Particle-laden flow through conduits is ubiquitous in both natural and industrial systems. In such flows, particles often migrate across the main fluid stream, resulting in non-uniform spatial distribution owing to particle–fluid and particle–particle interactions. The most relevant lateral particle migration mechanism by particle–fluid interaction is the Segré–Silberberg effect, which is induced by the inertial forces exerted on a particle, as the flow rate increases. However, methods to suppress it have not been suggested yet. Here, we demonstrate that adding a small amount of polymer to the particle-suspending solvent effectively suppresses the Segré–Silberberg effect in a square channel. To accurately determine the position of the particles within the channel cross-sections, we devised a dual-view imaging system applicable to microfluidic systems. Our analyses show that the Segré–Silberberg effect is effectively suppressed in a square microchannel due to the balance between the inertial and elastic forces at an optimal polymer concentration while maintaining nearly constant shear viscosity.

Numerical simulations are conducted to investigate particle suspension and deposition within turbidity currents. Utilizing Lagrangian particle tracking and a discrete element model, our numerical approach enables a detailed examination of autosuspension, deposition and bulk behaviours of turbidity current. We specifically focus on flow regimes where particle settling and buoyancy-induced hydrodynamics play equally important roles. Our discussion is divided into three parts. Firstly, we examine the main body of the current formed by suspended particles, revealing a temporal evolution consisting of initial slumping, propagation and dissipation stages. Our particle calculation allows for the tracking of autosuspended particles, enabling a deeper understanding of the connection between autosuspension and current propagation through energy budget analysis. In the second part, we delve into particle deposition, highlighting transverse and longitudinal variations. Transverse variations arise from lobe-and-cleft (LC) flow features, while longitudinal variations result from vortex detachment, particularly notable with large-sized particles. We observe that as particle size increases, leading to a particle Stokes number greater than 0.1, rapid particle settling suppresses the LC flow structure, resulting in wider lobes at the deposition height. Lastly, we propose a new scaling law for the propagation speed and current length. Our simulation results demonstrate close agreement with this new scaling law, providing valuable insights into turbidity current dynamics.

The added mass force resulting from the acceleration of a body in a fluid is of fundamental and practical interest in dispersed multiphase flows. Euler–Lagrange (EL) and Euler–Euler (EE) simulations require closure terms for the added mass force in order to accurately couple the conserved variables between phases. Presently, a more thorough understanding of the added mass force in a multi-particle system is developed based on potential flow resulting in a resistance matrix formulation analogous to Stokesian dynamics. This formulation is then used to generate a dataset of added mass resistance matrices for large systems of randomly generated particles. This methodology is used to create a volume fraction corrected binary model for predicting the added mass force in large systems as well as generate statistics of the added mass force in such systems. This work provides clarification to the theory of the added mass force for particle clouds, and modelling options that may be implemented in existing EL and EE codes.

In this paper, we study the disturbance velocity and density fields induced by a sphere translating vertically in a viscous density-stratified ambient. Specifically, we consider the limit of a vanishingly small Reynolds number $(Re = \rho U a/\mu \ll 1)$, a small but finite viscous Richardson number $(Ri_v = \gamma a^3g/\mu U\ll 1)$ and large Péclet number $(Pe = Ua/D\gg 1)$. Here, $a$ is the sphere's radius, $U$ its translational velocity, $\rho$ an appropriate reference density within the framework of the Boussinesq approximation, $\mu$ the ambient viscosity, $\gamma$ the absolute value of the background density gradient, g is acceleration due to gravity and $D$ the diffusivity of the stratifying agent. For the scenario where buoyancy forces first become comparable to viscous forces at large distances, corresponding to the Stokes-stratification regime defined by $Re \ll Ri_v^{1/3} \ll 1$ for $Pe \gg 1$, important flow features have been identified by Varanasi & Subramanian (J. Fluid Mech., vol. 949, 2022, A29) – these include a vertically oriented reverse jet, and a horizontal axisymmetric wake, on scales larger than the primary (stratification) screening length of ${O}(aRi_v^{-1/3})$. Here, we study the reverse-jet region in more detail, and show that it is only the central portion of a columnar structure with multiple annular cells concentric about the rear stagnation streamline. In the absence of diffusion, corresponding to $Pe = \infty$ $( \beta _\infty = Ri_v^{1/3}Pe^{-1} = 0)$, this columnar structure extends to downstream infinity with the number of annular cells diverging in this limit. We provide expressions for the boundary of the structure, and the number of cells within, as a function of the downstream distance. For small but finite $\beta _\infty$, two length scales emerge in addition to the primary screening length – a secondary screening length of ${O}(aRi_v^{-1/2}Pe^{1/2})$ where diffusion starts to smear out density variations across cells, leading to exponentially decaying density and velocity fields; and a tertiary screening length, $l_t \sim {O}(aRi_v^{-1/2}Pe^{1/2}[\zeta + \frac {13}{4}\ln {\zeta } + ({13^2}/{4^2})({\ln \zeta }/{\zeta })])$ with $\zeta = \frac {1}{2}\ln ({\sqrt {{\rm \pi} }Ri_v^{-1}Pe^3}/{2160})$, beyond which the columnar structure ceases to exist. The latter causes a transition from a vertical to a predominantly horizontal flow, with the downstream disturbance fields reverting from an exponential to an eventual algebraic decay, analogous to that prevalent at large distances upstream.

