Small finite-size particles suspended in fluid flow through an enclosed curved duct can focus to points or periodic orbits in the two-dimensional duct cross-section. This particle focusing is due to a balance between inertial lift forces arising from axial flow and drag forces arising from cross-sectional vortices. The inertial particle focusing phenomenon has been exploited in various industrial and medical applications to passively separate particles by size using purely hydrodynamic effects. A fixed size particle in a circular duct with a uniform rectangular cross-section can have a variety of particle attractors, such as stable nodes/spirals or limit cycles, depending on the radius of curvature of the duct. Bifurcations occur at different radii of curvature, such as pitchfork, saddle-node and saddle-node infinite period (SNIPER), which result in variations in the location, number and nature of these particle attractors. By using a quasi-steady approximation, we extend the theoretical model of Harding et al. (J. Fluid Mech., vol. 875, 2019, pp. 1–43) developed for the particle dynamics in circular ducts to spiral duct geometries with slowly varying curvature, and numerically explore the particle dynamics within. Bifurcations of particle attractors with respect to radius of curvature can be traversed within spiral ducts and give rise to a rich nonlinear particle dynamics and various types of tipping phenomena, such as bifurcation-induced tipping (B-tipping), rate-induced tipping (R-tipping) and a combination of both, which we explore in detail. We discuss implications of these unsteady dynamical behaviours for particle separation and propose novel mechanisms to separate particles by size in a non-equilibrium manner.

Comprehensive coherent structures around a surface-mounted low aspect ratio square cylinder in uniform flow with an oblique angle of $45^{\circ }$ were investigated for cylinder-width-based Reynolds numbers of 3000 and 10 000 by direct numerical simulation based on a topology-confined mesh refinement framework. High-resolution simulations and the critical-point concept were scrutinized to reveal for the first time the reasonable and compatible topologies of flow separation and complete near-wall structures, due to their extensive impact on various engineering applications. Large-scale horseshoe vortices are observed at two notable foci in the viscous sublayer. Within this layer, a wall-parallel jet is formed by downflow intruding into the bottom surface at the half-saddle point, then deflecting in the upstream direction and finally penetrating the bottom surface until the half-saddle point. A pair of conical vortices on the cylinder's top surface switch themselves on two sides of the square cylinder, where the switching frequency is identical with that of the sway of the side shear layer. The undulation of the Kelvin–Helmholtz instability is identified in the instantaneous development of a conical vortex and side shear layer, where the scaling of the ratio of the Kelvin–Helmholtz and von Kármán frequencies follows the power-law relation obtained by Lander et al. (J. Fluid Mech., vol. 849, 2018, pp. 1096–1119). Large-scale arch-shaped vortex is often detected in the intermediate wake region of a square cylinder, involving two interconnected portions, such as the leg portion separated from leeward surfaces and head portion rolled up from the top surface. The leg portion of the arch-shaped vortex was rooted by two foci near the bottom-surface plane.

This paper presents a comprehensive theoretical analysis of the interaction between two quasi-steady burning droplets with differing temperatures, sizes and distances, building upon the mass-flux-potential model and flame-sheet assumption. In contrast to existing research, this study introduces a fresh perspective on droplet interactions by considering the different temperatures of the droplets. Utilizing the bispherical coordinate approach, theoretical solutions for the Stefan flow, scalar fields, droplet evaporation/burning rates, interaction coefficients and flame positions have been derived successfully. A comparison with extensive numerical simulations indicates a good agreement between the analytical and numerical results under a variety of conditions. It is revealed that proximity between the droplets causes non-uniform evaporation rates on their surfaces, and in some cases, leads to condensation on the cooler droplet. Notably, when the temperatures of the two droplets differ, this results in an uneven temperature distribution across the flame surface, and increasing the temperature of one droplet substantially elevates the temperature of the nearby flame. This study also establishes a criterion for the transition between different combustion modes, specifically between group and separated combustion. The findings of this study are crucial in deepening our understanding of evaporation and combustion processes, as well as the dynamics of flame spreading, local ignition, and extinction in systems involving multiple droplets.

