We introduce a novel unsteady shear protocol, which we name rotary shear (RS), where the flow and vorticity directions are continuously rotated around the velocity-gradient direction by imposing two out-of-phase oscillatory shears (OSs) in orthogonal directions. We perform numerical simulations of dense suspensions of rigid non-Brownian spherical particles at volume fractions ($\phi$) between 0.40 and 0.55, subject to this new RS protocol, and compare with the classical OS protocol. We find that the suspension viscosity displays a similar non-monotonic response as the strain amplitude ($\gamma _0$) is increased: a minimum viscosity is found at an intermediate, volume-fraction-dependent strain amplitude. However, the suspension dynamics is different in the new protocol. Unlike the OS protocol, suspensions under RS do not show absorbing states at any $\gamma _0$ and do not undergo the reversible–irreversible transition: the stroboscopic particle dynamics is always diffusive, which we attribute to the fact that the RS protocol is inherently irreversible due to its design. To validate this hypothesis, we introduce a reversible-RS (RRS) protocol, a combination of RS and OS, where we rotate the shear direction (as in RS) until it is instantaneously reversed (as in OS), and find the resulting rheology and dynamics to be closer to OS. Detailed microstructure analysis shows that both the OS and RRS protocols result in a contact-free, isotropic to an in-contact, anisotropic microstructure at the dynamically reversible-to-irreversible transition. The RS protocol does not render such a transition, and the dynamics remains diffusive with an in-contact, anisotropic microstructure for all strain amplitudes.

The extensional rheology of dilute suspensions of spheres in viscoelastic/polymeric liquids is studied computationally. At low polymer concentration $c$ and Deborah number $\textit{De}$ (imposed extension rate times polymer relaxation time), a wake of highly stretched polymers forms downstream of the particles due to larger local velocity gradients than the imposed flow, indicated by $\Delta \textit{De}_{\textit{local}}\gt 0$. This increases the suspension’s extensional viscosity with time and $\textit{De}$ for $De \lt 0.5$. When $\textit{De}$ exceeds 0.5, the coil-stretch transition value, the fully stretched polymers from the far-field collapse in regions with $\Delta \textit{De}_{\textit{local}} \lt 0$ (lower velocity gradient) around the particle’s stagnation points, reducing suspension viscosity relative to the particle-free liquid. The interaction between local flow and polymers intensifies with increasing $c$. Highly stretched polymers impede local flow, reducing $\Delta \textit{De}_{\textit{local}}$, while $\Delta \textit{De}_{\textit{local}}$ increases in regions with collapsed polymers. Initially, increasing $c$ aligns $\Delta \textit{De}_{\textit{local}}$ and local polymer stretch with far-field values, diminishing particle–polymer interaction effects. However, beyond a certain $c$, a new mechanism emerges. At low $c$, fluid three particle radii upstream exhibits $\Delta \textit{De}_{\textit{local}} \gt 0$, stretching polymers beyond their undisturbed state. As $c$ increases, however, $\Delta \textit{De}_{\textit{local}}$ in this region becomes negative, collapsing polymers and resulting in increasingly negative stress from particle–polymer interactions at large $\textit{De}$ and time. At high $c$, this negative interaction stress scales as $c^2$, surpassing the linear increase of particle-free polymer stress, making dilute sphere concentrations more effective at reducing the viscosity of viscoelastic liquids at larger $\textit{De}$ and $c$.

We experimentally investigate the rotational dynamics of neutrally buoyant flat bodies of revolution (spheroids, disks and rings with different cross-sectional shapes) in shear flows. In the Stokes regime, the axis of revolution of these rigid particles moves in one of a family of closed periodic Jeffery orbits. Inertia is able to lift the orbit degeneracy and induces drift among several rotations towards limiting stable orbits. Furthermore, permanent alignment can be achieved for disks and rings with triangular cross-sectional shapes, provided the inertia is sufficiently high. The bifurcations between the different dynamics are compared with those predicted by small-inertia asymptotic theories and numerical simulations.

