This paper presents a theoretical model for the electro-osmotic flow (EOF) of semi-dilute polyelectrolyte (PE) solutions in nanochannels. We use mean-field theories to describe the properties of electric double layer and viscosity of PE solutions that are prerequisites for constructing the EOF model. The EOF model is validated via a good match to the existing experimental results. Based on the validated EOF model, we conduct a comprehensive analysis of EOF of semi-dilute PE solutions in nanochannels. First, we observe considerable EOF of PE solutions in the uncharged nanochannels, which is in stark contrast to EOF of simple electrolyte solutions. The analyses show that the EOF of PE solutions in uncharged nanochannels is triggered by the external electric field acting on the near-wall non-electroneutral regions resulting from the confinement-induced inhomogeneous distribution of PE monomers. Although the solutions are electroneutral as a whole, the presence of local non-electroneutral regions and the mismatch between non-electroneutral regions and high-viscosity regions lead to the net EOF in uncharged nanochannels. Furthermore, we reveal that the EOF mobility $\mu _{{eof}}$ in uncharged nanochannels exhibits a scaling law $\mu _{{eof}} \propto a^{-0.44}$ (wherein $a$ denotes monomer Kuhn length) and is inversely proportional to the PE chain length, while it decreases nonlinearly with the charge fraction of the PE chains. Moreover, the EOF mobility reaches its maximum at specific bulk monomer concentration, and increases with the nanochannel height before converging to that under no confinement. Second, we analyse the EOF of PE solutions in nanochannels with various wall effects, such as surface charge density, slip length and adsorption length. When the surface charge is absent, the adsorption length significantly influences the direction and magnitude of the EOF, whereas the slip length has no effect. When the wall becomes increasingly charged, the influence of adsorption length on EOF gradually diminishes, while the importance of the slip length progressively intensifies and the EOF is highly influenced by the co-action of various wall effects in a complicated manner. When the surface wall is oppositely charged to polymer monomers, the EOF mobility varies nonlinearly with the surface charge density, while a zero net flow of EOF followed by a direction reversal is discovered when the wall is likely charged to polymer monomers.

The settling of highly elastic non-Brownian closed fibres (called loops) under gravity in a viscous fluid is investigated numerically. The loops are represented using a bead–spring model with harmonic bending potential and finitely extensible nonlinear elastic stretching potential. Numerical solutions to the Stokes equations are obtained with the use of HYDROMULTIPOLE numerical codes, which are based on the multipole method corrected for lubrication to calculate hydrodynamic interactions between spherical particles with high precision. Depending on the elasto-gravitation number $B$, a ratio of gravitation to bending forces, the loop approaches different attracting dynamical modes, as described by Gruziel-Słomka et al. (2019 Soft Matt. 15, 7262–7274) with the use of the Rotne–Prager mobility of the elastic loop made of beads. Here, using a more precise method, we find and characterise a new mode, analyse typical time scales, velocities and orientations of all the modes, compare them and investigate their coexistence. We analyse numerically the transitions (bifurcations) to a different mode at certain critical values of the elasto-gravitation number.

The settling dynamics of finite size, slightly heavier-than-fluid Kolmogorov-scale particles in homogeneous, isotropic turbulence at moderate volume loadings is investigated. A thoroughly validated two-way-coupled, point-particle model based on the complete Maxey–Riley–Gatignol equation of motion is used with closure models for all forces, including the history force, together with corrections for the self-disturbance field created by the particle using a novel zonal-advection-diffusion-reaction method. Settling dynamics is investigated by varying turbulence intensities relative to the particle settling speed in quiescent flow for multiple Stokes numbers. The length scales associated with the turbulence structures that strongly interact with and influence the settling dynamics are investigated using multiscale statistical analysis of the fluid velocity and second invariant of the velocity gradient tensors sampled by the particles. The time scales are investigated using trajectory curvature angle statistics of inertial and fluid particles. Low-to-moderate Stokes number particles tend to sample strain-rate dominated regions of the flow, tend to follow the curvature of the flow paths and show enhanced settling at higher turbulence intensities due to fast tracking and preferential sampling. Higher Stokes number particles, on the other hand, have a tendency to travel in straight lines relative to the flow and result in reduced settling speeds due to loitering. For the low mass loadings considered in this work, there is minimal global effect on the turbulent flow characteristics; however, it is found that the Kolmogorov-scale particles interact with and locally modify flow structures approximately twice their size, whereas they sample flow velocities from scales up to ten times the particle size, influencing preferential sampling and settling characteristics.

