works (Tchieu et al. Reference Tchieu, Crowdy and Leonard2010; Gazzola et al. Reference Gazzola, Mimeau, Tchieu and Koumoutsakos2012a ), unless specified otherwise.

4.2 Flow analysis at the transport/non-transport boundary

Figure 5.

Analysis. Fixing $Re=82$ and increasing $\unicode[STIX]{x1D701}$ leads to (a) better transport due to flow acceleration in the wake. The cylinders and their trajectories are drawn until $T=14$ with the dashed black lines representing the baseline case. (b) Time-averaged velocity profiles (over 20 oscillation cycles) at the marked stations of (a). The observed accelerations are due to streamline contraction in the wake, visualized in (c) for representative cases without ( $\unicode[STIX]{x1D701}=0$ , white streamlines) and with ( $\unicode[STIX]{x1D701}=2$ , orange streamlines) streaming. Corresponding streaming-only fields – (d) streamlines and (e) velocities at the same stations as (a,b) – explain the behaviour of (a–c).

Here, we focus on the fluid mechanisms at play by drawing parallels between the system at the transitional $Re=82$ (in the case of $D_{s}/D_{m}=0.25$ ) with different $\unicode[STIX]{x1D701}$ and the corresponding pure streaming cases. Figure 5(a) depicts the slave trajectories for increasing $\unicode[STIX]{x1D701}$ , highlighting improvement in transport. This is due to a corresponding increase in the wake velocities as seen in figure 5(b), where we plot the time-averaged velocities, at the different stations marked in figure 5(a). The averaged streamlines (streamfunction iso-contours) of figure 5(c) for cases with ( $\unicode[STIX]{x1D701}=2$ ) and without ( $\unicode[STIX]{x1D701}=0$ ) oscillations explain this increase in wake velocity. Adding oscillations to the master displaces the streamlines such that their density is increased behind the master. This localized density increase (or alternatively ‘contraction’ of streamlines) corresponds to an increase in upstream flow velocities (figure 5 b). The degree of this contraction increases with $\unicode[STIX]{x1D701}$ . This behaviour is consistent with the corresponding streaming-only patterns of figure 5(d) where larger oscillation intensities increasingly push the outer eddies towards the mid-plane, causing streamline contraction and increasing flow velocities directed towards the master, as further quantified in figure 5(e). We thus identify streamline contraction as the primary cause for transport enhancement, with viscous streaming as driving mechanism.

Finally, this analysis helps explain the appearance of the adverse phase observed in figure 4 at low $R_{o}$ (hence at low $\unicode[STIX]{x1D701}$ and $Re$ , since $R_{o}=\unicode[STIX]{x1D701}\cdot Re$ ). Indeed, as we lower $R_{o}$ , the streaming flow field transitions from the FBL regime to the Stokes-like regime. This, in turn, causes an unfavourable change in the streaming flow direction, which in the Stokes-like regime is directed away from the master. Therefore, the adverse flow reduces the ability of the master to transport the slave. This is further manifested in a corresponding streamline dilatation, instead of contraction. More details and quantitative analysis can be found in the supplementary material, § 4.5.

4.3 Robustness

Given the sensitivity of the system (figure 3 c) to initial horizontal slave separation, it is important to further characterize the spatial robustness of streaming-enhanced transport. We expand our study by initializing (in separate simulations) slaves not only directly behind the master, but all around it (azimuthal positions $\unicode[STIX]{x1D703}$ ), while retaining the same surface-to-surface distance ( $0.1D_{m}$ ). We choose a representative $\unicode[STIX]{x1D701}=0.5$ and vary only $Re$ . Figure 6(a) illustrates the slave trajectories up to $T=14$ (for consistency) for representative $Re$ , coloured by their initial azimuthal positions. We observe that streaming is beneficial when compared to the baseline, even at longer times ( $2T=28$ ), as the number of transported slaves is observed to increase. Only in cases where the baseline already effectively traps (i.e. $Re=50$ ) does the number of slaves transported not improve (but also is not reduced). Nonetheless, streaming is observed to improve the rate at which slaves approach the master (i.e. their velocities), consistent with figure 3(d–g). More details of the quantification and analyses of robustness to initial radial and azimuthal position perturbations can be found in the supplementary material, §§ 4.6 and 4.7.

Figure 6.

Robustness. (a) Seeding slaves (suitably resized for visualization purposes) azimuthally around the master (oscillating with $\unicode[STIX]{x1D701}=0.5$ ) and plotting their trajectories (until $T=14$ ) reveal that the streaming-enhanced transport strategy is robust to changes in the initial azimuthal positions. Additionally, streaming is beneficial for transport when compared to the baseline ( $\unicode[STIX]{x1D701}=0$ ) cases across all $Re$ investigated, even at longer times ( $2T=28$ ). The supplementary material, § 4.6, contains additional details. Here we report the trajectories only until $T=14$ for consistency with the other results. The trajectories shown in (a) can be correlated to (b) the associated pure streaming fields (coloured by streamfunction with the solid lines indicating streamlines). Dashed lines indicate different stations at which we plot (c) streaming-induced velocities. We do not consider slaves in the comparison against pure streaming fields, with the implicit assumption that they do not markedly perturb the flow.

