4.1.1 Microstructure: linear and weakly nonlinear response

The linear response and weakly nonlinear microstructural distortions were computed in the present study utilizing two methods: analytically via a twin-multipole expansion for probe separations $d>2$ , and numerically via finite difference methods for arbitrary probe separations, as shown in appendices C–E. We begin with a discussion of the analytical results for the weakly nonlinear microstructure, followed by a comparison to our numerical results.

For separations $d>2$ , the $\mathit{O}(Pe)$ microstructural disturbance $f_{1}$ may be written as a twin-multipole expansion (cf. appendix C), comprising of an infinite series of multipole moments centred at probe I and a second expansion centred at probe II. While the full expansion is necessary to satisfy the no-flux boundary condition on each probe, computation of the $\mathit{O}(Pe)$ force on probe I requires only the dipole moment $(n=1)$ of the multipole expansion centred at probe I. This dipole moment is identical to the dipole moment centred about probe II, and is given in vector notation for arbitrary orientation as

where the subscripts $1,\text{I}$ correspond to the first perturbation to the structure about probe I. Expansion of the coefficients in a Taylor series about $d\rightarrow \infty$ then gives

consistent with the $\mathit{O}(1/d^{3})$ results of Krüger & Rauscher (Reference Krüger and Rauscher2007). Here, we have kept the three most physically significant terms: an isotropic term, a term decaying as $1/d^{3}$ and a term decaying as $1/d^{6}$ . Further terms in the Taylor expansion, as well as higher-order (quadrupole, octupole, etc.) moments are tabulated in appendix C.

As $d\rightarrow \infty$ , only the isotropic term remains, recovering the $\mathit{O}(Pe)$ microstructure of a single probe moving through a suspension (Squires & Brady Reference Squires and Brady2005). The terms scaling as $1/d^{3}$ and $1/d^{6}$ arise from probe interactions through the medium, and vary with probe orientation as $\cos \unicode[STIX]{x1D6FC}=\hat{\boldsymbol{d}}\boldsymbol{\cdot }\hat{\boldsymbol{U}}$ . Physically, the motion of probe II through the suspension distorts the surrounding microstructure, giving rise to a diffusive flux of bath particles that in turn interacts with probe I by inducing bath-particle collisions on its surface. Essentially, the bath particles act as an effective medium, influencing the motion of probe I via their flux. This is analogous to hydrodynamic entrainment, where a particle sets the fluid into motion, which in turn influences the motion of a second particle. As such, we term the interactions mediated via bath-particle flux entropic entrainment. Entropic entrainment can either hinder or enhance probe motion, depending on the direction of bath-particle flux, and manifests as a $1/d^{3}$ disturbance to the structure about probe I (second term in (4.14)). At the same time, the motion of probe I generates the same flux, entraining probe II with a $1/d^{3}$ disturbance to its structural deformation. Similar in ethos to reflection of hydrodynamic interactions through pure solvent between two spheres, this disturbance flux triggers an infinitude of reflected bath-particle interactions, leading to the $1/d^{6}$ and higher-order terms in (4.14).

Whether the disturbances hinder or facilitate probe motion depends on the orientation of the probes. In the tandem orientation ( $\hat{\boldsymbol{d}}\boldsymbol{\cdot }\hat{\boldsymbol{U}}=1$ ), the structural distortion about each probe is reduced relative to that of a single probe, i.e. the probes shield one another from the bath particles. Physically, the lead probe generates a depleted region behind it, reducing the structural drag on the trailing probe, while at the same time the trailing probe generates a flux of bath particles toward the leading probe, reducing the drag on the leading probe by an equivalent amount. In the side-by-side orientation $(\hat{\boldsymbol{d}}\boldsymbol{\cdot }\hat{\boldsymbol{U}}=0)$ , each probe accumulates particles in the upstream region into which the other probe must travel, with two consequences: increased structural drag and pressure on both probes, and more pronounced downstream depletion. Overall, the probes entrain one another via the bath particles to slow each other down.

Bath-particle advection arises in the second-order correction to the microstructure: the $\mathit{O}(Pe^{2})$ microstructural distortion $f_{2}$ . While the algebraic complexity of the multipole expansion of $f_{2}$ necessitates numerical solution for arbitrary probe separation (cf. appendix E), we have computed the $\mathit{O}(Pe^{2})$ multipole expansion about each probe in the limit of large separations. To illustrate the flux/motion interplay, the multipole expansion about probe I was found here to be

and about probe II to be

where $\boldsymbol{r}^{\prime }=\boldsymbol{r}-\boldsymbol{d}$ is the position of a bath particle relative to probe II. When $d\rightarrow \infty$ (no probe/probe interactions), the first terms in braces in (4.15) and (4.16) recover the $\mathit{O}(Pe^{2})$ weakly nonlinear correction to the microstructure about a single probe (Khair & Brady Reference Khair and Brady2006; Zia & Brady Reference Zia and Brady2010).