Capsules are widely used in bioengineering, chemical engineering and industry. The development of drug delivery systems using deformable capsules is progressing, yet the regulation of drug release within a capsule remains a challenge. Meanwhile, a microswimmer enclosed in a capsule can generate a large lubrication force on the capsule membrane, which could result in deformation and mechanical damage to the membrane. In this study, we numerically investigate how a capsule can be damaged by an enclosed microswimmer. The capsule membrane is modelled as a two-dimensional neo-Hookean material, with its deformability parametrised by capillary number. An isotropic brittle damage model is applied to express membrane rupture, with the Lighthill–Blake squirmer serving as the microswimmer model. In a sufficiently small capillary number regime, pusher-type squirmers exhibit stable swimming along the capsule membrane, while neutral-type and puller-type squirmers exhibit swimming towards the membrane and remain stationary. As capillary number increases, the damage to the membrane increases and rupture occurs in all swimming modes. For pusher-type squirmers, the critical capillary number leading to rupture is dependent on the initial incidence angle, whereas neutral-type and puller-type squirmers are independent of the initial value. Furthermore, we present methods for controlling membrane damage by magnetically orienting the microswimmer. The findings reveal that a static magnetic field can orient the microswimmer, leading to membrane damage and rupture even for a capsule that cannot be damaged by free swimming, while controlling the swimming path with a rotating magnetic field enables soft membranes to maintain deformation without rupture.

The flow-induced oscillations of a clamped flexible ring in a uniform flow were explored using the penalty immersed boundary method. Both inverted and conventional ring configurations were examined, with systematic analysis focused on the effects of bending rigidity and eccentricity. Four distinct oscillation modes were identified across parameter variations: flapping (F), deflected oscillation (DO), transverse oscillation (TO) and equilibrium (E) modes. Each mode exhibited a 2S wake pattern. The inverted ring sustained the DO mode under low bending rigidity with a deflected shape, transitioning to the TO mode at higher bending rigidity. In the TO mode, a lock-in phenomenon emerged, enabling the inverted ring to achieve a high power coefficient due to a simultaneous rise in both oscillation amplitude and frequency. By contrast, the conventional ring exhibited the F mode at low bending rigidity and transitioned to the E mode as rigidity increased, although its power coefficient remained lower because of reduced critical bending rigidity. For the inverted ring, low eccentricity enhanced oscillation intensity but limited the operational range of the TO mode. In contrast, for the conventional ring, reducing eccentricity led to an increase in oscillation amplitude. Among the investigated configurations, the inverted-clamped ring achieved the highest energy-harvesting efficiency, surpassing those of the conventional clamped ring and a buckled filament.

The work investigates the response dynamics of non-premixed jet flames to blast waves that are incident along the jet axis. In the present study, blast waves, generated using the wire-explosion technique, are forced to sweep across a non-premixed jet flame that is stabilised over a nozzle rim positioned at a distance of 264 mm from the source of the blast waves. The work spans a wide range of fuel-jet Reynolds numbers ($Re$; ranging from 267 to 800) and incident blast-wave Mach numbers ($M_{s,r}$; ranging from 1.025 to 1.075). The interaction imposes a characteristic flow field over the jet flame marked by a sharp discontinuity followed by a decaying profile and a delayed second spike. The second spike in the flow field profile corresponds to the induced flow that follows the blast front. While the response of the flame to the blast front was minimal, it was found to detach from the nozzle rim and lift off following the interaction with the induced flow. Subsequently, the lifted flame was found to reattach back at the nozzle or extinguish, contingent on the operating $Re$ and $M_{s,r}$. Alongside flame lift-off, flame-tip flickering was aggravated under the influence of the induced flow. A simplified theoretical model extending the vorticity transport equation was developed to estimate the change in flickering time scales and length scales owing to the interaction with the induced flow. The observed experimental trends were further compared against theoretical predictions from the model.

The nonlinear disturbance caused by either a localised pressure distribution moving at constant speed on the free surface of a liquid of finite depth or a flow over a topographic obstacle, is investigated using (i) the weakly nonlinear forced Kadomtsev–Petviashvili equation which is valid for depth-based Froude numbers near unity and (ii) the fully nonlinear free-surface Euler system. The presence of a steady v-shaped Kelvin wave pattern downstream of the forcing is established for this model equation, and the wedge angle is characterised as a function of the depth-based Froude number. Inspired by this analysis, it is shown that the wake can be eliminated via a careful choice of the forcing distribution and that, significantly, the corresponding nonlinear wave-free solution is stable so that it could potentially be seen in a physical experiment. The stability is demonstrated via the numerical solution of an initial value problem for both the model equation and the fully nonlinear Euler system in which the steady wave-free state is attained in the long-time limit.

