The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio, $\kappa$. A regular perturbation expansion in the polymer concentration, $c$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the $O(c)$ correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon $c\, De$ ($De$ is the imposed shear rate times the polymer relaxation time) and $\kappa$ and quantitatively on $c$. At a small but finite $c\, De$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing $\kappa$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate $c\, De$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller $c\, De$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing $c\, De$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing $c\, De$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.

To elucidate the effect of particle shape on the rheology of a dense, viscous suspension of frictional, non-Brownian particles, experimental measurements are presented for suspensions of polystyrene particles with different shapes in the same solvent. The first suspension is made of spheres whereas the particles which compose the second suspension are globular but with flattened faces. We present results from steady shear and shear-reversal rheological experiments for the two suspensions over a wide range of stresses in the viscous regime. Notably, we show that the rheology of the two suspensions is characterised by a shear-thinning behaviour, which is stronger in the case of the suspension of globular particles. Since the shear-reversal experiments indicate an absence of adhesive particle interactions, we attribute the shear thinning to a sliding friction coefficient which varies with stress as has been observed previously for systems similar to the first suspension. We observe that the viscosity of the two suspensions is similar at high shear stress where small sliding friction facilitates particle relative motion due to sliding. At lower shear stress, however, the sliding friction is expected to increase and the particle relative motion would be associated with rolling. The globular particles attain a higher viscosity at low shear stress than the spherical particles. We attribute this difference to a shape-induced resistance to particle rolling that is enhanced by the flattened faces. Image analysis is employed to identify features of the particle geometry that contribute to the resistance to rolling. It is shown that the apparent rolling friction coefficients inferred from the rheology are intermediate between the apparent dynamic and static rolling friction coefficients predicted on the basis of the image analysis. All three rolling resistance estimates are larger for the globular particles with flat faces than for the spherical particles and we argue that this difference yields the stronger shear thinning of the globular particle suspension.

We study the gravity-induced collisions of charged spheres of dielectric materials dispersed in a gaseous medium. When the gap thickness between the surfaces of two spheres is shorter than the mean free path of the surrounding fluid medium, continuum assumptions for the hydrodynamics interactions are no longer valid, and the non-continuum lubrication interactions result in surface-to-surface contact in finite time. Two like-charged dielectric spheres attract each other at close separations for a wide range of size and charge ratio values. We use trajectory analysis to calculate the collision rate and, thus, explore the role of electrostatic interactions in the collision dynamics of a pair of like-charged dielectric spheres. We present the modifications of pair trajectories due to electrostatic forces and show how collision efficiencies vary with the non-dimensional parameter capturing the relative strength of the electrostatic force to gravity as well as the charge ratio and size ratio.

A recent experiment by Wang et al. (Soft Matt., vol. 17, 2021, pp. 2985–2993) shows that a self-propelled compound drop in a surfactant-laden solution can autonomously change its motion from a straight line to a spiraling trajectory, enhancing its capability for chemical detection, catalytic reaction and pollutant removal in a large fluid region. To understand the underlying physics of this peculiar motion, we develop a two-dimensional minimal model to study the swimming dynamics of a compound droplet driven by a self-generated Marangoni stress. We find that, depending on the Péclet number ($Pe$) and the viscosity and volume ratios of the two compound phases, the drop can swim in a variety of trajectories, including straight lines, circles, zigzag curves and chaotic trajectories. The drop moves in circles when its two components have comparable volumes. Otherwise, it shows other types of motions depending on $Pe$. Our simulation results for the circular motion at small $Pe$ are qualitatively comparable to the experiment. The transition between zigzag and circular trajectories is mainly determined by the orientation of high-order modes with respect to the drop's swimming direction. For most compound drops, the speed decays as $Pe^{-1/3}$ at high Péclet numbers as it does for a single-phase drop. A drop with two equal components undergoes a run-and-reorient motion due to the competition between the even and odd modes.