Particle suspensions in confined geometries can become clogged, which can have a catastrophic effect on function in biological and industrial systems. Here, we investigate the macroscopic dynamics of dense suspensions in constricted geometries. We develop a minimal continuum two-phase model that allows for variation in particle volume fraction. The model comprises a ‘wet solid’ phase with material properties dependent on the particle volume fraction, and a seepage Darcy flow of fluid through the particles. We find that spatially varying geometry (or material properties) can induce emergent heterogeneity in the particle fraction and trigger the abrupt transition to a high-particle-fraction ‘clogged’ state.

Discontinuous shear-thickening (DST) fluids exhibit unique instability properties in a wide range of flow conditions. We present numerical simulations of a scalar model for DST fluids in a planar simple shear using the smoothed particle hydrodynamics approach. The model reproduces the spatially homogeneous instability mechanism based on the competition between the inertial and microstructural time scales, with good congruence to the theoretical predictions. Spatial inhomogeneities arising from a stress-splitting instability are rationalised within the context of local components of the microstructure evolution. Using this effect, the addition of non-locality in the model is found to produce an alternative mechanism of temporal instabilities, driven by the inhomogeneous pattern formation. The reported arrangement of the microstructure is generally in agreement with the experimental data on gradient pattern formation in DST. Simulations in a parameter space representative of realistic DST materials resulted in aperiodic oscillations in measured shear rate and stress, driven by formation of gap-spanning frictional structures.

Flows of particles through bottlenecks are ubiquitous in nature and industry, involving both dry granular materials and suspensions. However, difficulties in precisely controlling particle properties in conventional set-ups hinder the full understanding of these flows in confined geometries. Here, we present a microfluidic model set-up to investigate the flow of dense suspensions in a two-dimensional hopper channel. Particles with controlled properties such as shape and deformability are in situ fabricated with a photolithographic projection method and compacted at the channel constriction using a Quake valve. The set-up is characterised by examining the flow of a dense suspension of hard, monodisperse disks through constrictions of varying widths. We demonstrate that the microfluidic hopper discharges particles at a constant rate under both imposed pressure and flow rate. The discharge of particles under imposed flow rate follows a Beverloo-like scaling, while it varies nonlinearly with particle size under imposed pressure. Additionally, we show that the statistics of clog formation in our microfluidic hopper follow the same stochastic laws as reported in other systems. Finally, we show how the versatility of our microfluidic model system can be used to investigate the outflow and clogging of suspensions of more complex particles.

We investigate suspensions of non-Brownian, millimetric monodisperse spherical particles floating at quasi-two-dimensional fluid interfaces, from dilute to dense concentrations. Building upon the phase diagram in the capillary number ($Ca$) and areal fraction ($\phi$) constructed by Shin & Coletti (2024 J. Fluid Mech. 984, R7), we analyse the dynamics of both aggregation and dispersion. In the capillary-driven clustering regime ($Ca \lt 1$), strong inter-particle bonds yield large, fractal-like clusters that grow by hit-and-stick collisions. In the drag-driven break-up regime ($Ca \gt 1$, $\phi \lt 0.4$), turbulent fluctuations overcome capillarity and result in particles moving similarly to passive tracers and forming clusters by random adjacency. In the lubrication-driven clustering regime ($Ca \gt 1$, $\phi \gt 0.4$), the close inter-particle proximity amplifies lubrication forces and results in large, crystal-like clusters. Above a threshold concentration that depends on $Ca$, self-similar percolating clusters span the entire domain. The particle transport exhibits a classic ballistic-to-diffusive transition, with the long-time diffusivity hindered by the reduced fluctuating energy at high concentrations. Nearby particles separate at initially slow rates due to strong capillary attraction, and then follow a super-diffusive dispersion regime. In dense suspensions, the process is characterised by the time scale associated with inter-particle collisions and by the energy dissipation rate defined by the lubrication force between adjacent particles. Our results provide a framework for predicting particle aggregation in interfacial suspensions such as froth flotation and pollutant dispersion, and may inform the design of advanced materials through controlled colloidal self-assembly.