Flow and turbulent structures generated by the interaction of forced incoming oscillatory flow with a circular, vertical cylinder placed in an open channel with a horizontal bed are investigated using eddy-resolving simulations. Validation simulations performed with a Keulegan–Carpenter (KC) number of 20 and multiple-mode forcing corresponding to the laboratory experiment of Sumer et al. (J. Fluid Mech., vol. 332, 1997, pp. 41–70) show that detached eddy simulation (DES) predicts more accurately the amplification of the bed shear stress beneath the horseshoe vortex system (upstream side of the cylinder) and the maximum magnitude of the bed shear stress at the downstream (wake) side of the cylinder compared with unsteady Reynolds-averaged Navier–Stokes simulations. High-Reynolds-number DES simulations are then conducted with 1.5 ≤ KC ≤ 30.8 and one-mode sinusoidal forcing of the streamwise velocity in the approaching flow to investigate the changes in the wake vortex-flow regimes, the coherence of the horseshoe vortices and the generation of other near-bed coherent structures in the wake during the oscillatory cycle. The flow is periodic, no horseshoe vortices form and no vortices are shed in the wake for KC = 1.5. By contrast, for KC ≥ 8 horseshoe vortices are present over part of the oscillatory cycle and up to three wake vortices are shed over each half-cycle as KC is increased to 30.8. For an intermediate range of KC numbers, one (KC = 15.4) or two (KC = 8) of the vortices forming at the back of the cylinder during each half-cycle are washed around it when the flow reverses. The main horseshoe vortex and other horizontal near-bed vortices have a large capacity to amplify the bed shear stresses when the incoming velocity magnitude is significantly less than its peak value. Assuming the depth-averaged velocity in the incoming (undisturbed) oscillatory flow is the same in simulations conducted with different KC numbers, the peak values of the sediment entrainment potential measured by the mean (cycle-averaged) volumetric flux of sediment entrained from the bed over one oscillatory cycle occur for 8 ≤ KC ≤ 15.4. For all KC numbers, the in-line force variation over the oscillatory cycle is fairly well approximated by the Morison equation. For KC = 1.5, the in-line force is only due to inertia effects. For KC = 30.8, the maximum and minimum values of the phase-averaged in-line force are approximatively in phase with those of the incoming flow velocity. For KC ≥ 15, the phase-averaged in-line force coefficients vary between 0.8 and 1.1 during most of the oscillatory cycle (e.g. when the incoming flow velocity is not close to zero). This is different from cases with KC ≤ 8 where the in-line force coefficient is equal to zero twice during the oscillatory cycle as the in-line force becomes equal to zero for non-zero values of the incoming velocity. The largest cycle-to-cycle variations of the in-line force coefficient and in-line force are observed around KC = 8. For KC = 8 and 15.4, the cylinder is subject to relatively large phase-averaged spanwise drag forces that are comparable to the peak phase-averaged streamwise drag forces. As KC is increased to 30.8, the phase-averaged spanwise drag force becomes zero over the whole oscillatory cycle but the cylinder is still subject to large instantaneous spanwise forces over part of the oscillatory cycle.

Reduced-order models are often sought to efficiently represent key dynamical phenomena present among the broad range of temporal and spatial scales associated with unsteady and turbulent flow problems. Linear ‘input–output’ approaches and resolvent analyses reveal that important information about the most dangerous (most amplified) disturbances and the corresponding fluctuation response can be found with knowledge only of the base flow, or the turbulent mean field. In the work by Padovan et al. (J. Fluid Mech., vol. 900, 2020, A14), an important advance is made with regards to flows which have a periodically time-varying base flow, for example during unsteady vortex shedding from a body. By forming a harmonic resolvent relative to this base flow, limitations associated with the traditional linear resolvent are overcome to determine efficient bases for modelling of limit cycle flows and reveal novel information about key triadic (resonant) interactions.