We study the collisions in a gaseous medium of a dilute bidisperse suspension of non-Brownian spherical particles sedimenting along the flow axis of a simple shear flow. Continuum lubrication forces prevent particles from coming into contact in a finite time, thus collisions can occur only due to attractive interactions such as the van der Waals force. However, in a low-pressure medium, the lubrication forces are weaker than their continuum counterparts and allow particle pairs to collide, even without any attractive forces. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the interacting spheres, captures the significance of non-continuum interactions. We use uniformly valid hydrodynamic mobility functions, reflecting non-continuum lubrication at small separations and full continuum hydrodynamic interactions at moderate to larger separations. Due to the nature of the pair trajectory topology, the collision efficiency vanishes at a critical Knudsen number when simple shear flow alone drives the dynamics. Thus we perform collision calculations where particles experience the combined effects of van der Waals attraction and non-continuum hydrodynamics; van der Waals interactions enable collisions below the critical Knudsen number. Next, we calculate the collision efficiency for coupled differential sedimentation and simple shear driven motion in the presence of van der Waals interaction and non-continuum hydrodynamics. Finally, we explore the role of small particle inertia on relative trajectories and collision efficiencies in a non-continuum gas subject to a simple shear flow, ignoring the van der Waals force and gravity.

We study the effects of interfacial kinetics on the electro-hydrodynamics of ion transport near an ion-selective surface using a combination of linear stability analysis and numerical simulation. The finite kinetics of the electrolyte–electrode interface affects the ion transfer and electroconvection in many ways. On a surface of fixed topography, such as a metal surface of slow and stable ion deposition or covered by a polymer membrane, the finite kinetics reduces the current in one-dimensional ion diffusion/migration, increases the critical voltage for the onset of the electroconvective instability, changes the dynamics of the electroconvection and the overlimiting current, and enhances the lateral ion diffusion within the interfacial layer. The first three effects are indirectly caused by the reaction kinetics and can be characterized by an effective voltage difference across the liquid electrolyte. In comparison, the last effect is controlled by a direct interplay between kinetics and nonlinear electroconvection. Scaling laws for ion transport and features of electroconvection are proposed. We also analyse the linear stability of a surface which evolves under ion deposition and find that the finite kinetics decreases the growth rate of both electroconvective and morphological instabilities and therefore modifies the wavenumber of the most unstable mode.

As two particles approach each other, the continuum lubrication force diverges, with decreasing separation preventing contact. However, for separations comparable to the mean free path of the gas, $\lambda$, non-continuum effects cause the lubrication force to diverge more slowly with decreasing separation distance, allowing for contact in finite time. The first study of this phenomenon was done by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) for two particles moving along their line of centres. We extend their normal motion study to include tangential motions. For small Knudsen number $Kn=\lambda / a$, where $a$ is the harmonic mean of the two particle radii, we use a matched asymptotic expansion technique to obtain the non-continuum forces and torques for tangential motions of spheres separated by distances within the lubrication regime that are at or below the mean free path of the gas. The hydrodynamic resistivity functions are fitted to provide a uniformly valid approximation that smoothly transitions between the continuum multipole and non-continuum lubrication expressions for the forces and torques as the minimum gap between the particles $h_0$ varies from values of $O(a)$ to values of $O(\lambda )$. These functions, in combination with the result by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) and the classical work by Jeffrey & Onishi (J. Fluid Mech., vol. 139, 1984, pp. 261–290), yield a complete formulation for the hydrodynamic interactions of two spheres at all separations, from non-interacting spheres in the extreme far field through all the transitions that occur up to contact. We apply the new formulation to the classical case of a particle settling parallel to a vertical wall. The continuum Stokes equation predicts a settling speed that decreases with decreasing gap separation and vanishes at contact, whereas the non-continuum model developed herein predicts a finite settling speed at contact.

The collisions in a dilute polydisperse suspension of sub-Kolmogorov spheres with negligible inertia settling in a turbulent flow and interacting through hydrodynamics including continuum breakdown on close approach are studied. A statistically significant decrease in ideal collision rate without gravity is resolved via a Lagrangian stochastic velocity-gradient model at Taylor microscale Reynolds number larger than those accessible by current direct numerical simulation capabilities. This arises from the difference between the mean inward velocity and the root-mean-square particle relative velocity. Differential sedimentation, comparable to the turbulent shear relative velocity, but minimally influencing the sampling of the velocity gradient, diminishes the Reynolds number dependence and enhances the ideal collision rate i.e. the rate without interactions. The collision rate is retarded by hydrodynamic interactions between sphere pairs and is governed by non-continuum lubrication as well as full continuum hydrodynamic interactions at larger separations. The collision efficiency (ratio of actual to ideal collision rate) depends on the relative strength of differential sedimentation and turbulent shear, the size ratio of the interacting spheres and the Knudsen number (defined as the ratio of the mean-free path of the gas to the mean radius of the interacting spheres). We develop an analytical approximation to concisely report computed results across the parameter space. This accurate closed form expression could be a critical component in computing the evolution of the size distribution in applications such as water droplets in clouds or commercially valuable products in industrial aggregators.

