3. Results and discussion

The present system shows a marked tendency to form clusters, as clear from figure 1, which displays two realizations at the same $Re=970$ and $\phi =0.28$, but different interfacial conditions. For comparison, we define $\chi _{{cl}}$ as the number of clustered particles normalized by the total number of particles. Remarkably, in the SL configuration (figure 1a), the fraction of clustered particles $\chi _{{cl}}$ is more than five times larger than in the DL2 configuration (figure 1b). The sizes of the clusters, approximated as the diameter $D_{{cl}}$ of the circles encompassing them, also greatly differ between the two cases. To compare the dominant mechanisms at play, in the following we derive expressions for the main forces governing the particle motion: capillarity, drag and lubrication. In the considered range of parameters, electrostatic forces, e.g. due to asymmetrically distributed charges on the particles (Aveyard et al. Reference Aveyard2002), are several orders of magnitude weaker.

Figure 1.

Example snapshots with detected clusters obtained at the same $Re = 970$ and $\phi =0.28$ from different configurations: (a) SL, $\chi _{{cl}} = 0.99$; and (b) DL2, $\chi _{{cl}} = 0.23$. Scale bars correspond to $L_{{f}}=35~{\rm mm}$. The particles in each cluster are represented with the same colour. Circles encompassing each cluster are displayed.

The expression of the capillary attraction is determined by the nature of the distortion of the liquid interface around the particles. Assuming small slopes, the shape of the interface can be decomposed into Fourier modes. These expand into decaying multipoles, with each multipole excited by its corresponding Fourier mode at the contact line (Stamou, Duschl & Johannsmann Reference Stamou, Duschl and Johannsmann2000). When body forces and the torque on the particle are negligible, the leading-order interfacial disturbance $h$ at a distance $r$ from its centre is the quadrupolar distortion (Liu, Sharifi-Mood & Stebe Reference Liu, Sharifi-Mood and Stebe2018)

(3.1)\begin{equation} h(r) = h_{{qp}} {d_{{p}}^{2}}/({4r^{2}}), \end{equation}

where $h_{{qp}}$ is the mode's amplitude.

Conversely, when body forces on the particle are significant, the interfacial profile is described by the modified Bessel function of the second kind and order zero, whose asymptotic behaviour is logarithmic for small $r$. To first order, the vertical balance between buoyancy and interfacial tension then yields (Vella & Mahadevan Reference Vella and Mahadevan2005)

(3.2)\begin{equation} h(r) = \tfrac{1}{2} \varSigma d_{{p}} {Bo} \ln(l_{{c}}/r). \end{equation}

Here, $l_{c}=(\Delta \rho _{{f}} g/\gamma )^{-1/2}$ is the capillary length between fluids with density difference $\Delta \rho _{{f}}$ under the action of gravity $g$ and interfacial tension $\gamma$; the Bond number ${Bo} \equiv ({(\rho _{{f}}-\rho _{{p}})g d_{{p}}^{2}})/{4\gamma }$, with $\rho _{{f}}$ the density of the fluid on which the particle of density $\rho _{{p}}$ is floating; and

(3.3)\begin{equation} \varSigma \equiv \tfrac{1}{3}(2\rho_{{f}}/\rho_{{p}}-1) -\tfrac{1}{2}\cos{\theta}+\tfrac{1}{6}\cos^{3}{\theta} \end{equation}

is a dimensionless buoyancy-subtracted weight of the particle with the contact angle $\theta$.

In all the considered configurations, ${Bo} \ll 1$ (table 1), which indicates that the distortion induced by body forces is negligibly small. For example, for a particle in the SL configuration, where it is predominantly immersed ($\theta < {\rm \pi}/2$), such distortion at the contact-line radius would be ${<}1~\mathrm {\mu }{\rm m}$, much smaller than the quadrupolar distortion of ${\sim }14~\mathrm {\mu }{\rm m}$. Thus, while the slower-decaying contribution from body forces becomes significant at large separations, the quadrupolar contribution remains dominant for the groups of adjacent particles considered here. Therefore, we write (Stamou et al. Reference Stamou, Duschl and Johannsmann2000; Liu et al. Reference Liu, Sharifi-Mood and Stebe2018)

(3.4)\begin{equation} F_{{capillary}}={-}{3{\rm \pi}\gamma h_{{qp}}^{2}d_{{p}}^{4}}/{r^{5}}, \end{equation}

with $r$ the centre-to-centre interparticle distance.