These differences can be understood by analysing the corresponding pure streaming (figure 6 b,c) flow fields. Figure 6(b) shows the time-averaged streamlines, where we also depict the stations at which the corresponding streaming velocities (figure 6 c) are portrayed. At $Re=17$ , a thicker DC layer (with reverse flow, blue line in figure 6 c) ‘cushions’ the master and prevents the slaves from attaining close proximity. All the other cases have a DC layer of almost constant thickness, thus leading to qualitative differences, highlighted by slaves closely surrounding the master’s posterior. Comparing $Re=50$ and $Re=82$ , we notice that in the latter slaves with initial positions $|\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}|\gtrsim 15\unicode[STIX]{x03C0}/20$ (figure 6 a, colourbar) are transported further, due to more favourable upstream velocity profiles. Additionally, slaves almost directly behind the master (with initial positions $|\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}|\lesssim \unicode[STIX]{x03C0}/5$ ) are left farther behind for $Re=90$ as compared to $Re=82$ and $Re=50$ . This is explained by the fact that $Re=90$ is characterized by a baseline that cannot trap. Mild oscillations ( $\unicode[STIX]{x1D701}=0.5$ ) are then just barely capable of ‘pulling’ the slaves close enough to be trapped and transported, consistent with figure 3(d). We conclude that oscillation-based transport strategy is robust overall (even to radial perturbations, see supplementary material, § 4.7). Moreover the resulting slave trajectories are found to be consistent with streaming-induced velocities, reinforcing the hypothesis that this is indeed the responsible transport enhancement mechanism at play.

4.4 Design

Figure 7.

Design. (a) Inspired by streaming from triangles (Tatsuno Reference Tatsuno1975), we design the ‘bullet’ shape of (b) to induce a more favourable streaming field (more streamline contraction at $R_{o}=45$ ). We note that (b) compares the streamlines of pure streaming fields of the master (top, circular cylinder; bottom, bullet) without linear translation. (c) When the master linearly translates ( $Re=90$ ) and oscillates, the new design exhibits improved transport characteristics owing to its associated streamline contractions. This improvement in transport holds in spite of our initial configuration (inset) that penalizes the bullet by placing the slave further away.

If streaming is indeed the responsible agent for transport enhancement, we should be able to design new geometries that produce more favourable streaming fields that actually translate to improved slave transport once tested in our set-up of figure 1(b). We start by considering streaming only, and draw inspiration from the visual investigation of Tatsuno (Reference Tatsuno1975) on streaming triangles. These are shown to produce a large DC recirculation region (figure 7 a), which can be leveraged to trap and carry along passive cargoes. We then ‘borrow’ two key elements of this geometry – rear high-curvature tips and fore–aft symmetry breaking – to design a master with a ‘bullet’ cross-section (figure 7 c, inset; refer to supplementary material, § 4.8, for shape parametrization). This geometry is an intermediate morph between a circle and a triangle, and is chosen to facilitate the comparison in the case of linear transport ( $\unicode[STIX]{x1D701}=0$ ) between the cylinder and the bullet itself (similar front profile). Due to the break in the fore–aft symmetry of the bullet, the direction of oscillation is important. Here, consistent with the findings of Tatsuno (Reference Tatsuno1975) (figure 7 a), we switch from vertical (transverse) to horizontal (longitudinal) oscillations, so that the large DC recirculation region is located posteriorly, where the slaves are positioned. We note that this is an integral part of our streaming-field design strategy, and as long as $Re$ and $\unicode[STIX]{x1D701}$ are kept consistent, does not detract from the comparison with vertically oscillating cylinders. In figure 7(b), we show that the longitudinally oscillating bullet constricts the streamlines further than a transversely oscillating circle. This is reflected in its transport characteristics (figure 7 c): while the circular cylinder performs better than the bullet when streaming is not active (no oscillations, pure master linear motion), the bullet outperforms its circular counterpart when streaming is enabled, even for mild oscillation intensities ( $\unicode[STIX]{x1D701}=0.5$ ).

Figure 8.

Topology. We study the flow topological response to incremental shape changes (from the (a) circle to the (d) bullet) via line integral convolution of the streaming velocity field at $R_{o}=45$ . This reveals a route for the rational design of streaming-based transport devices. The key idea involves introducing rear high-curvature points and fore–aft symmetry breaking to enlarge/strengthen DC layer streaming, the extent of which is indicated by the saddle points and marked by solid black lines. (b,c) Intermediate shapes that illustrate the process.

The reason for the increased streamline contraction above lies in the flow topology change associated with the introduction of multiple curvatures and asymmetry in the master geometry (figure 7 b). To understand how this topological transition occurs, we progressively morph (figure 8) the circular cylinder into the bullet, and track the behaviour of critical points in the flow (these are saddle and half-saddle points (Hunt et al. Reference Hunt, Abell, Peterka and Woo1978), which provide a sparse yet complete representation of the global field (Perry & Chong Reference Perry and Chong1987)). In the context of streaming, saddle points indicate the physical extent of the DC layer, marked by solid black lines in figure 8. Morphing may create, destroy, merge or displace saddle and half-saddle points, which we leverage to manipulate the DC layer. Introducing high rear curvature points to the circle in figure 8(b) creates two new half-saddles, which allow the rear saddle point to move away from the surface, thus ‘unfolding’ the DC layer. The extent of this offset and the corresponding strength of the DC streaming region are related to the magnitude of the tip curvature. This is seen in figure 8(c) where tip curvature increase enlarges the DC layer. Streamlining the master geometry further strengthens the DC layer and gives rise to our final bullet (figure 8 d) design. Thus by manipulating the shape (and the oscillation direction) of the master – which dictates the streaming flow topological response – we can rationally design configurations that improve slave transport.

We conclude this section wherein we have conducted extensive two-dimensional investigations to analyse, understand and design streaming-based transport systems. In the next section, we carry forth these ideas to three dimensions.