Collisions in a dilute polydisperse suspension of spheres of negligible inertia interacting through non-continuum hydrodynamics and settling in a slow uniaxial compressional flow are studied. The ideal collision rate is evaluated as a function of the relative strength of gravity and uniaxial compressional flow and it deviates significantly from a linear superposition of these driving terms. This non-trivial behaviour is exacerbated by interparticle interactions based on uniformly valid non-continuum hydrodynamics, that capture non-continuum lubrication at small separations and full continuum hydrodynamic interactions at larger separations, retarding collisions driven purely by sedimentation significantly more than those driven purely by the linear flow. While the ideal collision rate is weakly dependent on the orientation of gravity with the axis of compression, the rate including hydrodynamic interactions varies by more than $100\,\%$ with orientation. This dramatic shift can be attributed to complex trajectories driven by interparticle interactions that prevent particle pairs from colliding or enable a circuitous path to collision. These and other important features of the collision process are studied in detail using trajectory analysis at near unity and significantly smaller than unity size ratios of the interacting spheres. For each case analysis is carried for a large range of relative strengths and orientations of gravity to the uniaxial compressional flow, and Knudsen numbers (ratio of mean free path of the media to mean radius).

We analyse the electrophoresis of a weakly charged particle with a thin double layer in a dilute polymer solution. The particle velocity in polymer solutions modelled with different constitutive equations is calculated using a regular perturbation in the polymer concentration and the generalized reciprocal theorem. The analysis shows that the polymer is strongly stretched in two regions, the birefringent strand and the high-shear region inside the double layer. The electrophoretic velocity of the particle always decreases with the addition of polymers due to both increased viscosity and fluid elasticity. At a small Weissenberg number ($Wi$), which is the product of the polymer relaxation time and the shear rate, the polymers inside the double layer contribute to most of the velocity reduction by increasing the fluid viscosity. With increasing $Wi$, viscoelasticity decreases and shear thinning increases the particle velocity. Polymer elasticity alters the fluid velocity disturbance outside the double layer from that of a neutral squirmer to a puller-type squirmer. At high $Wi$, the strong extensional stress inside the birefringent strand downstream of the particle dominates the velocity reduction. The scaling of the birefringent strand is used to estimate the particle velocity.

The hydrodynamic lift velocity of a neutrally buoyant fibre in a simple shear flow near a wall is determined for small, but non-zero, fibre Reynolds number, illustrating the role of non-sphericity in lift. The rotational motion and effects of fibre orientation on lift are treated for fibre positions that induce and do not induce solid-body wall contacts. When the fibre does not contact the wall its lift velocity can be obtained in terms of the Stokes flow field by using a generalized reciprocal theorem. The Stokes velocity field is determined using slender-body theory with the no-slip velocity at the wall enforced using the method of images. To leading order the lift velocity at distances large compared with the fibre length and small compared with the Oseen length is found to be $0.0303\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{2}a/(\unicode[STIX]{x1D707}\ln [2l/a])$, where $l$ and $a$ are the fibre half-length and radius, $\unicode[STIX]{x1D70C}$ is the density, $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate and $\unicode[STIX]{x1D707}$ is the viscosity of the fluid. When the fibre is close enough to the wall to make solid-body contact during its rotational motion, a process known as pole vaulting coupled with inertially induced changes of fibre orientation determines the lift velocity. The order of magnitude of the lift in this case is larger by a factor of $l/a$ than when the fibre does not contact the wall and it reaches a maximum of $0.013\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{3}/(\unicode[STIX]{x1D707}\ln [l/a])$ for the case of a highly frictional contact and about half that value for a frictionless contact. These results are used to illustrate how particle shape can contribute to separation methods such as those in microfluidic channels or cross-flow filtration processes.