The leakage from a fracture network to a surrounding medium during drainage, or backflow, driven by elastic relaxation, is considered. A network model is extended to include the effects of permeable boundaries, with the permeation through the wall assumed to be proportional to the local pressure. The regimes in which leakage is dominant relative to the parallel flow along the channel are evaluated at different times. Results show that, when the aperture of the channel is large enough, the parallel flow is greater than the permeation through the wall, and the channel thickness decreases in time, $t$, with a $t^{-1/3}$ behaviour, as reported previously. However, when the aperture is small, the channel thickness decreases exponentially in time. An asymptotic investigation of the solution for a single fracture is performed and extended to network systems. The study provides insight into the influence leakage may have on squeezing-induced flows, which is relevant to natural and engineering systems.

Depinning of liquid droplets on substrates by flow of a surrounding immiscible fluid is central to applications such as cross-flow microemulsification, oil recovery and waste cleanup. Surface roughness, either natural or engineered, can cause droplet pinning, so it is of both fundamental and practical interest to determine the flow strength of the surrounding fluid required for droplet depinning on rough substrates. Here, we develop a lubrication-theory-based model for droplet depinning on a substrate with topographical defects by flow of a surrounding immiscible fluid. The droplet and surrounding fluid are in a rectangular channel, a pressure gradient is imposed to drive flow and the defects are modelled as Gaussian-shaped bumps. Using a precursor-film/disjoining-pressure approach to capture contact-line motion, a nonlinear evolution equation is derived describing the droplet thickness as a function of distance along the channel and time. Numerical solutions of the evolution equation are used to investigate how the critical pressure gradient for droplet depinning depends on the viscosity ratio, surface wettability and droplet volume. Simple analytical models are able to account for many of the features observed in the numerical simulations. The influence of defect height is also investigated, and it is found that, when the maximum defect slope is larger than the receding contact angle of the droplet, smaller residual droplets are left behind at the defect after the original droplet depins and slides away. The model presented here yields considerably more information than commonly used models based on simple force balances, and provides a framework that can readily be extended to study more complicated situations involving chemical heterogeneity and three-dimensional effects.

Boundary-layer flow over a realistic porous wall might contain both the effects of wall-permeability and wall-roughness. These two effects are typically examined in the context of a rough-wall flow, i.e. by defining a ‘roughness’ length or equivalent to capture the effect of the surface on momentum deficit/drag. In this work, we examine the hypothesis of Esteban et al. (Phys. Rev. Fluids, vol. 7, no. 9, 2022, 094603), that a turbulent boundary layer over a porous wall could be modelled as a superposition of the roughness effects on the permeability effects by using independently obtained information on permeability and roughness. We carry out wind tunnel experiments at high Reynolds number ($14\ 400 \leq Re_{\tau } \leq 33\ 100$) on various combinations of porous walls where different roughnesses are overlaid over a given permeable wall. Measurements are also conducted on the permeable wall as well as the rough walls independently to obtain the corresponding length scales. Analysis of mean flow data across all these measurements suggests that an empirical formulation can be obtained where the momentum deficit ($\Delta U^+$) is modelled as a combination of independently obtained roughness and permeability length scales. This formulation assumes the presence of outer-layer similarity across these different surfaces, which is shown to be valid at high Reynolds numbers. Finally, this decoupling approach is equivalent to the area-weighted power-mean of the respective permeability and roughness length scales, consistent with the approach recently suggested by Hutchins et al. (Ocean Engng, vol. 271, 2023, 113454) to capture the effects of heterogeneous rough surfaces.

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Pattern-forming with externally imposed symmetry is ubiquitous in nature but little studied. We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. When periodic topographic structures are constructed on the heated boundary, they modulate the local temperature and velocity fields. Symmetric convection patterns in the form of regular vortex lattices are observed near the onset of convection, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which is well described by a Ginzburg–Landau-like model. We explore the phase diagram of buoyancy strength and periodicity of external forcing to find the optimal experimental settings for which the vortex patterns best match that of the external forcing.