Rather than characterizing $h_{{qp}}$ (which requires precise measurements of the contact angles), we perform separate experiments in which particle pairs are placed in the fluid layer configurations and approach each other due to capillary attraction. In this creeping flow regime, $F_{{capillary}} = -F_{{drag}} = -(3/2) {\rm \pi}\mu \dot {r} d_{{p}}$, where $\dot {r}/2$ is the velocity of each particle in the pair and $\mu$ is the dynamic viscosity of the conductive fluid in which the particles are mostly submerged. This leads to the power-law relation (Loudet et al. Reference Loudet, Alsayed, Zhang and Yodh2005; Liu et al. Reference Liu, Sharifi-Mood and Stebe2018)

(3.5)\begin{equation} r_{0}^{6}-r^{6}(t) = \varTheta t, \end{equation}

where $r_{0}$ is the initial interparticle distance at time $t=0$ and $\varTheta = {12\gamma h_{{qp}}^{2} d_{{p}}^{3}}/{\mu }$. We experimentally verify this relation by tracking particle pairs; see figure 2. The excellent agreement with (3.5) suggests that the interface deformation causes only marginal deviations from the Stokesian solution (Loudet et al. Reference Loudet, Qiu, Hemauer and Feng2020). Least-squares fits to (3.5) yield values of $\varTheta$, and thus $h_{{qp}}$, which are utilized for evaluating the capillary attraction in (3.4).

Figure 2.

(a) A time series of a pair of particles in DL1 configuration. The scale bar indicates 5 mm. (b) Plot of $\langle r_{0}^{6} - r^{6} \rangle$ versus $t$ in the three configurations. The dashed line indicates $\sim t$ scaling relation.

When the fluid flow is turbulent, the drag pulling a pair of particles away from each other is

(3.6)\begin{equation} F_{{drag}} = 3 {\rm \pi}\mu d_{{p}} \Delta u_{{f}}, \end{equation}

where $\Delta u_{{f}}$ is the velocity difference between two fluid locations corresponding to a pair of closely situated particles, evaluated along the separation vector between the pair. As we are interested in the interaction of adjacent particles, we approximate $\Delta u_{{f}} \sim d_{p} \dot {\epsilon }_{{max}}$, where $\dot {\epsilon }_{{max}}$ is the maximum principal strain rate. The Stokesian assumption is justified by the particle Reynolds number $Re_{{p}}=d_{{p}}\Delta u_{{f}}/\nu$ being relatively small. Indeed, as the density of the particles approximately matches that of the fluid, $Re_{{p}}$ is equivalent (within a geometric factor) to the Stokes number ${St} \equiv \tau _{{p}}u_{{rms}}/L_{{f}}$ (Ouellette, O'Malley & Gollub Reference Ouellette, O'Malley and Gollub2008), which is listed in table 1. The particle response time is taken as $\tau _{{p}}={\rho _{{p}}d_{{p}}^{2}}/(18\mu )$, neglecting finite-$Re_{{p}}$ corrections. In turn, the values of $St$ indicate that the particles have insignificant inertia with respect to the energetic scales of the flow.

The strain rate is estimated by schematizing the flow field as an array of Taylor–Green vortices, whose velocity components in directions $x$ and $y$ are, respectively,

(3.7)\begin{equation} \left. \begin{aligned} u & = U \cos \frac{{\rm \pi} x}{L_{{f}}} \sin\frac{{\rm \pi} y}{L_{{f}}},\\ v & ={-} U \sin \frac{{\rm \pi} x}{L_{{f}}} \cos\frac{{\rm \pi} y}{L_{{f}}}, \end{aligned} \right\} \end{equation}

for which $\dot {\epsilon }_{{max}}\approx ({\rm \pi} /\sqrt {2})(u_{{rms}}/L_{{f}})$. While simplified, such a representation of the flow yields excellent quantitative agreement with the statistical properties of 2D turbulence (Shin et al. Reference Shin, Coletti and Conlin2023). It follows that