This paper presents a theory to obtain the force per unit length acting on a slender filament with a non-circular cross-section moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has $O(1/\ln (2A))+O(\unicode[STIX]{x1D6FC}/\ln ^{2}(2A))$ contributions driven by the relative motion of the particle and the local fluid velocity and an $O(\unicode[STIX]{x1D6FC}/(\ln (2A)A))$ contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ($A=l/a_{0}$) is defined as the ratio of the particle size ($l$) to the cross-sectional dimension ($a_{0}$) and $\unicode[STIX]{x1D6FC}$ is the amplitude of the non-circular perturbation. Using thought experiments, we show that two-lobed and three-lobed cross-sections affect the response to relative motion and velocity gradients, respectively. A two-dimensional Stokes flow calculation is used to extend the perturbation analysis to cross-sections that deviate significantly from a circle (i.e. $\unicode[STIX]{x1D6FC}\sim O(1)$). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasi-periodically if they have two-lobed cross-section, and rotate chaotically and translate diffusively if they have a combination of two- and three-lobed cross-sections. Finally, we show the remarkable ability of our theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios (${\sim}10$) and cross-sections that deviate significantly from a circle, thereby making our theory a computationally inexpensive alternative to other Stokes flow solvers.

The plasma membranes of cells are thin viscous sheets in which some transmembrane proteins have two-dimensional mobility and some are immobilized. Previous studies have shown that immobile proteins retard the short-time diffusivity of mobile particles through hydrodynamic interactions and that steric effects of immobile proteins reduce the long-time diffusivity in a model that neglects hydrodynamic interactions. We present a rigorous derivation of the long-time diffusivity of a single mobile protein interacting hydrodynamically and thermodynamically with an array of immobile proteins subject to periodic boundary conditions. This method is based on a finite element method (FEM) solution of the probability density of the mobile protein diffusing with a position-dependent mobility determined through a multipole solution of Stokes equations. The simulated long-time diffusivity in square arrays decreases as the spacing in the array approaches the particle size in a manner consistent with a lubrication analysis. In random arrays, steric effects lead to a percolation threshold volume fraction above which long-time diffusion is arrested. The FEM/multipole approach is used to compute the long-time diffusivity far away from this threshold. An approximate analysis of mobile protein diffusion through a network of pores connected by bonds with resistances determined by the FEM/multipole calculations is then used to explore higher immobile area fractions and to evaluate the finite simulation cell size scaling behaviour of diffusion near the percolation threshold. Surprisingly, the ratio of the long-time diffusivity to the spatially averaged short-time diffusivity in these two-dimensional fixed arrays is higher in the presence of hydrodynamic interactions than in their absence. Finally, the implications of this work are discussed, including the possibility of using the methods developed here to investigate more complex diffusive phenomena observed in cell membranes.

In this study, we address the modification of sheared turbulence by dispersed inertial particles. The preferential sampling of the straining regions of the flow by inertial particles in turbulence leads to an inhomogeneous distribution of particles. The strong gravitational loading exerted by the highly concentrated regions results in anisotropic alteration of turbulence at small scales in the direction of gravity. These effects are investigated in a rapid distortion theory (RDT) extended for two-way coupled particle-laden flows. To make the analysis tractable, we assume that particles have small but non-zero inertia. In the classical results for single-phase flows, the RDT assumption of fast shearing compared to the turbulence time scales leads to the distortion and shear-induced production of turbulence. In particle-laden turbulence, the coupling between the two phases under rapid shearing induces number density fluctuations that convert gravitational potential energy to turbulent kinetic energy and modulate the turbulence spectrum in a manner that increases with mass loading. Turbulence statistics obtained from RDT are compared with Euler–Lagrange simulations of homogeneously sheared particle-laden turbulence.