In both (4.15) and (4.16), the first term, $\hat{\boldsymbol{U}}\boldsymbol{\cdot }\{\cdot \}\boldsymbol{\cdot }\hat{\boldsymbol{U}}$ , and the second term, $(\cdot )\boldsymbol{ : }\hat{\boldsymbol{U}}\hat{\boldsymbol{U}}$ , capture distinct physical phenomena. The former arises from the $\mathit{O}(1/d^{3})$ term of $f_{1}$ , which is identical about each probe, and thus cannot lead to net particle accumulation or depletion between the probes; depletion behind the leading probe will be balanced by accumulation in front of the trailing probe. In contrast, the latter term in each differs by a sign, indicating that accumulation (depletion) behind the leading probe arising from this term is enhanced by accumulation (depletion) in front of the trailing probe. Thus, the second term in (4.15) and (4.16) encode the strength of particle accumulation (or depletion) between the probes. Whether bath particles accumulate or are depleted depends on the probe orientation, $\unicode[STIX]{x1D6FC}$ . In the tandem orientation, $\unicode[STIX]{x1D6FC}=0$ , the leading probe shields the trailing probe, reducing the advective flux of bath particles toward the trailing probe, which in turn leads to bath-particle depletion between the probes. In the side-by-side orientation, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , the diffusion of bath particles passing between the two probes is sterically hindered, giving rise to bath-particle accumulation between the probes. In general, the magnitude of accumulation (or depletion) depends on the quantity

where positive values give rise to accumulation, and negative values to depletion. Thus, to leading order, there can be no net accumulation or depletion between the probes when $\unicode[STIX]{x1D6FC}=\cos ^{-1}(1/\sqrt{3})$ .

Analysis of the expressions obtained for linear and weakly nonlinear microstructural distortions leads to two key observations: microstructural distortions about two probes travelling in the tandem orientation are reduced relative to a single probe by shielding effects, while structural distortions about probes travelling in the side-by-side orientation are enhanced. For probe separation $d<2$ , the Smoluchowski equation must be solved numerically, as was shown in appendices C and D.

Figure 5.

Contour plots of two probes driven weakly ( $Pe\ll 1$ ) through the suspension. Top two rows: parallel and perpendicular components of the $\mathit{O}(Pe)$ microstructure row separations $d=3$ (a,b) and $d=1$ (c,d). Bottom three rows: parallel, perpendicular and skew components of the $\mathit{O}(Pe^{2})$ microstructure for separations $d=3$ (e–g) and $d=1$ (h–j). The microstructure varies smoothly from strong depletion in blue to strong accumulation in red, with green representing the equilibrium, unperturbed structure, while the black circles represent the excluded-volume surfaces of the probes.

To obtain solutions for the perturbed microstructure for arbitrary separations $d$ , we note that the weakly nonlinear microstructure may be written in terms of five orientation-independent functions (cf. appendices C and D):

Contour plots of each of these five components of the microstructural distortion are shown in figure 5 for two separations, $d=1$ and $d=3$ . These two values of $d$ are chosen to illustrate the influence of excluded-volume interactions as probe separation is decreased from $d=3$ to $d=1$ . Contour plots in the left column correspond to $d=3$ ; the parallel microstructural contributions $f_{1}^{\Vert }$ and $f_{2}^{\Vert }$ , shown in plots (a,e), exhibit a decrease in bath-particle density in the region between the probes, while the perpendicular components $f_{1}^{\bot }$ and $f_{2}^{\bot }$ , shown in plots (b,f), exhibit an increase, in agreement with our analytical expressions. In contrast, the microstructural distortion contour plot for $d=1$ (figure 5 c,d,h,i,j) differ qualitatively, because, when $d<2$ , the excluded-volume surfaces of the probes overlap and bath particles cannot enter the intervening gap, giving a microstructural distortion that resembles that of an isolated, non-spherical probe. This highlights the two competing mechanisms of probe interaction: for closely spaced probes, excluded-volume effects exert a much stronger influence on the microstructure, and hence the force exerted on the probes, than probe–probe interactions through the bath.

4.1.2 Drag force in the linear-response regime

For asymptotically weak forcing, the $\mathit{O}(Pe)$ (linear response) force exerted on each of the probes by the bath particles is linearly related to probe velocity by the resistance tensors $\hat{\unicode[STIX]{x1D64D}}_{11}$ and $\hat{\unicode[STIX]{x1D64D}}_{12}$ . These tensors exhibit the same symmetries of Stokes flow, with the important implication that the leading-order interactions can produce no relative force between two same-size, same velocity probes. That is, there is no $\mathit{O}(Pe)$ tendency for the probes to attract or repel one another; the linear-response force is a drag force that acts to slow centre-of-mass probe motion. Insertion of (4.4) and (4.5) into (3.24) gives the drag force in the linear-response regime in terms of these tensors as

We derived analytical expressions for the four resistance functions in (4.19), which can be found in appendix C. The drag force $|\boldsymbol{F}^{drag}|$ , normalized on the single-probe drag $6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}_{U,0}^{micro}U$ , is plotted along with the four resistance functions as a function of probe separation in figure 6 for several orientations $\unicode[STIX]{x1D6FC}$ .

Figure 6.

Drag force in linear-response regime as it varies with probe separation. Main figure: force obtained from numerical solutions of (4.2) for several orientations $\unicode[STIX]{x1D6FC}$ . Inset: scalar resistance functions coupling drag to velocity. (Solid lines are analytical solutions; open symbols are numerical solutions.)