The interaction between elastic structures and fluid interfaces, known as ‘hydroelastic’ problems, presents unique challenges to classical frameworks established for rigid spheres and liquid droplets. In this work, we experimentally demonstrate an intriguing phenomenon where ultrasoft hydrogel spheres rebound from a water surface at high impact speeds, even when their density exceeds that of water. We further propose a theoretical force-balance model, incorporating energy redistribution and potential flow theory, to predict the critical impact speed for the transition from sinking to rebounding, as well as the temporal evolution of both spreading diameter and cavity expansion. Our findings extend the classical Weber- and Bond-number-dominated paradigms for rigid spheres and liquid droplets, demonstrating that hydrogel dynamics is controlled by a modified elastocapillary Mach number, with rebound achievable even for hydrophilic spheres. These findings improve the understanding of soft-impact hydrodynamics and offer design principles for applications in biomimetic robotics and energy-absorbing materials.

We investigate the dynamics of a pair of rigid rotating helices in a viscous fluid, as a model for bacterial flagellar bundle and a prototype of microfluidic pumps. Combining experiments with hydrodynamic modelling, we examine how spacing and phase difference between the two helices affect their torque, flow field and fluid transport capacity at low Reynolds numbers. Hydrodynamic coupling reduces the torque when the helices rotate in phase at constant angular speed, but increases the torque when they rotate out of phase. We identify a critical phase difference, at which the hydrodynamic coupling vanishes despite the close spacing between the helices. A simple model, based on the flow characteristics and positioning of a single helix, is constructed, which quantitatively predicts the torque of the helical pair in both unbounded and confined systems. Finally, we show the influence of spacing and phase difference on the axial flux and the pump efficiency of the helices. Our findings shed light on the function of bacterial flagella and provide design principles for efficient low-Reynolds-number pumps.

The investigation of shock/blast wave diffraction over various objects has garnered significant attention in recent decades on account of the catastrophic changes that these waves inflict on the environment. Equally important flow phenomena can occur when the moving expansion waves diffract over bodies, which has been hardly investigated. To investigate the effect of expansion wave diffraction over different bodies, we conducted shock tube experiments and numerical simulations to visualise the intricate wave interactions that occur during this process. The current investigation focuses on the phenomenon of expansion wave diffraction across three distinct diffracting configurations, namely the bluff, wedge and ogive bodies. The diffraction phenomenon is subsequently investigated under varying expansion wave strengths through the control of the initial diaphragm rupture pressure ratios. The shock waves generated by the expansion wave diffraction in the driver side of the shock tube, which was initially identified in numerical simulations by Mahomed & Skews (2014 J. Fluid Mech., vol. 757, pp. 649–664), have been visualised in the experiments. Interesting flow features, such as unsteady shock generation, transition, and symmetric/asymmetric vortex breakdown, have been observed in these expansion flows. An in-depth analysis of such intricate flow features resulting from expansion wave diffraction is performed and characterised in the current study.

Mixing-induced reactions play an important role in a wide range of porous media processes. Recent advances have shown that fluid flow through porous media leads to chaotic advection at the pore scale. However, how this impacts Darcy-scale reaction rates is unknown. Here, we measure the reaction rates in steady mixing fronts using a chemiluminescence reaction in index-matched three-dimensional porous media. We consider two common mixing scenarios for reacting species, flowing either in parallel in a uniform flow or towards each other in a converging flow. We study the reactive properties of these fronts for a range of Péclet numbers. In both scenarios, we find that the reaction rates significantly depart from the prediction of hydrodynamic dispersion models, which obey different scaling laws. We attribute this departure to incomplete mixing effects at the pore scale, and propose a mechanistic model describing the pore-scale deformations of the front triggered by chaotic advection and their impact on the reaction kinetics. The model shows good agreement with the effective Darcy-scale reaction kinetics observed in both uniform and converging flows, opening new perspectives for upscaling reactive transport in porous media.