We investigate the deformation, dynamics and rheology of a single and a suspension of elastic capsules in inertial shear flow using high-fidelity particle-resolved simulations. For a single capsule in the shear flow, we elucidate the interplay of flow inertia and viscosity ratio, revealing the mechanism behind the stretching of capsule surface during tank-treading motion and the sign changes in normal stress differences with increasing inertia. When examining capsule suspensions, we thoroughly discuss the impact of volume fraction on average deformation, diffusion and rheology. Notably, we observe the formation of bridge structures due to hydrodynamic interactions, which enhance the inhomogeneity of the microstructure and alter the surface stress distribution within the suspension. We identify a critical Reynolds number range that marks the transition of capsule diffusion from non-inertial to inertial regimes. Furthermore, we reveal close connections between the behaviour of individual capsules and dense suspensions, particularly regarding capsule deformation and dynamics. Additionally, we propose multiple new empirical correlations for predicting the deformation factor of a single capsule and the relative viscosity of the suspension. These findings provide valuable insights into the complex behaviour of elastic capsules in inertial flows, informing the design of more accurate and efficient inertial microfluidic systems.

The present study investigates the gravity-driven settling dynamics of non-Brownian suspensions consisting of spherical and cubic particles within a triply periodic domain. We numerically examine the impact of solid volume fraction on the evolving microstructure of the suspension using the rigid multiblob method under Stokes flow conditions. Our simulations match macroscopic trends observed in experiments, and align well with established semi-empirical correlations across a broad range of volume fractions. At low to moderate solid volume fractions, the settling mechanism is governed primarily by hydrodynamic interactions between the particles and the surrounding fluid. However, frequent collisions between particles in a highly packed space tend to suppress velocity fluctuations at denser regimes. For dilute suspensions, transport properties are shaped predominantly by an anisotropic microstructure, though this anisotropy diminishes as many-body interactions intensify at higher volume fractions. Notably, cubic particles exhibit lower anisotropy in velocity fluctuations compared to spherical particles, owing to more efficient momentum and energy transfer from the gravity-driven direction to transverse directions. Finally, bidisperse suspensions with mixed particle shapes show enhanced velocity fluctuations, driven by shape-induced variations in drag and increased hydrodynamic disturbances. These fluctuations in turn affect the local sedimentation velocity field, leading to the segregation of particles in the mixture.

The orientational trajectories of rod-like particles suspended in a liquid are influenced by their surroundings, such as the type of flow and nearby walls, and deviate from the well-known Jeffery orbits in shear flows. We consider two types of shear flows between two parallel planar walls: wall-driven simple shear flow (C-flow), and parabolic flow driven by an external body force (P-flow). We simulated hydrodynamically interacting rod-like particles using a chain-of-spheres model immersed in a lattice Boltzmann fluid within a confined channel. As these particles in shear flows approach the wall, their orbits become flattened, exhibiting a ‘swinging motion’ on a plane parallel to the wall. Near the wall, the influence of the wall on the orbital motion varies depending on the flow type. In P-flow, the particles maintain their periodic swinging motions, whereas in C-flow, they stop swinging and align with the flow direction. This difference arises due to distinct hydrodynamic interactions with the wall in each flow type. Simulations also replicated the ‘pole-vaulting’ motion, where particles move away from the wall during their tumbling motion. For weakly sedimenting particles under shear flows, both flow types showed behaviour similar to that of neutrally buoyant particles. However, in P-flow, driven by gravity towards the wall, the particles cease their swinging motion and align perpendicularly to the flow direction, consistent with experimental observations.

We report an anomalous capillary phenomenon that reverses typical capillary trapping via nanoparticle suspension and leads to a counterintuitive self-removal of non-aqueous fluid from dead-end structures under weakly hydrophilic conditions. Fluid interfacial energy drives the trapped liquid out by multiscale surfaces: the nanoscopic structure formed by nanoparticle adsorption transfers the molecular-level adsorption film to hydrodynamic film by capillary condensation, and maintains its robust connectivity, then the capillary pressure gradient in the dead-end structures drives trapped fluid motion out of the pore continuously. The developed mathematical models agree well with the measured evolution dynamics of the released fluid. This reversing capillary trapping phenomenon via nanoparticle suspension can be a general event in a random porous medium and could dramatically increase displacement efficiency. Our findings have implications for manipulating capillary pressure gradient direction via nanoparticle suspensions to trap or release the trapped fluid from complex geometries, especially for site-specific delivery, self-cleaning, or self-recover systems.