In the tandem orientation ( $\unicode[STIX]{x1D6FC}=0$ , solid curve in figure 6), the drag force exerted on each of the probes by the deformed microstructure is strictly weaker than that exerted on a single probe driven through the suspension by the same external force, showing that tandem motion of two probes is easier for each than for one alone. Physically, this drag reduction arises when the trailing probe travels into the region of reduced bath-particle density following the leading probe (cf. figure 5). This shielding effect becomes more pronounced with decreased probe separation, leading to a corresponding monotonic reduction of drag with decreasing probe separation. In fact, the rate of drag reduction speeds up as the probes get closer, until $d=2$ , after which the reduction slows.

Side-by-side ( $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ ) probe translation produces qualitatively different behaviour. When the pair are not close, $d>2$ , the microstructural drag on each probe is greater than that on a single probe translating under the same external force. The drag increases with decreasing separation, owing to an increase in bath-particle density about each probe, particularly in the region between them (cf. figure 5). However, this trend reverses as the probes get closer: for $d<2$ , the drag force decreases with decreasing probe separation, due to the exclusion of bath particles from the intervening gap between the probes (cf. figure 5); the reduction of collisions mitigates drag. Finally, for separations $d\lesssim 1.1$ it becomes easier for the two probes to move together than one alone. Overall, it is harder for two probes in the side-by-side orientation to move together than if they travelled alone through the suspension, except when they travel so close together that they shield one another from collisions.

Comparison of the two limits, $\unicode[STIX]{x1D6FC}=0$ and $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ , reveals that it is always easier for a pair of probes to move in the tandem orientation than in the side-by-side orientation; that is, there is an energetic penalty associated with side-by-side motion as evidenced by the higher rate of effective viscous dissipation. The penalty is small for widely separated probes, and increases as the probes get closer until a maximum is reached near $d\simeq 1.8$ . The probes can reduce this energetic cost in one of two ways: by travelling together or apart, or by reorientation toward the tandem configuration, during which they will each experience a monotonic decrease of the drag force (cf. figure 6). As will be discussed next in § 4.1.3, both of these behaviours mitigate the energetic penalty of side-by-side translation in the weakly nonlinear-response regime.

4.1.3 Interactive force in the weakly nonlinear regime: relative motion

Relative probe motion – reorientation and attraction or repulsion – first emerges in the weakly nonlinear regime. The $\mathit{O}(Pe^{2})$ bath particle-mediated force is given by the third-rank weakly nonlinear resistance tensor $\hat{\unicode[STIX]{x1D64F}}_{1}$ which couples the interactive force to weakly nonlinear motion. We showed that, after an orthogonal decomposition, $\hat{\unicode[STIX]{x1D64F}}_{1}$ is described by three scalar coupling coefficients, which propagate the parallel and perpendicular interactive force as given by (4.11)–(4.12). We obtained analytical expressions for the three scalar coefficients appearing in (4.11) and (4.12), which can be found in appendix D.

Figure 7.

Attractive/repulsive interactive force in weakly nonlinear regime as it varies with probe separation. Main figure: force obtained from numerical solutions of (4.3) for several orientations $\unicode[STIX]{x1D6FC}$ . Inset: scalar resistance functions coupling interactive force to velocity. (Solid lines are analytical solutions; open symbols are numerical solutions.)

The parallel interactive force that tends to produce relative motion – attraction or repulsion – as well as the resistance coefficients $T^{\Vert }$ and $T^{\bot }$ , are plotted in figure 7, which shows that the attractive or repulsive nature of the parallel interactive force depends both on probe separation $d$ and orientation $\unicode[STIX]{x1D6FC}$ . Interactions in the tandem orientations $(\unicode[STIX]{x1D6FC}=0)$ are strictly attractive for all separations; physically, this arises due to collision shielding of the trailing probe by the leading probe. Because the lead probe experiences more head-on collisions with bath particles than does the trailing probe, the result is an effective non-equilibrium attraction. However, the attraction strength is non-monotonic in probe separation; it first increases with decreasing separation (moving right to left in figure 7) until a local maximum is reached at $d\simeq 2.4$ . At this close proximity, the space available for bath particles is severely restricted by the two nearby no-flux surfaces of the probes, and they accumulate there. For separations $d<2$ , bath particles are excluded entirely from the gap between the probes, and the resulting imbalance in osmotic pressure leads to a pronounced increase in attraction strength, in a manner similar to the equilibrium depletion force.

The interactive force in the side-by-side orientation $(\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2)$ is qualitatively different from that in the tandem orientation and also from the equilibrium depletion force, both of which are strictly attractive. In the side-by-side orientation, first the probes repel, then they attract as they approach one another. Physically, bath particles near the outboard surfaces of the probes may diffuse freely away, while particles between the probes accumulate as they struggle to migrate through the constrained gap between the probes, and the increased osmotic pressure acts to drive the probes apart. This intensifies as the gap shrinks until it becomes small enough to exclude bath particles entirely $(d=2)$ . Beyond this, the gap is dilute relative to the bath, reducing the repulsion strength until interactions become attractive for separations $d\lesssim 1.8$ . This produces a $Pe^{2}$ -small enhancement to or reduction of the $\mathit{O}(1)$ equilibrium depletion force. However, it acts over a much longer range of many particle lengths. This long-ranged repulsion, when coupled with the long-ranged attractions in the tandem orientation, may give rise to the formation of particle-rich columns or long particle chains in dense suspensions.