We study the formation of dust-free regions above hot horizontal surfaces of uniform temperature and propose relations for its height in the limit of small particle inertia and gravitational effects. By including particle inertia, thermophoretic, gravitational and viscous effects, we conduct Lagrangian simulations of particle dynamics in a natural convection boundary layer over a horizontal surface. Trajectory analysis of the particles inside the boundary layer on the surface reveals the existence of two separatrices originating from a saddle point, which form the boundary of the dust-free region. These separatrices for low gravitational effects follow the boundary layer thickness, but are of much lower height and also depend on the dimensionless thermophoretic number ($Th$) and Prandtl number ($Pr$). We obtain a relation for the dimensionless height of the dust-free region ($\eta _{df}$) as a function of $Pr$ and $Th$, for low dimensionless gravitational number ($Gn$); the numerical solution of this equation gives us the dust-free region height for any $Th$ and $Pr$. We then obtain scaling laws for $\eta _{df}$ using the boundary layer equations corresponding to the $Pr \gg 1$ and $Pr \ll 1$ cases; these scaling laws are shown to be valid respectively for $Pr>1$ and $Pr<1$, except in the large $\eta$ limit for $Pr>1$, where $\eta$ is the boundary layer similarity variable. We then obtain an empirical relation in this large $\eta$ limit using the numerical solutions of the boundary layer equations for the intermediate $Pr$ case to obtain scaling laws for dust-free region height for the whole range of $Pr \ll 1$ to $Pr \gg 1$.

We study the instability of a dusty simple shear flow where the dust particles are distributed non-uniformly. A simple shear flow is modally stable to infinitesimal perturbations. Also, a band of particles remains unaffected in the absence of any background flow. However, we demonstrate that the combined scenario – comprising a simple shear flow with a localized band of particles – can exhibit destabilization due to their two-way interaction. The instability originates solely from the momentum feedback from the particle phase to the fluid phase. Eulerian–Lagrangian simulations are employed to illustrate the existence of this instability. Furthermore, the results are compared with a linear stability analysis of the system using an Eulerian–Eulerian model. Our findings indicate that the instability has an inviscid origin and is characterized by a critical wavelength below which it is not persistent. We have observed that increasing particle inertia dampens the unstable modes, whereas the strength of the instability increases with the strength of the coupling between the fluid and particle phases.

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.

In the astrophysics community it is common practice to model collisionless dust, entrained in a gas flow, as a pressureless fluid. However, a pressureless fluid is fundamentally different from a collisionless fluid – the latter of which generically possess a non-zero anisotropic pressure or stress tensor. In this paper we derive a fluid model for collisionless dust, entrained in a turbulent gas, starting from the equations describing the motion of individual dust grains. We adopt a covariant formulation of our model to allow for the geometry and coordinate systems prevalent in astrophysics, and provide a closure valid for the accretion disc context. We show that the continuum mechanics properties of a dust fluid corresponds to a higher-dimensional anisotropic Maxwell fluid, after the extra dimensions are averaged out, with a dynamically important rheological stress tensor. This higher-dimensional treatment has the advantage of keeping the dust velocity and velocity of the fluid seen, and their respective moments, on the same footing. This results in a simplification of the constitutive relation describing the evolution of the dust rheological stress.