We revisit the model problem of Squires & Brady (Phys. Fluids, vol. 17, 2005, 073101), where a Brownian probe is dragged through a dilute dispersion of Brownian bath particles. In this problem, the microrheology due to excluded-volume interactions is represented by an effective viscosity, with the nonlinearity in the driving force entering via the dependence of the viscosity increment (relative to the viscosity of a pure solvent) upon the deformation of the dispersion microstructure. Our interest is in the limit of large Péclet numbers, $ P{\kern-1pt}e\gg 1$, where the microstructural deformation adopts the form of a boundary layer about the upstream hemisphere of the probe. We show that the boundary-layer solution breaks down at the equator of the probe and identify a transition region about the equator, connecting the layer to a downstream wake. The microstructural deformation in this region is governed by a universal boundary-value problem in a semi-bounded two-dimensional domain. The equatorial region continues downstream as a transition layer, which separates the wake of the probe from the undisturbed ambient; in that layer, the microstructure is governed by a one-dimensional heat-like equation. Accounting for the combined contributions from the respective asymptotic provinces we find the approximation $ ({1}/{2})[1+ (\ln P{\kern-1pt}e + 1.046)/ P{\kern-1pt}e]$ for the ratio of the large-$ P{\kern-1pt}e$ viscosity increment to the corresponding linear-response increment. Our asymptotic approximation is in excellent agreement with the increment predicted by a finite-difference numerical calculation of the microstructure deformation, tailored to the large-$ P{\kern-1pt}e$ topology.

Active suspensions encompass a wide range of complex fluids containing microscale energy-injecting particles, such as cells, bacteria or artificially powered active colloids. Because they are intrinsically non-equilibrium, active suspensions can display a number of fascinating phenomena, including turbulent-like large-scale coherent motion and enhanced diffusion. Here, using a recently developed active fast Stokesian dynamics method, we present a detailed numerical study of the hydrodynamic diffusion in apolar active suspensions of squirmers. Specifically, we simulate suspensions of active but non-self-propelling spherical squirmers (or ‘shakers’), of either puller type or pusher type, at volume fractions from 0.5 % to 55 %. Our results show little difference between pulling and pushing shakers in their instantaneous and long-time dynamics, where the translational dynamics varies non-monotonically with the volume fraction, with a peak diffusivity at around 10 % to 20 %, in stark contrast to suspensions of self-propelling particles. On the other hand, the rotational dynamics tends to increase with the volume fraction as is the case for self-propelling particles. To explain these dynamics, we provide detailed scaling and statistical analyses based on the activity-induced hydrodynamic interactions and the observed microstructural correlations, which display a weak local order. Overall, these results elucidate and highlight the different effects of particle activity versus motility on the collective dynamics and transport phenomena in active fluids.

The shear-induced diffusivity of non-Brownian spheres in monodisperse suspensions undergoing viscous flow was calculated using simulations that account for particle roughness and friction as independent parameters. The diffusivity increases significantly as the friction coefficient is increased, and the effect is largest on rougher particles. Roughness reduces the transverse diffusivities relative to smoother particles for sufficiently concentrated suspensions of frictionless and low-friction particles. However, the diffusivity of roughened particles is larger than smoother ones at high values of the friction coefficient. The increase of the diffusivity with friction is associated with a significant broadening of the variance of the rotational velocities. The most prevalent observation, when correlating the microstructure to changes in diffusivity for frictionless particles, is that less diffusive systems, with larger roughness, form layers along the flow direction. These results confirm previous experimental and simulation results that roughness can decrease diffusivity at large concentrations using a more detailed model. Also, comparisons of the simulation results with previously published experimental measurements indicate that friction improves the alignment of the results with experiments.

