The confined Stokesian dynamics (CSD) algorithm recently reported equilibrium properties but was missing hydrodynamic functions required for suspension stress and non-equilibrium properties. In this first of a two-part series, we expand the CSD algorithm to model the traceless part of the stress tensor. To obtain quantities needed to solve the integral expressions for the stress, we developed a general method to solve Stokes’ equations in bispherical coordinates. We calculate the traceless stress tensor for arbitrary particle-to-enclosure size ratio. We next compute rheology of a confined suspension by implementing the stresslet hydrodynamic coefficients into CSD, yielding the deviatoric part of the many-body hydrodynamic stresslet. We employed energy methods to relate this stresslet to the high-frequency viscosity of the confined suspension, finding an increase in viscous dissipation with crowding and confinement well beyond the unconfined value. We show that confinement effects on viscosity are dominated by near-field interactions between the particles that reside very near the cavity wall (rather than particle–wall interactions). Surprisingly, this near-field effect is stronger than the viscosity of an unconfined suspension, showing that entropic exclusion driven by the wall sets up many lubrication interactions that then generate strong viscous dissipation. The limiting case of a particle near a flat wall reveals a correction to prior literature. The theory presented in this work can be expanded to study the Brownian contribution to the viscosity of confined suspensions in and away from equilibrium. In part 2, we report the osmotic pressure, via the trace of the stress tensor.

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared with diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection–diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly spherical particles, weak flow and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

We revisit Brenner's seminal work on the Stokes resistance of a slightly deformed sphere (Chem. Engng Sci., vol. 19, 1964, p. 519), evaluate its range of validity and extend its applicability to higher deformations for axisymmetric particles, using hydrodynamic radius as the measure of Stokes resistance. Brenner's method solves the flow around a slightly deformed sphere through two mapping steps: the first mapping translates the surface velocity on the deformed sphere to that over a reference sphere of arbitrary radius using an asymptotic expansion of the flow field in terms of deformation amplitude and a Taylor expansion of the velocity field around the surface of the reference sphere. Subsequently, the second mapping extrapolates the velocity field from the surface of the reference sphere to any point in the fluid using Lamb's general solution for Stokes flow. While the original work addresses slightly deformed spheres to a linear order in deformation amplitude, we demonstrate that the first mapping, in combination with axisymmetric spectral modes (J. Fluid Mech., vol. 936, 2022, R1), can accommodate significant deformations to arbitrary orders of perturbation, and thus is not limited to slightly deformed spheres. Also, while first-order analysis is suitable for nearly spherical particles, second-order terms can provide a reasonable range for significantly higher deformations.

The mobility of externally driven phoretic propulsion of particles is evaluated by simultaneously solving the solute conservation equation, interaction potential equation and the modified Stokes equation. While accurate, this approach is cumbersome, especially when the interaction potential decays slowly compared with the particle size. In contrast to external phoresis, the motion of self-phoretic particles is typically estimated by relating the translation and rotation velocities with the local slip velocity. While this approach is convenient and thus widely used, it is only valid when the interaction decay length is significantly smaller than the particle size. Here, by taking inspiration from Brady (J. Fluid Mech., vol. 922, 2021, A10), which combines the benefits of two approaches, we reproduce their unified mobility expressions with arbitrary interaction potentials and show that these expressions can conveniently recover the well-known mobility relationships of external electrophoresis and diffusiophoresis for arbitrary double-layer thickness. Additionally, we show that for a spherical microswimmer, the derived expressions relax to the slip velocity calculations in the limit of the thin interaction length scales. We also employ the derived mobility expressions to calculate the velocities of an autophoretic Janus particle. We find that there is significant dampening in the translation velocity even when the interaction length is an order of magnitude larger than the particle size. Finally, we study the motion of a catalytically self-propelled particle, while it also propels due to external concentration gradients, and demonstrate how the two propulsion modes compete with each other.