Between these two limits, $0<\unicode[STIX]{x1D6FC}<\unicode[STIX]{x03C0}/2$ , the curves show a smooth transition from pure attraction behaviour to the separation-dependent repulsive/attractive behaviour. Of note is the tendency for the most repulsive configurations to become the most attractive as $d\rightarrow 1$ , suggesting a tendency to reorient, discussed next.

Figure 8.

Reorientation force in the weakly nonlinear regime, scaled on $\sin 2\unicode[STIX]{x1D6FC}$ , as it varies with probe separations. (Solid lines obtained from analytical solution of (4.3); open symbols are numerical solutions.)

Beyond attraction and repulsion, a tendency to reorient emerges at $\mathit{O}(Pe^{2})$ , and this force, (cf. (4.12)), is projected from the velocity onto the force by the scalar resistance function $T^{\times }$ . As such, varying $\unicode[STIX]{x1D6FC}$ only serves to change the magnitude of the reorientation force. The reorientation force $F_{\bot }^{int}$ is plotted in figure 8, scaled on $\sin 2\unicode[STIX]{x1D6FC}$ . The force is strictly negative for all separations. This implies that the tandem orientation is stable for all probe separations in the weakly nonlinear regime, while the side-by-side orientation is unstable: any small perturbation from this arrangement results in a force that pushes the probes toward the tandem alignment. Physically, this arises from a self-reinforcing recirculating bath-particle flux from the accumulation region of each probe toward the depletion region of the other. This can be seen mathematically from expression (4.15), whereupon insertion into (3.17) shows that the reorientation force on each probe emerges only from the disturbed structure, $f_{2,\text{I}/\text{II}}$ about the other. For widely separated probes, the strength of bath particle flux from one probe to the other increases as the probes grow closer, which in turn increases the reorientation force. When bath particles can no longer fit between the probes $(d<2)$ , the overlap of the excluded volume reduces the flux of bath particles from one probe to the other, which in turn leads to a reduction in the reorientation force as the separation is decreased.

4.1.4 Comparison with continuum theory

At wide separations, the probes interact over length scales much larger than the size of the bath particles, making it natural to view the weakly deformed intervening suspension as an effective continuum. Recall that the linear-response pair resistance functions that couple microstructural drag to velocity give configuration-independent effective viscosity. At wide separations, these decay as $R_{12}^{\Vert }=-2R_{12}^{\bot }\sim 1/d^{3}$ (cf. appendix C), reminiscent of Brinkman’s continuum theory applied to interactions between a pair interacting through a porous medium (Kim & Russel Reference Kim and Russel1985) or in a bed of fixed particles (Durlofsky & Brady Reference Durlofsky and Brady1987). This behaviour is characteristic of the interactions between mass dipoles in Stokes flow (Diamant Reference Diamant2007) as seen also in interactions between aerosol particles (Keh & Chen Reference Keh and Chen1995), and interactions between electrophoretically driven colloids (Long & Ajdari Reference Long and Ajdari2001). The origin of this correspondence between continuum mechanics and our micromechanical model can be seen by examining the linear-response microstructure (4.14). In the linear-response regime, the probes behave as diffusive dipoles: bath particles accumulate in front of the probes and are depleted from behind, creating a disturbance sink and source. From the perspective of a second probe located far from the first, the first probe behaves as a mass dipole: there is an effective mass source at the front of the probe, and an equivalent sink in the back.

This analysis would change slightly with the inclusion of hydrodynamic interactions. In particulate matter, there are two sources of long-ranged interactions, the mass dipole computed here arising from mass conservation of the microstructure, and a point force (Stokeslet) arising from momentum conservation in the fluid (Diamant Reference Diamant2007). In our system, for finite values of $\unicode[STIX]{x1D705}_{a},\unicode[STIX]{x1D705}_{b}$ , the point force will give rise to a hydrodynamic resistance tensor equivalent to two probes interacting through a Newtonian fluid of effective viscosity $5\unicode[STIX]{x1D702}\unicode[STIX]{x1D719}_{b_{h}}/2$ :

where $\unicode[STIX]{x1D719}_{b_{h}}$ is the volume fraction of bath particles based on the hydrodynamic radius $b_{h}$ and the factor $a_{h}/(a+b)$ arises because $d$ is made dimensionless on $a+b$ . The point force decays as $1/d$ , and as such will be much larger than the $1/d^{3}$ mass dipole as $d\rightarrow \infty$ . The relative importance of the two terms at intermediate probe separations is seen by equating the entropic resistance tensor in (4.8) to that in equation (4.20), revealing that, for equal-sized probe and bath particles, the mass dipole is larger than the point force when $d\ll (1+\unicode[STIX]{x1D705}_{a})^{2}$ . Thus, in the full hydrodynamics limit $(\unicode[STIX]{x1D705}_{a}\rightarrow 0)$ the point force is always dominant, while in the freely draining limit considered here $(\unicode[STIX]{x1D705}_{a}\rightarrow \infty )$ , the point force may be neglected entirely. That is, the Einstein viscosity correction vanishes in freely draining suspensions. For finite values of $\unicode[STIX]{x1D705}_{a}$ , the separation of length scales between the thermodynamic and hydrodynamic radii of the probes gives rise to mass dipole-like interactions for small to intermediate probes separations until the point force ‘kicks in’ at $d\sim (1+\unicode[STIX]{x1D705}_{a})^{2}$ . This separation of length scales is similar to behaviours observed in two-point microrheology experiments in viscoelastic networks, where the tracer beads interact as mass dipoles at length scales less than the dynamic correlation length of the network, and as point forces for larger separations (Sonn-Segev et al. Reference Sonn-Segev, Bernheim-Groswasser, Diamant and Roichman2014).