Settling velocity statistics for dilute, non-Brownian homogeneous suspensions of polydisperse spheres having a log-normal size distribution are generated from Stokesian dynamics simulations, as a function of the total volume fraction $\phi$ and normalised width $\alpha$ of the particle size distribution. Several hundred instantaneous configurations are averaged to obtain reliable statistics. The paper reports data for the average and fluctuating settling velocity of each particle class in a suspension that is widely polydisperse – previous work was limited to only two or three classes, and the average settling velocity of each particle class was in most cases not reported – and provides an assessment of the accuracy of the analytical models proposed by Batchelor, Richardson & Zaki, Davis & Gecol and Masliyah–Lockett–Bassoon in predicting the simulation data. A limited comparison with dynamic simulations in which the particle microstructure is allowed to evolve in time is also included.

The motion of a sphere freely rising or falling in a 5d (d is the diameter of the sphere) square tube was numerically studied for the sphere-to-fluid density ratio ranging from 0.1 to 2.3 (0.1 ≤ ρs/ρ ≤ 2.3, ρs is the density of spheres and ρ the fluid density) and Galileo number from 140 to 230 (140 ≤ Ga ≤ 230). We report that Hopf bifurcation occurs at Gacrit ≈ 157, where both the heavy and light spheres lose stability. The helical motion is widely seen for all spheres at Ga > 160 resulting from a double-threaded vortex interacting with the tube walls, which becomes irregular at Ga ≥ 190 where heavy spheres act differently from their counterparts; that is, heavy spheres change their helical directions alternately while light spheres exhibit helical trajectories with jaggedness in connection with the shedding of the double-threaded vortices. This is because of the difference in inertia between the heavy and light spheres. We also checked the oscillation periods for the helical motion of the spheres. They show opposite variations with ρs/ρ for the two types of spheres. Light spheres (ρs/ρ ≤ 0.7) reach a zigzagging regime at Ga ≥ 200 where a vortex loop (hairpin-like vortical structure) is formed which may develop into a vortex ring downstream at small ρs/ρ. This might be the first time a transition from the helical motion to the zigzagging motion for heavy spheres (ρs/ρ ≥ 1.8) has been reported. Finally, we examined the dependence of both the terminal Reynolds number and the drag coefficient of the spheres on the Galileo number.

Colloidal gels have strong industrial relevance as they can behave as liquids or solids. The latter allows them to support a buoyant weight against gravity. However, the system is intrinsically out of equilibrium, which means that the colloids must eventually settle out of the suspension. The process of settling has been captured theoretically, but the presence of a delay time during which the gel appears relatively unaffected by gravity has not. Here, we modify existing frameworks to capture this delay, by treating the gel as a continuum with viscous response that is based on the local bond density. We can solve our model numerically to obtain the evolution of the colloid density profile and recover qualitatively the accumulation of a dense layer on top of the settling gel, as is observed experimentally in depletion gels. This numerical study is complemented by a theoretical analysis that allows us to identify an emergent time and length scale that set the dynamics of the gel. Our model provides a solid foundation for future studies that incorporate hydrodynamic erosion and tackle industrially relevant geometries.

The featured article ‘Break-up of a falling drop containing dispersed particles’ (Nitsche and Batchelor, J. Fluid Mech., 1997, vol. 340, pp. 161–175) is G. K. Batchelor's last published paper with his former postdoctoral associate J. M. Nitsche. The objective of the study was to investigate the randomness of the velocities of interacting rigid particles falling under gravity through a viscous fluid at a small Reynolds number and its consequence for the breakup of a falling cloud of particles. The study focused on a quintessential problem of the collective dynamics of interacting particles and has been an inspiration for subsequent work.

Shear-induced self-diffusion is a fundamental mode of transport in granular flows. Yet its critical behaviour and dependence on the particle solid fraction are still unclear. Here, we rationalize these dependencies by performing two-dimensional pressure-imposed numerical simulations of dense non-Brownian frictional suspensions. Our results, combined with existing numerical data on inertial granular flows, show that the shear-induced diffusion coefficients of both systems can be captured by a single function of the distance to jamming. They further show that the grain diffusive behaviour is underpinned by a specific random walk process, having a constant elementary step length driven at a frequency that increases with the solid fraction. The proposed scaling laws pave the way for a better understanding of mixing processes in granular media.