Experimental measurements of the force and torque on freely settling fibres are compared with predictions of the slender-body theory of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462). Although the flow is viscous dominated at the scale of the fibre diameter, fluid inertia is important on the scale of the fibre length, leading to inertial torques which tend to rotate symmetric fibres toward horizontal orientations. Experimentally, the torque on symmetric fibres is inferred from the measured rate of rotation of the fibres using a quasi-steady torque balance. It is shown theoretically that fibres with an asymmetric radius or mass density distribution undergo a supercritical pitch-fork bifurcation from vertical to oblique settling with increasing Archimedes number, increasing Reynolds number or decreasing asymmetry. This transition is observed in experiments with asymmetric mass density and we find good agreement with the predicted symmetry breaking transition. In these experiments, the steady orientation of the oblique settling fibres provides a means to measure the inertial torque in the absence of transient effects since it is balanced by the known gravitational torque.

Part 1 (Rani et al. J. Fluid Mech., vol. 871, 2019, pp. 450–476) of this study presented a stochastic theory for the clustering of monodisperse, rapidly settling, low-Stokes-number particle pairs in homogeneous isotropic turbulence. The theory involved the development of closure approximations for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions $\boldsymbol{r}$. In this part 2 paper, the theory is quantitatively analysed by comparing its predictions of particle clustering with data from direct numerical simulations (DNS) of isotropic turbulence containing particles settling under gravity. The simulations were performed at a Taylor micro-scale Reynolds number $Re_{\unicode[STIX]{x1D706}}=77.76$ for three Froude numbers $Fr=\infty ,0.052,0.006$, where $Fr$ is the ratio of the Kolmogorov scale of acceleration and the magnitude of gravitational acceleration. Thus, $Fr=\infty$ corresponds to zero gravity, and $Fr=0.006$ to the highest magnitude of gravity among the three DNS cases. For each $Fr$, particles of Stokes numbers in the range $0.01\leqslant St_{\unicode[STIX]{x1D702}}\leqslant 0.2$ were tracked in the DNS, and particle clustering quantified both as a function of separation and the spherical polar angle. We compared the DNS and theory values for the exponent $\unicode[STIX]{x1D6FD}$ characterizing the power-law dependence of clustering on separation. The $\unicode[STIX]{x1D6FD}$ from the $Fr=0.006$ DNS case are in reasonable agreement with the theoretical predictions obtained using the second drift closure (referred to as DF2). To quantify the anisotropy in clustering, we calculated the leading–order coefficient in the spherical harmonics expansion of the p.d.f. of pair relative positions. The coefficients predicted by the theory (DF2) again show reasonable agreement with those calculated from the DNS clustering data for $Fr=0.006$. However, we note that in spite of the high magnitude of gravity, the clustering is only marginally anisotropic both in DNS and theory. The theory predicts that the spherical harmonic coefficient scales with $\unicode[STIX]{x1D6FD}(=\unicode[STIX]{x1D6FD}_{2}St_{\unicode[STIX]{x1D702}}^{2})$, where $\unicode[STIX]{x1D6FD}_{2}$ is the ratio of the drift and diffusion flux coefficients. Since the drift flux, and thereby $\unicode[STIX]{x1D6FD}_{2}$, is seen to decrease with gravity for $St_{\unicode[STIX]{x1D702}}<1$, the anisotropy is also correspondingly diminished.

In this two-part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr\ll St_{\unicode[STIX]{x1D702}}\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e. for pair separations smaller than the Kolmogorov length scale. In this regime, the theory approximates the fluid velocity field in a reference frame following the primary particle as locally linear. In this part 1 paper, we present the derivation of closure approximations for the drift and diffusion fluxes in the p.d.f. equation for pair relative positions $\boldsymbol{r}$. The drift flux contains the time integral of the third and fourth moments of the ‘seen’ fluid velocity gradients along the trajectories of primary particles. These moments may be analytically resolved by making approximations regarding the ‘seen’ velocity gradient. Accordingly, two closure forms are derived specifically for the drift flux. The first invokes the assumption that the fluid velocity gradient along particle trajectories has a Gaussian distribution. In the second drift closure, we account for the correlation time scales of dissipation rate and enstrophy by decomposing the velocity gradient into the strain-rate and rotation-rate tensors scaled by the turbulent dissipation rate and enstrophy, respectively. An analytical solution to the p.d.f. $\langle P\rangle (r,\unicode[STIX]{x1D703})$ is then derived, where $\unicode[STIX]{x1D703}$ is the spherical polar angle. It is seen that the p.d.f. has a power-law dependence on separation $r$ of the form $\langle P\rangle (r,\unicode[STIX]{x1D703})\sim r^{\unicode[STIX]{x1D6FD}}$ with $\unicode[STIX]{x1D6FD}\sim St_{\unicode[STIX]{x1D702}}^{2}$ and $\unicode[STIX]{x1D6FD}<0$, analogous to that for the radial distribution function of non-settling pairs. An explicit expression is derived for $\unicode[STIX]{x1D6FD}$ in terms of the drift and diffusion closures. The $\langle P\rangle (r,\unicode[STIX]{x1D703})$ solution also shows that, for a given $r$, the clustering of $St_{\unicode[STIX]{x1D702}}\ll 1$ particles is only weakly anisotropic, which is in conformity with prior observations from direct numerical simulations of isotropic turbulence containing settling particles.