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.

In this study we investigate the sedimentation of prolate spheroids in a quiescent fluid by means of the particle-resolved direct numerical simulation. With the increase of the particle volume fraction $\phi$ from $0.1\,\%$ to $10\,\%$, we observe a non-monotonic variation of the mean settling velocity of particles, $\langle V_s \rangle$. By virtue of the Voronoi analysis, we find that the degree of particle clustering is highest when $\langle V_s \rangle$ reaches the local maximum at $\phi =1\,\%$. Under the swarm effect, clustered particles are found to preferentially sample downward fluid flows in the wake regions, leading to the enhancement of the settling speed. As for lower or higher volume fractions, the tendency of particle clustering and the preferential sampling of downward flows are attenuated. The hindrance effect becomes predominant when the volume fraction exceeds 5 % and reduces $\langle V_s \rangle$ to less than the isolated settling velocity. Particle orientation plays a minor role in the mean settling velocity, although individual prolate particles still tend to settle faster in suspensions when they deviate more from the broad-side-on alignment. Moreover, we also demonstrate that particles are prone to form column-like microstructures in dilute suspensions under the effect of wake-induced hydrodynamic attractions. The radial distribution function is higher at a lower volume fraction. As a result, the collision rate scaled by the particle number density decreases with the increasing volume fraction. By contrast, as another contribution to the particle collision rate, the relative radial velocity for nearby particles shows a minor degree of variation due to the lubrication effect.

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.

Experiments and numerical simulations of inertial particles in underexpanded jets are performed. The structure of the jet is controlled by varying the nozzle pressure ratio, while the influence of particles on emerging shocks and rarefaction patterns is controlled by varying the particle size and mass loading. Ultra-high-speed schlieren and Lagrangian particle tracking are used to experimentally determine the two-phase flow quantities. Three-dimensional simulations are performed using a high-order, low-dissipative discretization of the gas phase while particles are tracked individually in a Lagrangian manner. A simple two-way coupling strategy is proposed to handle interphase exchange in the vicinity of shocks. Velocity statistics of each phase are reported for a wide range of pressure ratios, particle sizes and volume fractions. An upstream shift of the Mach disk in the presence of particles reveals significant two-way coupling even at low mass loading. A semi-analytic model that predicts the extent of the Mach disk shift is presented based on a one-dimensional Fanno flow that takes into account volume displacement by particles and interphase exchange due to drag and heat transfer. The per cent shift in Mach disk is found to scale with the mass loading, nozzle pressure ratio and interphase slip velocity and inversely with the particle diameter.

This paper presents numerical simulations of the free fall of homogenous cylinders of length-to-diameter ratios $2$, $3$ and 5 and solid-to-fluid-density ratios $\rho _s/\rho$ going from 0 to 10 in transitional regimes. The path instabilities are shown to be due to two types of transitional states. The well-known fluttering state is a solid mode, characterised by significant oscillations of the cylinder axis due to a strong interaction between the vortex shedding in the wake and the solid degrees of freedom. Weakly oscillating, mostly irregular trajectories, are fluid modes, associated with purely fluid instabilities in the wake. The interplay of solid and fluid modes leads to a varying scenario in which the length-to-diameter and density ratios play an important role. The description is accompanied by the presentation of the identified transitional states in terms of path characteristics and vorticity structure of the wakes and by bifurcation diagrams showing the evolution of asymptotic states with increasing Galileo numbers. There appears to be a strong difference between the behaviour of cylinders of aspect ratio $L/d=3$ and 5. A similar contrast is stated between light cylinders of density ratios $\rho _s/\rho \le 2$ and dense cylinders of density ratios 5 and 10. Finally, the question of the scatter of values of the drag coefficient and of the frequency of oscillations raised in the literature is addressed. It is shown, that in addition to external parameters (Galileo number, density and aspect ratio) the amplitude of oscillations characterising the instability development is to be taken into account to explain this scatter. Fits of the simulation results to simple correlations are proposed. Namely that of the drag coefficient proves to be accurate (better than 1 % of accuracy) but also that of the Strouhal number (a few per cent of accuracy) may be of practical use.