(3.8)\begin{equation} F_{{drag}} = \left(\frac{3 {\rm \pi}^{2}}{\sqrt{2}}\right) \frac{\mu d_{{p}}^{2} u_{{rms}}}{L_{{f}}}. \end{equation}

We then define the capillary number given by the ratio of $F_{{drag}}$ and $F_{{capillary}}$ evaluated for $r=d_{{p}}$:

(3.9)\begin{equation} {Ca} = \frac{6\sqrt{2} {\rm \pi}d_{{p}}^{6} u_{{rms}}}{\varTheta L_{{f}}}. \end{equation}

In figures 1(a) and 1(b), $Ca = 0.16$ and 2.49, respectively, which accounts for the much stronger clustering in the former: drag from the straining flow dominates, and clusters are disrupted for ${Ca} \gg 1$, and vice versa for ${Ca} \ll 1$. This is confirmed in figure 3(a), showing $\chi _{{cl}}$ versus $Ca$: collectively, the data adhere to a unified trend, confirming the role of $Ca$ as the controlling parameter. This applies at aerial fractions $\phi \leq 0.20$, for which $F_{{drag}}$ and $F_{{capillary}}$ are the main forces.

Figure 3.

(a) The fraction of clustered particles ($\chi _{{cl}}$) versus ${Ca}$ at $\phi <0.20$. (b) The normalized mean cluster diameter ($\langle D_{{cl}} \rangle / L_{{f}}$, black) and the normalized particle kinetic energy ($\langle E_{{k,p}} \rangle / \langle E_{{k,f}} \rangle$, red) as functions of $\phi$ at ${Ca} > 1$.

For large $\phi$, on the other hand, we observe stronger clustering and a reduction in the root-mean-square particle velocity, which we attribute to additional viscous dissipation by the lubrication force (Wagner & Brady Reference Wagner and Brady2009; Morris Reference Morris2020a). For a pair of nearby particles moving at relative speed $\Delta u_{{p}}$, this is

(3.10)\begin{equation} F_{{lubrication}} = \frac{3 {\rm \pi}\mu d_{{p}}^{2} \Delta u_{{p}}}{8(r-d_{{p}})}. \end{equation}

Relating $\Delta u_{{p}} \approx r \dot {\epsilon }_{{max}} \approx r ({\rm \pi} /\sqrt {2}) (u_{{rms}}/L_{{f}})$, we approximate

(3.11)\begin{equation} F_{{lubrication}} = \frac{3 {\rm \pi}^{2} \mu d_{{p}}^{2} u_{{rms}} r }{8 \sqrt{2} (r - d_{{p}}) L_{{f}}}. \end{equation}

Balancing $F_{{lubrication}}$ and $F_{{drag}}$ yields the separation $r/d_{{p}} = 9/8$: lubrication prevails when neighbouring particles are, on average, closer than such distance, dissipating their kinetic energy. To obtain the corresponding aerial fraction, we employ random sequential adsorption (Evans Reference Evans1993) to distribute in 2D space non-overlapping circles with a range of diameters comparable to those of our particles, i.e. assuming a normal distribution of diameters with the same standard deviations as in table 1. This synthetic method indicates that the average nearest-neighbour separation reaches $(9/8)d_{{p}}$, for $\phi \sim 0.4$. This approximate threshold is confirmed in figure 3(b), showing the mean cluster diameter $\langle D_{{cl}} \rangle$ normalized by the forcing length scale $L_{{f}}$, and the particle kinetic energy $\langle E_{{k,p}} \rangle$ normalized by the fluid kinetic energy $\langle E_{{k,f}} \rangle$ (measured in the unladen case under the same forcing). For cases in which drag overcomes capillarity ($Ca > 1$), the trends for different interfacial conditions are quantitatively similar: for $\phi > 0.4$, $\langle D_{{cl}} \rangle$ grows rapidly and $\langle E_{{k,p}} \rangle / \langle E_{{k,f}} \rangle$ dips. The cluster size appears to plateau at ${\sim }8.5L_{{f}}$, corresponding to the diagonal of the square FOV of size $6L_{{f}}$, i.e. the maximum size detectable by the imaging system. That is, above approximately $\phi =0.6$, the system percolates forming a macrocluster of vanishing small kinetic energy that spans the entire domain.