Keeping a tube from being plugged by a fluid is an important process in applications. An interesting re-entrant phenomenon for the capillary state with the occluding state sandwiching the non-occluding state from both the high- and low-Bond-number regions can appear by inserting a rod into a horizontal tube at an eccentric position (Tan et al., J. Fluid Mech, vol. 946, 2022, A7). Containers with rounded corners are very common. We theoretically investigate a situation for a horizontal open tube with rounded corner(s). The results show that a re-entrant non-occlusion at a contact angle can also appear without the insertion of any object. The competition between the rounded corner wetting/non-wetting effect and gravity effect can lead to a re-entrant non-occlusion. The re-entrant non-occlusion is affected by the shape and orientation of the rounded corner(s). For a tube with only one rounded corner, the re-entrant non-occlusion exists when the rounded corner has a not-so-large corner radius and is not in a landscape orientation. For a tube with two (or more) rounded corners, the corner(s) with the strongest corner effect will determine the existence or non-existence of the re-entrant non-occlusion. This paper provides an effective scheme for designing a high-performance capillary with corners that are not easily occluded by a fluid and removing fluid blockage from a capillary in optofluidic/microfluidic applications.

The stability and postcritical behaviour of a horizontal flag undergoing gravity-induced deformation and periodic contact with a nearby horizontal rigid wall are experimentally investigated. The results elucidate the combined effects of gravity and contact on flutter, and reveal design principles for application to triboelectric energy harvesting. By varying the free-stream velocity, flag thickness and distance between the flagpole and the wall, the dynamics of the flag are classified into quasistatic equilibrium, flutter, partial contact and saturated contact modes. Considering the significance of gravitational effects, a new dimensionless flow velocity is proposed to identify the distribution of the dynamic modes, and its definition varies according to whether the wall is placed above or below the flag. The critical conditions for transitions between the dynamic modes are determined from the balance of fluid dynamic and gravitational effects. The distance from the flagpole to the wall is found to be more critical for transitions in the lower-wall configuration than in the upper-wall configuration. The peak contact force as well as the oscillation amplitude and frequency at postequilibrium exhibits remarkably different trends depending on the location of the wall. The peak contact force imposed on the wall by the fluttering flag weakens as the distance to the wall increases in the case of an upper wall, whereas it becomes stronger in the case of a lower wall.

An experimental study on stratified particle-laden plumes is presented and five steady-state flow regimes have been identified. The steady-state behaviour of the plume is directly related to the magnitude of the convective velocity associated with particle-induced instabilities, $U_c$, in relation to the terminal settling velocity of each individual particle, $u_{st}$. When $u_{st}>U_c$, the ratio of particle to fluid buoyancy flux at the source, $P$, becomes important. For $P<0.2$, the plume dynamics appears very similar to a single-phase plume as particle recycling has minimal impact on the steady-state plume height. When $P>0.2$, the plume height decreases significantly, creating an anvil-shaped intrusion similar to those associated with explosive volcanic eruptions. Importantly, the measured steady-state heights of plumes within this settling regime validate the collapse model of Apsley & Lane-Serff (J. Fluid Mech., vol. 865, 2019, pp. 904–927). When $u_{st}\leqslant U_c$, particle re-entrainment behaviour changes significantly and the plume dynamics becomes independent of $P$. When $u_{st}\approx U_c$, a trough of fluid becomes present in the sedimenting veil due to a significant flux of descending particles at the edge of the plume. Once $u_{st}< U_c$, the particles spreading in the intrusion become confined to a defined radius around the plume due to the significant ambient convection occurring beneath the current. For $u_{st}\ll U_c$, or in the case of these experiments, when $U_c\geqslant 1\ \text{cm s}^{-1}$, ambient convection becomes so strong that intrusion fluid is pulled down to the plume source, creating a flow reminiscent of a stratified fountain with secondary intrusions developing between the original current and the tank floor. Through an extension of the work of Cardoso & Zarrebini (Chem. Engng Sci., vol. 56, issue 11, 2001a, pp. 3365–3375), an analytical expression is developed to determine the onset of convection in the environment beyond the edge of the plume, which for a known particle settling velocity, can be used to characterise a plume's expected settling regime. In all plume regimes, the intrusion fluid is observed to rise in the environment following the sedimentation of particles and a simple model for the change in intrusion fluid height has been developed using the steady-state particle concentration at the spreading level.