This comparison with continuum mechanics may be extended to the weakly nonlinear regime. Our micromechanical model predicts that the third-rank resistance tensor coupling the interactive force to weakly nonlinear motion $\hat{\unicode[STIX]{x1D64F}}_{1}$ decays as $1/d^{2}$ for widely separated probes (cf. appendix D). Pair interactions in the continuum second-order fluid model are described by an analogous third-rank tensor that also decays as $1/d^{2}$ (Brunn Reference Brunn1977). Furthermore, the second-order fluid model predicts that the side-by-side configuration is unstable (Brunn Reference Brunn1977), much as in our model. However, qualitative differences arise when considering attraction and repulsion. In the continuum theory, a pair of probes are attracted toward one another regardless of their separation and orientation (Brunn Reference Brunn1977; Khair & Squires Reference Khair and Squires2010). In contrast, when accounting for the detailed microstructural evolution of the bath, the attractive or repulsive character of the interactive force depends on probe separation and orientation (cf. figure 7). This discrepancy arises largely from the neglect of local microstructural heterogeneity by the second-order fluid model; while the second-order fluid model can account for the evolution of conformational degrees of freedom of individual polymers as the slow-flow limit of polymer dumbbell models (Bird et al. Reference Bird, Armstrong, Hassager and Curtiss1977), it cannot account for microstructural evolution at length scales larger than a single particle. In our micromechanical model, the linear-response microstructure is locally homogenous, as each region of bath-particle accumulation is exactly balanced by depletion, and there is a correspondence between continuum mechanics and micromechanics, as evidenced by the identical $1/d^{3}$ scalings of the linear resistance tensors in each approach. However, in the weakly nonlinear regime, advective flux leads to a build-up of bath particles around and between the probes, leading to local spatial heterogeneities that produce the repulsive interactions not captured by continuum theory.

4.2.1 Microstructure

Figure 9.

Contour plots for two probes driven strongly ( $Pe=10$ ) through the suspension for various separations $d=1$ (a), $d=1.95$ (b) and $d=3$ (c). The microstructure varies smoothly from strong depletion in blue to strong accumulation in red, with green representing the equilibrium, unperturbed structure. For each separation, five orientations are shown: $\unicode[STIX]{x1D6FC}=0,\unicode[STIX]{x03C0}/6,\unicode[STIX]{x03C0}/4,\unicode[STIX]{x03C0}/3$ and $\unicode[STIX]{x03C0}/2$ .

Here we study the microstructural response to arbitrary strength of forcing. Beyond the weakly nonlinear regime, the microstructural symmetries that produces Stokes-flow-like resistance tensors coupling applied force to probe velocity are lost. To obtain the effective force exerted on the probes as they travel through the bath, the configuration of bath particles $g(\boldsymbol{r}\mid \boldsymbol{d};\unicode[STIX]{x1D6FC},Pe)$ must be determined for each probe separation $d$ , orientation $\unicode[STIX]{x1D6FC}$ and velocity $Pe$ . To do so, we solve (3.10), the conditional pair Smoluchowski equation, over a wide space of these parameters, via a numerical finite-difference method (cf. appendix E). An example set of contour plots of the microstructure thus obtained for $Pe=10$ is given in figure 9 for a range of probe separations $d$ and orientations $\unicode[STIX]{x1D6FC}$ ; contour plots for $Pe=5$ , $20$ and $50$ are given in the supplementary material available at https://doi.org/10.1017/jfm.2017.789. Colours range from dark red for strong accumulation to green for undisturbed structure to dark blue for strong depletion of bath-particle density. In this regime, bath particles are strongly advected to the surfaces of the probes; bath particles accumulate in an upstream boundary layer, and are depleted in a trailing wake. The boundary layers and wakes of the two probes interact, giving rise to a structural distortion that depends strongly on probe separation and orientation.

When $d=1$ (figure 9 a), the microstructural distortion around the probes resembles that caused by the motion of a single, non-spherical probe. Bath particles are excluded from the gap between the probes, and accumulate on their outboard surfaces. Upon increasing the separation to $d=1.95$ (figure 9 b), the exclusion region shrinks, increasing the volume accessible to bath particles. Particles are advected into the gap, but now must diffuse through to pass by the probes. The no-flux probe surfaces dominate the region, restricting the escape route of the bath particles and causing them to accumulate until they slowly diffuse through. Accumulation is most pronounced in the side-by-side orientation $(\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2)$ , where the upstream faces of both probes are entirely exposed to collisions, forcing bath particles into the gap; accumulation is weakest in the tandem orientation $(\unicode[STIX]{x1D6FC}=0)$ , where the leading probe shields the trailing probe from collisions, and the gap from bath-particle entry.