Heavy particles suspended in turbulent flow possess inertia and are ejected from violent vortical structures by centrifugal forces. Once piled up along particle paths, this small-scale mechanism leads to an effective large-scale drift. This phenomenon, known as ‘turbophoresis’, causes particles to leave highly turbulent regions and migrate towards calmer regions, explaining why particles transported by non-homogeneous flows tend to concentrate near the minima of turbulent kinetic energy. It is demonstrated here that turbophoretic effects are just as crucial in statistically homogeneous flows. Although the average turbulent activity is uniform, instantaneous spatial fluctuations are responsible for inertial-range inhomogeneities in the particle distribution. Direct numerical simulations are used to probe particle accelerations, specifically how they correlate to local turbulent activity, yielding an effective coarse-grained dynamics that accounts for particle detachment from the fluid and ejection from excited regions through a space- and time-dependent non-Fickian diffusion. This leads to cast fluctuations in particle distributions in terms of a scale-dependent Péclet number ${\textit {Pe}}_\ell$, which measures the importance of turbulent advection compared with inertial turbophoresis at a given scale $\ell$. Multifractal statistics of energy dissipation indicate that $ {\textit {Pe}}_\ell \sim \ell ^\delta /\tau _{p}$ with $\delta \approx 0.84$. Numerical simulations support this behaviour and emphasise the relevance of the turbophoretic Péclet number in characterising how particle distributions, including their radial distribution function, depends on $\ell$. This approach also explains the presence of voids with inertial-range sizes, and the fact that their volumes have a non-trivial distribution with a power-law tail $p(\mathcal {V}) \propto \mathcal {V}^{-\alpha }$, with an exponent $\alpha$ that tends to 2 as ${\textit {Pe}}_\ell \to 0$.

The gas dynamics of shock-induced gas filtration through densely packed granular columns with vastly varying shock intensity and the structural parameters are numerically investigated using a coupled Eulerian–Lagrangian approach. The results shed fundamental light on the thermal effects of the shock-induced gas filtration manifested by a distinctive self-heating hot gas layer traversing the medium. The characteristics of the thermal effects in terms of the thermal intensity and uniformity are found to vary with the shock Mach number, Ms, and the filtration coefficient of the granular media, Π. As the incident shock transitions from weak to strong, and (or) the filtration coefficient increases from O(10−5) to O(104), the heating mechanisms transition between three distinct heating modes. A phase diagram of heating modes is established on the parameter space (Ms, Π), which enables us to predict the characteristics of the thermal effect in different shock-induced gas filtrations. The thermal effects markedly accelerate the pressure diffusion due to the additional heat influx when the time scale of the former is smaller than or comparable to the latter. Based on the contour map displaying the coupling degree of the thermal effects and the pressure diffusion, we identify a decoupling criterion whereby the isothermal assumption holds if only the pressure diffusion is concerned. The thermal effects may well bring about considerable thermal shocks which pose a great threat to the integrity of the solid skeleton and further reduce the overall shock resistance performance of the porous media.

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.

Three-dimensional (3-D) clustering characteristics of large-Stokes-number sprays interacting with turbulent swirling co-flows are investigated experimentally. The astigmatic interferometric particle imaging (AIPI) technique is used for simultaneous measurement of the spray droplets position in 3-D space and their corresponding diameter. The Stokes number estimated based on the Kolmogorov time scale varies from 34 to 142. The results show that the degree of droplet clustering plateaus at about 0.4 and at large Stokes numbers. It is obtained that the mean length scale of the clusters normalized by the Kolmogorov length scale follows a power-law relation, and the mean void length scale normalized by the integral length scale plateaus at about 1.5 and at large Stokes numbers. It is shown that the ratio of the number density of the droplets residing within the clusters to the global number density increases with increasing Stokes number and is about 8 for the largest Stokes number examined in this study. The joint characteristics of cluster's normalized volume and the mean diameter of droplets residing within the clusters show that small-volume clusters accommodate droplets with a relatively broad range of diameters. However, large clusters carry droplets with the most probable diameter. The developed AIPI technique in the present study and the corresponding spray characteristics are of importance for engineering applications that aim to understand the 3-D clustering characteristics of large-Stokes-number droplets sprayed into turbulent swirling co-flows.