In this paper, we propose a series expansion of the baroclinic torque in low-Mach-number flows, so that the accuracy and universality of any buoyancy term could be examined analytically, and new types of buoyancy terms could be constructed and validated. We first demonstrate that the purpose of introducing a buoyancy term is to approximate the baroclinic torque, and straightforwardly the error of any buoyancy term could be defined with the deviation of its curl from the corresponding baroclinic torque. Then a regular perturbation method is introduced for the elliptic equation of the hydrodynamic pressure in low-Mach-number flows, resulting in a sequence of Poisson equations, whose solutions lead to the series representation of the baroclinic torque and the new types of buoyancy terms. It is found that the frame invariance of the momentum equation is maintained with one of the new types of buoyancy terms. With the error definition of buoyancy terms and the series representation of the baroclinic torque, the validity and accuracy of previous and new buoyancy terms are examined. Finally, numerical simulations confirm that, with a decreasing density variation or an increasing order of our new buoyancy term, the simplified equations can converge to the original low-Mach-number equations.

Particulate flows at moderate particle Reynolds numbers are important in critical engineering and geological applications. This experimental study explores neutrally buoyant suspensions in an outer-rotating coaxial rheometer for solid fractions, $\phi$, from 0.1 to 0.5, and particle Reynolds number, $Re$, from 0.5 to 800, covering laminar, transitional and turbulent regimes; $Re$ is defined in terms of the square of the particle diameter and the shear rate. For $0.1 < \phi < 0.4$ and $0.5 < Re <10$, the direct torque measurements normalised by the laminar flow torque, $M/M_{lam}$, are independent of $Re$, but depend on $\phi$. For the same range of $\phi$ and for $10< Re<100$, the normalised torques depend on both $\phi$ and $Re$, and show an increasing dependence on $Re$. As $Re$ increases, the flow transitions to turbulence. Small particles delay the turbulent transition for $\phi \leqslant 0.3$, while large particles augment the transition. A modified Reynolds number, $Re^\prime$, that depends linearly on the particle diameter and the maximum velocity, $U_{o}$, is introduced for both laminar and turbulent flows and shows a better correlation of the results as compared with $Re$. For $\phi = 50\,\%$, the normalised torque minus the torque at zero rotational speed is nearly independent of $Re^\prime$. Rheological models based on $Re^\prime$ and the Krieger–Dougherty relative viscosity are proposed in the laminar regime for $10< Re^\prime <500$; in the turbulent regime, a correlation is proposed in terms of $Re^\prime$ and $\phi$ for $1000< Re^\prime < 6000$.

The composition of the Earth's mantle, built by impacts of metallic core planetesimals, depends on the time offered to metal–silicate equilibration (concentration equalization by mixing) as the impactor fragments settle down. Smaller fragments equilibrate while bigger ones remain segregated and accumulate in the – therefore iron rich – Earth's core. To understand the primary instability setting the interpenetration depth and wavelength between the impactor and the mantle, Lherm et al. (J. Fluid Mech., vol. 937, 2022, A20) analyse the intermingled phenomena occurring when a drop impacts a substrate of a pool of another liquid, digging a crater. The density difference between the phases triggers a Rayleigh–Taylor instability when the drop is denser than the pool. The overall planet chemical composition relies on a tradeoff between fragmentation and mixing; this study provides elements to decipher their relative importance.