At still larger separations, $d=3$ (figure 9 c), bath particles may pass freely through the gap between the probes. A key feature of the distorted microstructure at this separation is the shedding of the detached boundary layer of the leading probe towards the trailing probe. Whether this increases or decreases bath particle density around the trailing probe depends on probe orientation. In the tandem orientation $(\unicode[STIX]{x1D6FC}=0)$ , the trailing probe travels through the wake created by the translation of the leading probe, a phenomenon first noted by Khair & Brady (Reference Khair and Brady2007). Essentially, the trailing probe moves through a tunnel devoid of bath particles in a manner reminiscent of the reduced viscosity corridors proposed to explain experimentally observed pair attractions (Joseph et al. Reference Joseph, Liu, Poletto and Feng1994). The bath-particle density about the trailing probe increases when it is not travelling directly behind the leading probe ( $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/6,\unicode[STIX]{x03C0}/4$ , and $\unicode[STIX]{x03C0}/3$ ). At these orientations, bath particles are shed from the forward-most probe toward the upstream face of the trailing probe, where they then accumulate. Finally, in the side-by side orientation, bath particles may move freely between the probes, leading to probe microstructures that are largely independent. We now analyse the drag and interactive forces produced by these micromechanics.

4.2.2 Drag force

Figure 10.

Magnitude of the drag force on each probe for $Pe=10$ , scaled on the low- $Pe$ single-particle drag force $6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}_{U}^{micro}a_{h}U$ at that same value of $Pe$ , plotted as a function of probe separation $d$ for various orientations $\unicode[STIX]{x1D6FC}$ . Symbols: numerical results obtained from solution of (3.10).

The magnitude of the drag force is obtained from (3.24) by inserting the microstructures obtained in § 4.2.1 into (3.17), and is plotted as a function of probe separation $d$ in figure 10, for $Pe=10$ for several probe orientations $\unicode[STIX]{x1D6FC}$ ; corresponding plots for $Pe=5$ , 20 and 50 are given in the supplementary material. The drag force is normalized by the microstructural drag of a single, fixed velocity probe at the same value of $Pe$ (Squires & Brady Reference Squires and Brady2005; Swan & Zia Reference Swan and Zia2013). At small separations, $|\boldsymbol{F}^{drag}|<6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}_{U}^{micro}a_{h}U$ , showing that each probe experiences a reduction in drag relative to that of a single probe travelling at that same velocity, confirming the microstructural picture that the two probes behave as a single, non-spherical particle (cf. figure 9). Drag decreases even further as $\unicode[STIX]{x1D6FC}$ decreases from the side-by-side ( $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ ) to the tandem $(\unicode[STIX]{x1D6FC}=0)$ orientation, because the trailing probe is increasingly shielded from bath-particle collisions by the leading probe. As the probes begin to separate, the drag on each increases regardless of orientation; and for some orientations, $\unicode[STIX]{x03C0}/4\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/2$ , the drag force on each is greater even than on a single probe moving through the suspension. Once the probes near the tandem orientation, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/6$ , this behaviour is delayed to separations $d>2.5$ , where the wake from the leading probe is shed directly toward the upstream face of the trailing probe (cf. figure 9), increasing the total microstructural drag above that of any other orientation. Finally, for all probe separations, microstructural drag force is smallest in the tandem orientation, $\unicode[STIX]{x1D6FC}=0$ , where the leading probe fully shields the trailing probe from collisions.

Next, we look for evidence of the non-Newtonian rheology observed in shear flow (Bergenholtz et al. Reference Bergenholtz, Brady and Vicic2002) or singe-probe active microrheology (Squires & Brady Reference Squires and Brady2005; Swan & Zia Reference Swan and Zia2013), e.g. shear or velocity thinning of the effective viscosity with increasing flow strength, $Pe$ . To do so, we plot the magnitude of the drag force acting on each probe in the side-by-side orientation in figure 11 as a function of $Pe$ . The drag force is scaled advectively on the single-particle entropic drag force $6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}_{U,0}^{micro}a_{h}U$ to give the particle drag viscosity, analogous to the single-probe microviscosity (Squires & Brady Reference Squires and Brady2005; Swan & Zia Reference Swan and Zia2013). The solid curve for $d\rightarrow \infty$ recovers behaviour identical to single-probe non-Newtonian viscosity (Swan & Zia Reference Swan and Zia2013), with a low- $Pe$ Newtonian plateau, the onset of velocity thinning at $Pe\sim 1$ , and decay toward a high- $Pe$ Newtonian plateau of $\unicode[STIX]{x1D702}_{U}^{micro}=\unicode[STIX]{x1D702}_{U,0}^{micro}/2$ . For all separations, the curves exhibit qualitatively similar behaviour; at $Pe\ll 1$ , the asymptotic results of § 4.1.2 are recovered, where the drag force is linear in the forcing. The separation dependence of the drag force observed in figure 10 for $Pe=10$ and $d<2$ holds across the entire range of $Pe$ examined: very closely spaced probes each experience a reduction in drag relative to that on a single probe, while larger separations $(1.5\leqslant d\leqslant 1.8)$ experience an increase in drag due to strong bath-particle accumulation between the probes. Qualitatively different behaviour is observed when the probe separation is large enough for bath particles to fit between them $(d>2)$ . When forcing is weak, the drag force for the separation $d=2.5$ is greater than that of a single probe, as bath-particle accumulation between the probes increases the total resistance. As $Pe$ is increased, boundary layers formed around each probe compress and narrow with the growing flow strength, until ultimately the detached boundary layers cease to interact, and the drag force then approaches the single-probe limit.