We present new unconstrained simulations and constrained experiments of a pair of pitching hydrofoils in a leader–follower in-line arrangement. Free-swimming simulations with matched pitching amplitudes show self-organisation into stable formations at a constant gap distance without any control. Over a wide range of phase synchronisation, amplitude and Lighthill number typical of biology, we discover that the stable gap distance scales with the actual wake wavelength of an isolated foil rather than the nominal wake wavelength. A scaling law for the actual wake wavelength is derived and shown to collapse data across a wide Reynolds number range of $200 \leqslant Re \leqslant 59\,000$. Additionally, vortex analysis uncovers that the leader’s wake wavelength-to-chord ratio, $\lambda /c$, is the key dimensionless variable to maximise the follower’s/collective efficiency. When $\lambda /c \approx 2$ it ensures that the follower’s leading edge suction force and the net force from a nearby vortex pair act in the direction with the foil’s motion thereby reducing the follower’s power. Moreover, in both simulations and experiments mismatched foil amplitudes are discovered to increase the efficiency of hydrofoil schools by 70 % while maintaining a stable formation without closed-loop control. This occurs by (i) increasing the stable gap distance between foils to push them into a high-efficiency zone and (ii) raising the level of efficiency in these zones. This study bridges the gap between constrained and unconstrained studies of in-line schooling by showing that constrained-foil measurements can map out the potential efficiency benefits of schooling. These findings can aid in the design of high-efficiency biorobot schools.

Aerodynamic loads play a central role in many fluid dynamics applications, and we present a method for identifying the structures (or modes) in a flow that make dominant contributions to the time-varying aerodynamic loads in a flow. The method results from the combination of the force partitioning method (Menon & Mittal, 2021, J. Fluid Mech., vol. 907, A37) and modal decomposition techniques such as Reynolds decomposition, triple decomposition and proper orthogonal decomposition, and is applied here to three distinct flows – two-dimensional flows past a circular cylinder and an aerofoil, and the three-dimensional flow over a revolving rectangular wing. We show that the force partitioning method applied to modal decomposition of velocity fields results in complex, and difficult to interpret inter-modal interactions. We therefore propose and apply modal decomposition directly to the $Q$-field associated with these flows. The variable $Q$ is a nonlinear observable that is typically used to identify vortices in a flow, and we find that the direct decomposition of $Q$ leads to results that are more amenable to interpretation. We also demonstrate that this modal force partitioning can be extended to provide insights into the far-field aeroacoustic loading noise of these flows.

In this study, we explore the evolution of instabilities in magneto-quasi-geostrophic (MQG) modons on the $f$-plane using a magnetohydrodynamic rotating shallow water model. The numerical experiments have been conducted using a recently proposed second-order flux-globalisation-based path-conservative central-upwind scheme. Our focus is on the evolution and interactions of three key configurations: singular, regular and hollow MQG modons, which represent cases where the magnetic field is confined within the separatrix, evenly distributed inside and outside the separatrix and localised outside the separatrix, respectively. The singular MQG modon emerges as the most stable configuration, demonstrating the greatest resilience to destabilising forces. A notable observation is its transition from a quadrupolar to a tripolar magnetic field structure before reverting to a quadrupole adjusted magnetic modon, accompanied by a clockwise rotation of the system. In terms of stability, singular modons are the most stable ones, while hollow modons are the least stable. As instabilities develop, southward or northward displacements become significantly more pronounced than eastward or westward movements, primarily due to the Coriolis force. Among the configurations, the hollow (singular) modons experience the biggest (smallest) displacements. Additionally, we investigate modon collisions and highlight three scenarios: interactions between cyclonic and anticyclonic components that form a composite modon with meridional bifurcation; collisions of cyclonic vortices that produce a tripolar structure with counterclockwise rotation; and collisions between anticyclonic components that result in a stable, quasi-stationary tripolar configuration. The resulting magnetic poles exhibit a checkered pattern, with their amplitude decreasing with increasing distance from the central vortex.

Viscous flow through high-permeability channels occurs in many environmental and industrial applications, including carbon sequestration, groundwater flow and enhanced oil recovery. In this work, we study the displacement of a less-viscous fluid by a more-viscous fluid in a layered porous medium in a rectilinear configuration, where two low-permeability layers sandwich a higher-permeability layer. We derive a theoretical model that is validated using corroborative laboratory experiments, when the influence of the density difference is negligible. We find that the location of the propagating front increases with time according to a power-law form $x_f \propto t^{1/2}$, while the fluid–fluid interface exhibits a self-similar shape, when the motion of the displaced fluid is negligible in an unconfined porous medium. In the experimental set-up, distinct permeability layers were constructed using various sizes of spherical glass beads. The working fluids comprised fresh water as the less-viscous ambient fluid, and a glycerine–water mixture as the more-viscous injecting fluid. Our experimental measurement show a better match with the theory for the experiments performed at low Reynolds numbers and with permeable boundaries in the far field.