We study the sedimentation of U-shaped circular disks in the Stokes limit of vanishing inertia. We simulate the flow past such disks using a finite-element-based solution of the three-dimensional Stokes equations, accounting for the integrable singularities that develop along their edges. We show that the purely vertical sedimentation of such disks in their upright (upside-down) U orientation is unstable to perturbations about their pitching (rolling) axes. The instability is found to depend only weakly on the size of the container in which the disks sediment, allowing us to analyse their behaviour based on the resistance matrix which governs the evolution of the disk's six rigid-body degrees of freedom in an unbounded fluid. We show that the governing equations can be reduced to two ordinary differential equations which describe the disk's inclination against the direction of gravity. A phase-plane analysis, the results of which are in good agreement with experiments, reveals that the two instabilities generally cause the disk to sediment along complex spiral trajectories while it alternates between pitching- and rolling-dominated motions. The chirality of the trajectories is set by the initial conditions rather than the (non-chiral) shape of the disk. For certain initial orientations, the disk retains its inclination and sediments along a perfectly helical path. The observed behaviour is fundamentally different from that displayed by flat circular disks which sediment without any reorientation. We therefore study the effect of variations in the disk's curvature to show how in the limit of vanishing curvature the behaviour of a flat disk is recovered.

A stream of free-falling particles from a rectangular hopper, hereafter called a ‘curtain’, was characterised systematically using a well-resolved, non-intrusive optical shadowgraphic method, to reveal both an additional axial region and an additional dilute region distributed laterally on either side of the curtain, relative to those identified previously. The effects of particle size and hopper outlet thickness on the evolution of the particle curtain were separately isolated, whilst measuring particle mass flow rate. The curtains were characterised into four distinct axial regions, namely a near-field expansion region near to the hopper exit, a neck zone where the curtain contracts, a region of intermediate-field expansion and a far field with particles reaching terminal velocity. The initial expansion half-angle, $2.3^{\circ } \le \alpha \le 4.8^{\circ }$, was found to be insensitive to particle size, but to increase with hopper outlet thickness. The ‘trough’ in the neck zone was deduced to be caused by a pressure gradient driven by particle acceleration. The curtain expansion rate at the intermediate field was found to increase with a decrease in particle size and hopper outlet thickness. The outermost dilute-particle region was deduced to be caused by collisions, induced by gradients in the velocity profile near to the hopper exit. New dimensionless analysis reveals that the dynamics of curtains can be characterised broadly into two regimes, one in which the aerodynamics is dominant and the other where it is weak. Curtain transmittance was found to scale with the Froude number, highlighting the importance of particle momentum.

Explosive dispersal of granular media widely occurs in nature across various length scales, enabling engineering applications ranging from commercial or military explosive systems to the loss prevention industry. However, the correlation between the explosive dispersal behaviour and the structure of dispersal system is far from completely understood, thereby compromising the prediction of the explosive dispersal outcome resulting from a specific dispersal system. Here, we investigate the dispersal behaviours of densely packed particle rings driven by the enclosed pressurized gases using coarse-grained computational fluid dynamics–discrete parcel method. Distinct dispersal modes emerge from the dispersal systems with vastly varying sets of the macro- and micro-scale structural parameters in terms of the dispersal completeness and the spatial uniformity of the dispersed mass. Further investigation reveals the variation in the dispersal modes arises from the collective effects of multiscale gas–particle coupling relationships. Specifically, the macroscale coupling dictates the cyclic momentum/energy transfer between gases and particle ring as an entirety. The mesoscale coupling relates to the inter-pore gas filtration through the thickness of the particle ring, leading to the mass/energy reduction of the explosive source. The microscale coupling involves the individual particle dynamics influenced by the local flow parameters. A persistent macroscale coupling results in an incomplete dispersal which takes the form of an aggregated annular band, whereas the meso- and micro-scale couplings alter the macroscale coupling to a different extent. By incorporating the effects of the variety of structural parameters on the multiscale gas–particle coupling relationships, a non-dimensional parameter referred to as the modified mass ratio is constructed, which shows an explicit correlation with the dispersal mode. We proceed to establish a dispersal ring model in the continuum frame which accounts for the macro and meso-scale coupling effects. This model proves to be capable of successfully predicting the ideal and validated failed dispersal modes.