It is known that the dispersion of colloidal particles in porous media is determined by medium structure, pore-scale flow variability and diffusion. However, much less is known about how diffusiophoresis, that is, the motion of colloidal particles along salt gradients, impacts large-scale particle dispersion in porous media. To shed light on this question, we perform detailed pore-scale simulations of fluid flow, solute transport and diffusiophoretic particle transport in a two-dimensional hyper-uniform porous medium. Particles and solute are initially uniformly distributed throughout the medium. The medium is flushed at constant flow rate, and particle breakthrough curves are recorded at the outlet to assess the macroscopic effects of diffusiophoresis. Particle breakthrough curves show non-Fickian behaviour manifested by strong tailing that is controlled by the diffusiophoretic mobility. Although diffusiophoresis is a short-time, microscopic phenomenon owing to the fast attenuation of salt gradients, it governs macroscopic colloid dispersion through the partitioning of particles into transmitting and dead-end pores. We quantify these behaviours by an upscaled analytical model that describes both the retention and release of colloids in dead-end pores and the observed long-time tailings. Our results suggest that diffusiophoresis is an efficient tool to control particle dispersion and filtration through porous media.

We consider the dynamic wetting and dewetting processes of films and droplets of complex liquids on planar surfaces, focusing on the case of colloidal suspensions, where the particle interactions can be sufficiently attractive to cause agglomeration of the colloids within the film. This leads to an interesting array of dynamic behaviours within the liquid and of the liquid–air interface. Incorporating concepts from thermodynamics and using the thin-film approximation, we construct a model consisting of a pair of coupled partial differential equations that represent the evolution of the liquid film and the effective colloidal height profiles. We determine the relevant phase behaviour of the uniform system, including finding associated binodal and spinodal curves, helping to uncover how the emerging behaviour depends on the particle interactions. Performing a linear stability analysis of our system enables us to identify parameter regimes where agglomerates form, which we independently confirm through numerical simulations and continuation of steady states, to construct bifurcation diagrams. We obtain various dynamics such as uniform colloidal profiles in an unstable situation evolving into agglomerates and thus elucidate the interplay between dewetting and particle aggregation in complex liquids on surfaces.

While the problem governing Stokes flow about a single particle that is subject to an external force is ill posed in two dimensions (the ‘Stokes paradox’), the related problem of two mutually repellent particles is well posed. Motivated by self-assembly phenomena in thin viscous membranes, we consider this problem in the limit of remote particles. Such limits are typically handled in the literature using reflection techniques, which provide successive approximations to the mutual hydrodynamic interactions. Since their starting point is a single particle in an unbounded fluid domain, these techniques are futile in the present two-dimensional problem. We show how this apparent contradiction is resolved via use of singular perturbations. We obtain a two-term approximation for the velocity acquired by circular disks, considering both rigid and free particle surfaces. We also illustrate our perturbation scheme for elliptic disks, deriving a renormalised single-particle velocity. The utility of our asymptotic scheme is illustrated in the general problem of hydrodynamic interaction between a cluster of remote disks.

Dense suspensions of solid particles in viscous liquid are ubiquitous in both industry and nature, and there is a clear need for efficient numerical routines to simulate their rheology and microstructure. Particles of micron size present a particular challenge: at low shear rates, colloidal interactions control their dynamics while at high rates, granular-like contacts dominate. While there are established particle-based simulation schemes for large-scale non-Brownian suspensions using only pairwise lubrication and contact forces, common schemes for colloidal suspensions generally are more computationally costly and thus restricted to relatively small system sizes. Here, we present a minimal particle-based numerical model for dense colloidal suspensions that incorporates Brownian forces in pairwise form alongside contact and lubrication forces. We show that this scheme reproduces key features of dense suspension rheology near the colloidal-to-granular transition, including both shear thinning due to entropic forces at low rates and shear thickening at high rates due to contact formation. This scheme is implemented in LAMMPS, a widely used open source code for parallelised particle-based simulations, with a runtime that scales linearly with the number of particles, making it amenable for large-scale simulations.

We examined the influences of slip parameters on the velocity and thermal characteristics of a ferrofluid film of fixed thickness. The flow is generated on a rough and inclined whirling surface that is positioned in an external magnetic (dipole) field. The similarity transformation reduces the model equations (continuity, momentum, energy and concentration), and the solution of the normalized coupled ordinary differential equations is carried out through the finite element process. The influences of slip effects, Brownian motion, thermophoresis and a heat source on the velocity (radial, tangential and axial), gravity (drainage, induced), temperature profile and concentration profile are determined. The tangential flow and temperature are both decreased by an increase in the velocity slip parameter, whereas drainage, induced, radial and axial flows are increased. Enlarging the thermal slip parameter decreases the temperature. Improving slip parameters (velocity and thermal) also improves the concentration profile. Both Nusselt and Sherwood numbers are found to improve on improving the velocity slip parameter, while they decrease on decreasing the thermal slip parameter. The results and insights from this work could be applied to a wide range of medicinal fields, such as targeted medication therapy and delivery, tissue engineering, etc. as well as different industrial processes including coating, lubrication, heat transfer, etc.