We study the homogeneous isotropic turbulence of a shear-thinning fluid modelled by the Carreau model, and show how the variable viscosity affects the multiscale behaviour of the turbulent flow. We show that Kolmogorov theory can be extended to such non-Newtonian fluids, provided that the correct choice of average is taken when defining the mean Kolmogorov scale and dissipation rate, to properly capture the effect of the variable viscosity. Thus the classical phenomenology à la Kolmogorov can be observed in the inertial range of scale, with the energy spectra decaying as $k^{-5/3}$, with $k$ being the wavenumber, and the third-order structure function obeying the $4/5$ law. The changing viscosity instead strongly alters the small scale of turbulence, leading to an enhanced intermittent behaviour of the velocity field.

A nonlinear Schrödinger equation for pure capillary waves propagating at the free surface of a vertically sheared current has been used to study the stability and bifurcation of capillary Stokes waves on arbitrary depth. A linear stability analysis of weakly nonlinear capillary Stokes waves on arbitrary depth has shown that (i) the growth rate of modulational instability increases as the vorticity decreases whatever the dispersive parameter $kh$, where $k$ is the carrier wavenumber and $h$ the depth; (ii) the growth rate is significantly amplified for shallow water depths; and (iii) the instability bandwidth widens as the vorticity decreases. Particular attention has been paid to damping due to viscosity and forcing effects on modulational instability. In addition, a linear stability analysis to transverse perturbations in deep water has been carried out, demonstrating that the dominant modulational instability is two-dimensional whatever the vorticity. Near the minimum of linear phase velocity in deep water, we have shown that generalised capillary solitary waves bifurcate from linear capillary Stokes waves when the vorticity is positive. Moreover, we have shown that the envelope of pure capillary waves in deep water is unstable to transverse perturbations. Consequently, deep-water generalised capillary solitary waves are expected to be unstable to transverse perturbations.

Görtler vortices induced by concave curvature in supersonic turbulent flows are investigated using resolvent analysis and large-eddy simulations at Mach 2.95 and Reynolds number $ Re_{\delta }=63\,500$ based on the boundary-layer thickness $ \delta$. Resolvent analysis reveals that the most amplified coherent structures manifest as streamwise counter-rotating vortices with optimal spanwise wavelength $ 2.4\delta$ at cut-off frequency $f\delta /{u}_{\infty } =0.036$, where $ {u}_{\infty }$ is the freestream velocity. The leading spectral proper orthogonal decomposition modes with spanwise wavelength approximately $ 2\delta$ align well with the predicted coherent structures from resolvent analysis at $f\delta /{u}_{\infty } =0.036$. These predicted and extracted coherent structures are identified as Görtler vortices, driven by the Görtler instability. The preferential spanwise scale of the Görtler vortices is further examined under varying geometric and freestream parameters. The optimal spanwise wavelength is insensitive to the total turning angle beyond a critical value, but sensitive to the concave curvature $ K$ at the same turning angle. A limit spanwise wavelength $ 1.96\delta$, corresponding to an infinite concave curvature as $ K\rightarrow \infty$, is identified and validated. Increasing the freestream Mach number or decreasing the ratio of wall temperature to freestream temperature reduces the optimal wavelength normalised by $ \delta$, while variations in freestream Reynolds number have negligible impact. Additionally, a modified definition of the turbulent Görtler number $ G_{T}$ based on the peak eddy viscosity in boundary layers is proposed and employed to assess the occurrence of Görtler instability.

The supersonic wake of a circular cylinder in Mach 3 flow was studied through spectral proper orthogonal decomposition (SPOD) of high-speed focussing schlieren datasets. A wavenumber decomposition of the SPOD eigenvectors was found to be an effective tool for isolating imaging artefacts from the flow features, resulting in a clearer interpretation of the SPOD modes. The cylinder wake consists of both symmetric and antisymmetric instabilities, with the former being the dominant type. The free shear layers that form after the flow separates from the cylinder surface radiate strong Mach waves that interact with the recompression shocks to release significant disturbances into the wake. The wake shows a bimodal vortex shedding behaviour with a purely hydrodynamic instability mode around a Strouhal number of 0.2 and an aeroacoustic instability mode around Strouhal number of 0.42. The hydrodynamic mode, which is presumably the same as the incompressible case, is weaker and decays rapidly as the wake accelerates due to increasing compressibility. The aeroacoustic mode is the dominant shedding mode and persists farther into the wake because of an indirect energy input received through free-stream acoustic waves. A simple aeroacoustic feedback model based on an interaction between downstream propagating shear-layer instabilities and upstream propagating acoustic waves within the recirculation region is shown to accurately predict the shedding frequency. Based on this model, the vortex shedding in supersonic flows over a circular cylinder occurs at a universal Strouhal number (based on approach free-stream velocity and feedback path length) of approximately 0.3.