Figure 11.

Magnitude of the drag force acting each probe in the side-by-side orientation ( $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ ), scaled on the low- $Pe$ single-particle drag force $6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}_{U,0}^{micro}a_{h}U$ , plotted as a function $Pe$ for various probe separations $d$ . Lines: numerical results obtained from solution of (3.10). The $d\rightarrow \infty$ result was first obtained by Squires & Brady (Reference Squires and Brady2005).

4.2.3 Parallel interactive force: attraction and repulsion

Figure 12.

Attractive/repulsive parallel interactive force, scaled on the characteristic osmotic force $n_{b}kT(a+b)^{2}$ , plotted as a function of probe separation $d$ for $Pe=10$ for various orientations $\unicode[STIX]{x1D6FC}$ . Symbols: numerical results obtained from solution of (3.10). Solid red line: equilibrium depletion force (3.1).

The parallel component of the interactive force – the non-equilibrium attractive or repulsive force – is plotted in figure 12 as a function of probe separation at fixed $Pe=10$ for several probe orientations; corresponding plots for $Pe=5$ , 20 and 50 are given in the supplementary material. The equilibrium depletion force, given by (3.1), is also plotted for comparison. In § 4.1.3, we showed that in the weakly nonlinear regime, pair interactions give rise to an $\mathit{O}(Pe^{2})$ small enhancement or reduction of the $\mathit{O}(1)$ equilibrium depletion force. Here, in the nonlinear regime, we see that non-equilibrium interactions are stronger than the equilibrium force. For small separations, bath particles are excluded from the space between the probes (cf. figure 9), and the resultant imbalance in osmotic pressure leads to non-equilibrium attractions stronger than the equilibrium depletion force for all orientations $\unicode[STIX]{x1D6FC}$ . Increasing the separation reduces this imbalance, decreasing the attraction strength. For orientations at or near the side-by-side orientation $(\unicode[STIX]{x03C0}/4\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/2)$ , bath particles accumulate strongly between the probes (cf. figure 9), leading to repulsive interactions between the probes, a reversal from the attraction at or near the tandem orientation $(0\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x03C0}/6)$ , where the leading probe shields the trailing probe from collisions. For separations $d>2$ , the force quickly decays to zero for all orientations other than tandem $(\unicode[STIX]{x1D6FC}=0)$ . Away from the tandem orientation, interactions are largely mediated by the $\mathit{O}(1/Pe)$ thick boundary layer about each probe, leading to interactions that quickly decrease with increasing separation. In the tandem orientations, the trailing probe is immersed in the $\mathit{O}(Pe)$ long wake of the leading probe, giving rise to long-range attractions, as previously found by Khair & Brady (Reference Khair and Brady2007).

To examine the effects of the strength of forcing on the interactive component of the effective force, the parallel interactive force between two probes in the side-by-side orientation is plotted as a function of $Pe$ in figure 13. To elucidate the separation dependence of the interactive force across several decades in $Pe$ , the interactive force has been scaled advectively on $Pe$ . For all separations, the advectively scaled interactive force increases in magnitude with increasing $Pe$ until $Pe\sim \mathit{O}(1)$ , reflecting the $Pe^{2}$ scaling of the interactive force in the weakly nonlinear regime. Further increase of $Pe$ decreases the advectively scaled interactive force in a manner analogous to the flow thinning observed in the drag force (cf. figure 10). As $Pe$ grows large, $Pe\gg 1$ , the interactive force depends on probe separation. For separations $d<2$ , bath particles are excluded from between the probes, and the resultant imbalance in osmotic pressure gives rise to a non-zero interactive force as $Pe\rightarrow \infty$ . However, the attractive or repulsive nature of the interactive force depends on both separation and $Pe$ . Interactions are strictly attractive for $d=1$ , as bath-particle accumulation on the outboard surfaces of the probes drives them together (cf. figure 9). At larger separations, bath-particle accumulation between the probes can drive them apart. For $d=1.5$ , this behaviour is non-monotonic: the force is attractive for weak forcing, becoming repulsive as the forcing strength increases and bath particles accumulate in the cleft between the probes, and finally transitions to a second repulsive regime at $Pe\simeq 30$ as the boundary layer narrows. The force for $d=1.8$ is strictly repulsive due to strong accumulation between the probes. For larger separations $(d>2)$ , the probes will cease to interact as $Pe\rightarrow \infty$ , as the shed boundary layers of the probes narrow. This behaviour is realized for the separation distance $d=2.5$ ; the probes repel when $Pe$ is small, with the magnitude increasing with $Pe$ until $Pe\sim \mathit{O}(1)$ , after which the interactive force vanishes.

Figure 13.

Attractive/repulsive parallel interactive force acting on two side-by-side probes ( $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/2$ ), scaled on the characteristic osmotic force $n_{b}kT(a+b)^{2}$ and advectively on $Pe$ , plotted as a function $Pe$ for various probe separations $d$ . Lines: numerical results obtained from solution of (3.10).

4.2.4 Perpendicular interactive force: reorientation

Figure 14.