The colloidal state (stable, coagulated, or gel-like) and the rheological properties of Na-rich montmorillonite (Wyoming) dispersions are strongly influenced by organic cations. This effect is shown for homologous organic cations: alkyl trimethylammonium ions, paraquat, diquat, alkyl bispyridinium ions, and the triphenylmethane dyes crystal violet, methyl green, and tris (tri-methylammonium phenyl) methane chloride. The critical coagulation concentrations, cK, are small (often < 1 mmol/L) because the cations are enriched in the Stern layer and influence the solvent structure near the surface. The strong adsorption of the counterions at the clay-mineral surface causes cK values to increase with the solid content. Charge reversal (recharging) of the particles was observed with the longer chain alkyl trimethyl-ammonium ions, dodecyl bispyridinium ions, and crystal violet. Other cations reduced the electrophoretic mobility to zero but positive particle charges were not observed.

The plastic viscosity increased sharply at the critical coagulation concentration and showed a minimum slightly below cK, which was caused by the electroviscous effect. Yield values were developed at concentrations above cK. In most cases, yield values reached a plateau where the amount of organic cations was ∼0.5 mmol/g, i.e., about half of the cation-exchange capacity. The cK values decreased with increasing hydrophobicity of homologous compounds, but the yield value showed maxima at intermediate chain lengths. The yield value of several 0.5% dispersions was high, e.g., dodecyl trimethylammonium ions, 71 Pa; paraquat, 100 Pa; diquat, 42 Pa; hexyl bispyridinium ions, 53 Pa (vs. Ca2+, 0.2 Pa; Al3+, 0.7 Pa). The storage modulus as a function of the number of organic cations changed in a similar way as the yield value, and high values were observed (e.g., dodecyl trimethylammonium ions, hexyl bispyridinium ions: 1000 Pa, paraquat: >4000 Pa). Thus, dispersions with high viscosity, yield value, and pronounced viscoelasticity are obtained by coagulating Na-rich montmorillonite dispersions with organic cations.

Flow generation by colloidal motors activated by external stimuli is an important issue for active matter physics and several nanotechnological or biomedical applications. For instance, flow recirculation generated by rotating magnetic self-assemblies allows effective ‘pumping’ of a thrombolytic drug towards a blood clot along a blocked vessel. However, the physics of the flow generation in this case remains still poorly explored. This study is focused on the generation of a recirculation flow of a magnetic colloid (aqueous suspension of iron oxide nanoparticles with partially screened electrostatic repulsion) within a closed microfluidic channel via application of an external rotating magnetic field. The colloid undergoes reversible phase separation manifested through the appearance of micron-sized elongated aggregates. They synchronously rotate with the magnetic field and can generate macroscopic flows only in the presence of gradients of the aggregate concentration across the channel induced by superposition of a weak magnetic field gradient to the homogeneous rotating field. We achieve recirculation flows with a characteristic speed ${\sim} 5{-}8\;{\rm \mu}\textrm{m}\;{\textrm{s}^{ - \textrm{1}}}$ at low magnetic field amplitude and frequency (${H_0} \approx 3{-}10\;\textrm{kA}\;{\textrm{m}^{ - 1}}$, ${f = 5{-}15\ \textrm{Hz}}$) at low nanoparticle volume fraction ${\varphi _p} = (1.6{-}3.2) \times {10^{ - 3}}$. The concentration and velocity profiles have been assessed experimentally through particle tracking and particle image velocimetry, and have also been computed using the hydrodynamic diffusion approach coupled with the momentum balance equation with a magnetic torque term. The model correctly reproduces the shape of the experimental concentration and velocity fields and explains complex behaviours of the average recirculation speed as a function of governing parameters (${H_0}$, f, ${\varphi _p}$, channel size).