In summary, our analysis suggests three distinct regimes depending on the balance of three factors: capillarity, viscous drag and lubrication. When $Ca < 1$, capillary attraction holds particles together and leads to the formation of tightly bound clusters. When $Ca > 1$, the strain field of the fluid acts on the particles through viscous drag and breaks up the clusters, as long as $\phi <0.4$; while lubrication preserves slowly moving clusters at higher concentrations. This is schematically illustrated in the phase diagram in figure 4(a) and by the snapshots in figure 4(b) corresponding to the three regimes: capillary-driven clustering, drag-driven break-up and lubrication-driven clustering. We remark that the identified values of the control parameters $Ca$ and $\phi$ are not expected to be critical thresholds, as the considered process does not entail abrupt transitions. In fact, within each one of these regimes, quantitative changes are still observed as the control parameters are varied. For example, in the capillary-driven clustering regime, the aggregates grow bigger with increasing $\phi$, as more particles stick to larger groups and these merge together; this is evident by comparing the two snapshots at $Ca = 0.23$ in figure 4(b). However, those differences appear quantitative rather than qualitative, and the main mechanisms responsible for the behaviour in each regime are not altered.

Figure 4.

(a) Three predicted clustering/break-up regimes in ${Ca}$–$\phi$ space. (b) Snapshots from different configurations, illustrating varying degrees of clustering with $\chi _{{cl}}$ values of 0.991 and 0.624 (i), (ii), and 0.990 and 0.249 (iii), (iv). Insets provide magnified views to highlight differences. Particles within each cluster are represented with the same colour.

The proposed phase diagram is supported by the contour maps over the $Ca$–$\phi$ space displayed in figure 5, which depicts the behaviour of the key observables discussed above for the parameter space we explored. In particular, in the map of $\chi _{{cl}}$ (figure 5a), dense vertical contour lines at $Ca < 1$ give way to horizontal ones at $Ca > 1$, bounding the drag-driven break-up regime. The trend of the normalized mean cluster size $\langle D_{{cl}} \rangle /L_{{f}}$ (figure 5b) is similar, with a sharp increase for $\phi \geq 0.4$. Over this range of concentrations, larger clusters are often centred at the core of vortical structures, where the local strain rate is insufficient to tear them apart (see the supplementary movies available at https://doi.org/10.1017/jfm.2024.246). Furthermore, comparing this map with that of the particle kinetic energy $\langle E_{{k,p}} \rangle / \langle E_{{k,f}} \rangle$, we notice that the motion of clustered particles is greatly inhibited when $\langle D_{{cl}} \rangle$ becomes much larger than $L_{{f}}$, which is the typical size of the forced turbulent eddies (Boffetta & Ecke Reference Boffetta and Ecke2012; Shin et al. Reference Shin, Coletti and Conlin2023): as a cluster spans multiple counter-rotating eddies, the latter are not effective in moving it around. Lastly, the energy maps reveal that even giant clusters spanning the entire domain (e.g. at $\phi \geq 0.6$) have different dynamics depending on $Ca$: they are effectively rigid and static when capillarity dominates ($Ca < 1$), while they are softer and dynamic when drag takes over ($Ca > 1$); see the supplementary movies.

Figure 5.

Phase diagram in ${Ca}$–$\phi$ space to characterize the particle behaviour, mapping (a) $\chi _{{cl}}$, (b) $\langle D_{{cl}} \rangle /L_{{f}}$ and (c) $\langle E_{{k,p}} \rangle / \langle E_{{k,f}} \rangle$.

We remark that the fluid kinetic energy is not directly measured, as the millimetre-sized particles impede sufficient optical access. The values we use, evaluated at the same forcing and $\phi =0$, are, however, expected to be a close approximation of the fluid energy in the suspensions. The submerged particles only occupy a minor portion of the conductive fluid layer and therefore do not significantly alter the Lorentz force. The frictional dissipation on the particle surfaces is also a fraction of that experienced by the fluid on the floor of the tray, which dominates the dissipation rate (Shin et al. Reference Shin, Coletti and Conlin2023), except when giant rigid clusters of vanishing kinetic energy are formed.