Particle-laden flows of sedimenting solid particles or droplets in a carrier gas have strong inter-phase coupling. Even at low particle volume fractions, the two-way coupling can be significant due to the large particle to gas density ratio. In this semi-dilute regime, the slip velocity between phases leads to sustained clustering that strongly modulates the overall flow. The analysis of perturbations in homogeneous shear reveals the process by which clusters form: (i) the preferential concentration of inertial particles in the stretching regions of the flow leads to the formation of highly concentrated particle sheets, (ii) the thickness of the latter is controlled by particle-trajectory crossing, which causes a local dispersion of particles, (iii) a transverse Rayleigh–Taylor instability, aided by the shear-induced rotation of the particle sheets towards the gravity normal direction, breaks the planar structure into smaller clusters. Simulations in the Euler–Lagrange formalism are compared to Euler–Euler simulations with the two-fluid and anisotropic-Gaussian methods. It is found that the two-fluid method is unable to capture the particle dispersion due to particle-trajectory crossing and leads instead to the formation of discontinuities. These are removed with the anisotropic-Gaussian method which derives from a kinetic approach with particle-trajectory crossing in mind.

The formation of a cluster of activated fractures when fluid is injected in a low permeability rock is analysed. A fractured rock is modelled as a dual porosity medium that consists of a growing cluster of activated fractures and the rock’s intrinsic porosity. An integro-differential equation for fluid pressure in the developing cluster of fractures is introduced to account for the pressure-driven flow through the cluster, the loss of fluid into the porous matrix and the evolution of the cluster’s permeability and porosity as the fractures are activated. Conditions under which the dependence of the permeability and porosity on the fluid pressure can be derived from percolation theory are discussed. It is shown that the integro-differential equation admits a similarity solution for the fluid pressure and that the cluster radius grows as a power law of time in two regimes: (i) a short-time regime, when many fractures are activated but pressure-driven flow in the network still dominates over fluid loss; and (ii) a long-time regime, when fluid loss dominates. The power law exponents in the two regimes are functions of the Euclidean dimension of the cluster, percolation universal exponents and the injection protocol. The model predicts that the effects of the fluid properties on the morphology of the formed network are different in the two similarity regimes. For example, increasing the injection rate with time, in the flow dominant regime, produces a smaller cluster of activated fractures than that formed by injecting the fluid at a constant rate. In the fluid loss dominated regime, however, ramping up the injection rate produces a larger cluster.

A ring with a cross-section that has a blunt inner and sharper outer edge can attain an equilibrium orientation in a Newtonian fluid subject to a low Reynolds number simple shear flow. This may be contrasted with the continuous rotation exhibited by most rigid bodies. Such rings align along an orientation when the rotation due to fluid vorticity balances the counter-rotation due to the extensional component of the simple shear flow. While the viscous stress on the particle tries to rotate it, the pressure can generate a counter-vorticity torque that aligns the particle. Using boundary integral computations, we demonstrate ways to effectively control this pressure by altering the geometry of the ring cross-section, thus leading to alignment at moderate particle aspect ratios. Aligning rings that lack fore–aft symmetry can migrate indefinitely along the gradient direction. This differs from the periodic spatial trajectories of fore–aft asymmetric axisymmetric particles that rotate in periodic orbits. The mechanism for migration of aligned rings along the gradient direction is elucidated in this work. The migration speed can be controlled by varying the cross-sectional shape and size of the ring. Our results provide new insights into controlling motion of individual particles and thereby open new pathways towards manipulating macroscopic properties of a suspension.