We have developed a parameter-free, two-phase, volume-averaged approach to predictively describe the spin-up flow of dilute, cluster-free ferrofluids excited by low-frequency rotating magnetic fields. Predictive validation of the model was performed through a thorough comparison with local velocity profile measurements, and it demonstrated its ability to capture the spin-up flow dynamics without the need for parameter tuning by carefully delineating the validity domain of the ferrofluid dilutedness conditions. To gain insight into the underlying flow mechanisms, we performed a systematic parametric analysis examining the effects of the induced magnetic field, the dipolar interactions between magnetic nanoparticles and the demagnetizing field. How these mechanisms shape the flow of dilute ferrofluids excited by low-frequency rotating fields in a standard spin-up flow geometry has been addressed using probabilistic nanoparticle orientational dynamics, combining Faxén's laws and the Smoluchowski equation to describe the transport of particle magnetic moments. Our findings revealed that the induced magnetic field is the primary driving force of ferrofluid spin-up flow. The dipole interactions and demagnetizing field, on the other hand, contribute only as secondary phenomena to the overall flow behaviour. Furthermore, we have discussed the potential extension of the two-phase approach, in particular with respect to the formation of chain-like aggregates under the influence of strong magnetic fields. Overall, our study provides valuable insights into the complex dynamics of ferrofluid flow and contributes to a comprehensive understanding of the key mechanisms governing the spin-up flow of dilute ferrofluids excited by low-frequency rotating magnetic fields.

Motile bacteria play essential roles in biology that rely on their dynamic behaviours, including their ability to navigate, interact and self-organize. However, bacteria dynamics on fluid interfaces are not well understood. Swimmers adsorbed on fluid interfaces remain highly motile, and fluid interfaces are highly non-ideal domains that alter swimming behaviour. To understand these effects, we study flow fields generated by Pseudomonas aeruginosa PA01 in the pusher mode. Analysis of correlated displacements of tracers and bacteria reveals dipolar flow fields with unexpected asymmetries that differ significantly from their counterparts in bulk fluids. We decompose the flow field into fundamental hydrodynamic modes for swimmers in incompressible fluid interfaces. We find an expected force-doublet mode corresponding to propulsion and drag at the interface plane, and a second dipolar mode, associated with forces exerted by the flagellum on the cell body in the aqueous phase that are countered by Marangoni stresses in the interface. The balance of these modes depends on the bacteria's trapped interfacial configurations. Understanding these flows is broadly important in nature and in the design of biomimetic swimmers.

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength $\beta =a^*e^*E_\infty ^*/k_B^*T^*$, defined as the ratio of the product of the applied electric field magnitude $E_\infty ^*$ and particle radius $a^*$, to the thermal voltage $k_B^*T^*/e^*$, where $k_B^*$ is Boltzmann's constant, $T^*$ is the absolute temperature, and $e^*$ is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density $\sigma$ over a wide range of $\beta$. Here, $\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$, where $\sigma ^*$ is the dimensional surface charge density, and $\epsilon ^*$ is the permittivity of the electrolyte. For moderately charged particles ($\sigma ={O}(1)$), the electrophoretic velocity is linear in $\beta$ when $\beta \ll 1$, and its dependence on the ratio of the Debye length ($1/\kappa ^*$) to particle radius (denoted by $\delta =1/(\kappa ^*a^*)$) agrees with Henry's formula. As $\beta$ increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is $\delta$-dependent. For $\beta \gg 1$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all $\delta$. For highly charged particles ($\sigma \gg 1$) in the thin-Debye-layer limit ($\delta \ll 1$), our computations are in good agreement with recent experimental and asymptotic results.

Diffusiophoresis refers to the motion of a colloidal particle in a solute concentration gradient, animated by particle–solute interactions. We present a theoretical analysis of the diffusiophoretic motion of a viscous drop in a gradient of neutral solute at zero Reynolds number. In a spatially uniform gradient, the translational velocity of a spherical drop was found by Anderson et al. (J. Fluid Mech., vol. 117, 1982, pp. 107–121). Here, we show additionally that the drop experiences no tendency to deform, regardless of the magnitude of the interfacial tension at the interface of the drop and suspending fluid. Next, we consider a non-uniform gradient, where the ambient solute concentration takes the form of a quadrupole around the drop centroid. This gradient does not induce drop translation, due to symmetry, but does induce a deformation in the drop shape, which is spheroidal to first order in the capillary number $Ca=\beta k_B T R^2 K/\gamma$, where $\beta$ is the magnitude of the quadrupolar variation in solute concentration, $k_B T$ is the thermal energy, $R$ is the drop radius, $K$ is the Gibbs adsorption length, and $\gamma$ is the interfacial tension. Whether the drop becomes prolate or oblate depends on whether the solute–drop interaction is attractive or repulsive. Therefore, our work shows that in principle, a drop could undergo deformation during diffusiophoresis in a non-uniform solute gradient.