(a) Reorientational perpendicular interactive force, scaled on the characteristic osmotic force $n_{b}kT(a+b)^{2}$ , plotted as a function of probe separation $d$ for $Pe=10$ and various orientations $\unicode[STIX]{x1D6FC}$ . Symbols: numerical results obtained from solution of (3.10). (b) Reorientation force, for $Pe$ , plotted versus probe orientation $\unicode[STIX]{x1D6FC}$ for fixed separation $d$ .

The perpendicular component of the interactive force, the reorientation force, is plotted in figure 14(a) for $Pe=10$ as a function of separation as probe orientation is varied; corresponding plots for $Pe=5$ , 20 and 50 are given in the supplementary material. In the weakly nonlinear regime, this force always acts to move the probes toward the tandem orientation, indicating that the side-by-side configuration is always unstable (cf. § 4.1.3). Far from equilibrium, the perpendicular interactive force exhibits qualitatively different behaviour. For separations $d>2.3$ , the reorientation force at each separation $d$ may be strongly positive or weakly negative, depending on orientation $\unicode[STIX]{x1D6FC}$ . This behaviour is explored in figure 14(b), where the reorientation force is plotted versus probe orientation $\unicode[STIX]{x1D6FC}$ for fixed separation $d$ . The orientation for which $F_{\bot }^{int}$ reverses from positive to negative at a given separation $d$ and flow strength $Pe$ may be used to define a critical orientation, $0<\unicode[STIX]{x1D6FC}_{c}(d,Pe)<\unicode[STIX]{x03C0}/2$ . When $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{c}$ , the reorientation force is negative, and the probes are pushed toward the tandem orientation; the probes are pushed toward the side-by-side orientation when $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$ . Physically, this behaviour arises from interactions mediated by the detached boundary layer of the leading probe. Bath particles from the leading probe are shed toward the upstream face of the trailing probe (cf. figure 9). When $\unicode[STIX]{x1D6FC}<\unicode[STIX]{x1D6FC}_{c}$ , the majority of this density falls on the exterior surface of the trailing probe, pushing it into the wake behind the first probe. In contrast, when $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FC}_{c}$ , the majority of the particle density is on the interior face of the second probe, pushing it away from the wake.

The stability of the tandem and side-by-side orientations as a function of probe separation may be examined through the framework of this critical orientation. The critical orientation is plotted versus probe separations for fixed values of $Pe$ in figure 15, giving a ‘stability map’ for the probes. Probes with orientations and separations falling below the critical orientations experience a force pushing them toward tandem orientation, while the force on probes falling above the line pushes them toward side-by-side orientation. Thus, for separations where $\unicode[STIX]{x1D6FC}_{c}(d)=\unicode[STIX]{x03C0}/2$ , only the tandem orientation is stable, while both orientations are stable when $\unicode[STIX]{x1D6FC}_{c}(d)<\unicode[STIX]{x03C0}/2$ . The separation distance for which the side-by-side orientation is first stable, i.e. where $\unicode[STIX]{x1D6FC}_{c}$ is first less than $\unicode[STIX]{x03C0}/2$ , decreases as the strength of forcing decreases. This is because as $Pe$ increases, diffusion is weakened relative to advection, and the boundary layer shed from the leading probe narrows, which in turn decreases the density of bath particles shed upon the trailing probe.

Figure 15.

Critical orientation for which the reorientation force $F^{int}=0$ plotted as a function of probe separation $d$ for various values of $Pe$ .

4.2.5 Comparison with experiments

In the nonlinear-response regime, our model predicts that probes sedimenting side-by-side under gravity will attract and reorient if their initial spacing is small, and will fall without reorienting when widely separated, in qualitative agreement with experimental observations (Joseph et al. Reference Joseph, Liu, Poletto and Feng1994; Gumulya et al. Reference Gumulya, Horsley, Pareek and Lichti2011a ). This highlights the utility of our micromechanical model, where prior theoretical approaches (Brunn Reference Brunn1977) predict that the side-by-side orientation is always unstable. However, there are discrepancies between our model and experimental results, largely owing to the large size of the probes in experiments $(a\sim 1{-}10~\text{mm})$ . In experiments, the separation distance for which the side-by-side orientation is stable is set by the probe size, where in our model it is set by the bath-particle size. This discrepancy can be understood by recognizing that hydrodynamic interactions, neglected in the present model, can produce changes in the strength and range of the interactive force (Sriram & Furst Reference Sriram and Furst2012, Reference Sriram and Furst2015). In the side-by-side orientation, repulsive interactions are suppressed and the force becomes longer ranged, while the strength of attraction is reduced in the tandem orientation.

However, the qualitative agreement between our results and experiments should not be viewed as a fortunate coincidence. Our micromechanical model is valid in the microscopic limit, where the probes interact (on average) with a single bath particle at a time, whereas the large, macroscopic probes in experiments interact with many polymer molecules. Extensive investigation in the field of microrheology has shown a correspondence between the microscopic and macroscopic interrogation, owing to similarities in structural asymmetry (Squires & Brady Reference Squires and Brady2005; Khair & Brady Reference Khair and Brady2006; Zia & Brady Reference Zia and Brady2012). Thus, while direct experimental comparison with our results would require probes small enough to respond to depletion forces $(a\sim 1{-}50~\unicode[STIX]{x03BC}\text{m})$ , our findings are applicable to more general cases.