We study how channel width variations influence the dynamics of free-surface granular flows. For this purpose, we extend a continuum model framework to granular flows passing through channels that narrow or widen. Our theory uses a linearized approximation to an established dense granular flow rheology and a Coulomb friction law to model interaction between flow and sidewalls. We test the theoretical predictions using two novel 40 cm-diameter drums (convex and concave) filled halfway with 2 mm diameter particles rotated at rates in which the shear layer remains shallow and dense. We apply particle tracking velocimetry to enable quantitative comparisons between experimental data and theoretical predictions. We find that our experimental kinematics and energy profiles largely agree with the theoretical predictions. In general, flows through narrowing channels are faster and deeper than flows through widening channels. The influence of width variations grows with increasing flow speed, and the form of the rate dependence changes fundamentally as the regime changes from one in which kinetic energy is dissipated locally to one in which it is advected downstream. For both regimes, theoretical scaling analysis leads us to experimentally validated power laws, in which the exponent depends on the flow regime, and the multiplicative coefficient depends on channel geometry alone. Finally, we discuss how the differences between theoretical predictions and experimental data may be useful for improving our understanding of flows through non-uniform channels.

Shock-tube experiments on various two-bubble and two-spike interfaces are performed to examine the dependence of bubble competition and spike competition on the initial spectra and density ratio of the interface. The differences in the influences of bubble competition and spike competition on the Richtmyer–Meshkov instability are highlighted for the first time. The bubble-competition effect is mainly dependent on the initial spectra of the two-bubble configuration. In contrast, the spike-competition effect is determined by both the initial spectra and density ratio. The extended buoyancy–drag model is adopted to explain the variation of the drag force imposed on the long-wavelength and short-wavelength structures as the initial conditions change. Based on the spectrum analysis, it is found that the constituent modes of two-bubble and two-spike interfaces have different contributions to the long-wavelength and short-wavelength perturbation growths. A generalised, nonlinear, analytical model is then established to quantify the bubble-competition effect and spike-competition effect considering arbitrary initial spectra and density ratio. The bubble-competition effect is believed to be stronger than the spike-competition effect at a high density ratio because it suppresses the high-frequency perturbation growth more evidently.

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.

We present a theoretical study of viscous slug motion inside a microscopically rough capillary tube, where pronounced stick–slip motion can emerge at slow displacement rates. The mathematical description of this intermittent motion can be reduced to a system of ordinary differential equations, which also describe the motion of a pendulum inside a fluid-filled rotating drum. We use this analogy to show that the stick–slip motion transitions to steady sliding at high displacement rates. We characterize this crossover with a simple scaling relation and show that the crossover is accompanied by a shift in the dominant energy dissipation mechanisms within the system.

Motivated by contradicting or insufficient information regarding the large-scale flow dynamics around surface-mounted finite-height square prisms of small aspect ratio, the present study investigates the dominant vortex shedding and low-frequency dynamics around a surface-mounted cube. These flow modes were obtained from the spectral proper orthogonal decomposition of large-eddy simulation results, at a Reynolds number of $\textit {Re}=1\times 10^4$ and two different types of boundary layer: a thin and laminar boundary layer with thickness $\delta /D=0.2$ and a thick and turbulent boundary layer with $\delta /D=0.8$. The main antisymmetric mode pair revealed a new flow pattern with the alternate shedding of streamwise flow structures, indicating a transition from the half-loops of taller prisms to only streamwise strands (i.e. no vertical core) for smaller aspect ratio. The formation process of the streamwise structures is due to a reorientation of the vorticity of the arch vortex in the streamwise direction characteristic of the shed structures. The low-frequency drift mode affected the length of the recirculation region, the strength of vortex shedding, and the near-wall flow field and pressure distribution on the cube's faces, leading to low-frequency variations in the fluctuating drag and normal force coefficients. These large-scale flow dynamics were similar for both boundary layers, but minor differences were identified, related mostly to the occurrence of flow attachment and the formation of a headband vortex for the thicker boundary layer.

Surfactant-like impurities are omnipresent in multiphase emulsions and may substantially affect the motion of small droplets by altering their interfacial properties. Usually these surfactants are soluble in the bulk and undergo adsorption–desorption onto the interface which modifies their surface concentration and hence their overall influence on droplet motion. Yet, the impact of the bulk solubility and transport of surfactants on droplet dynamics, especially in the presence of bounding walls, remains poorly understood. As such, in this article, we assess the impact of bulk soluble surfactants on the settling of a spherical drop towards a plane wall. We consider coupled bulk and interfacial transport of surfactants, mediated by adsorption–desorption processes and construct a semi-analytical framework for arbitrary values of ‘bulk interaction parameter’, which dictates the strength of adsorption–desorption kinetics compared with bulk diffusion. Our results indicate that while mass exchange between the bulk and the interface can remobilize the drop, a finite bulk diffusion rate restricts this process and therefore slows down the drop. This also results in bulk concentration depletion near the south pole and accumulation near the north pole, the extent of which becomes strongly asymmetric with an enhanced intensity of depletion, as the drop approaches the wall. Presence of the wall and bulk solubility are found to aid each other towards remobilizing the drop by aptly modifying the interfacial concentration. Our results may provide fundamental insights into the kinetics of surfactant-laden drops, with potential applications in food and pharmaceutical industries, separation processes, etc.

The effects of thermal convection on turbulence in accretion discs, and particularly its interplay with the magnetorotational instability (MRI), are of significant astrophysical interest. Despite extensive theoretical and numerical studies, such an interplay has not been explored experimentally. We conduct linear analysis of the azimuthal version of MRI (AMRI) in the presence of thermal convection and compare the results with our experimental data published before. We show that the critical Hartmann number ($Ha$) for the onset of AMRI is reduced by convection. Importantly, convection breaks symmetry between $m = \pm 1$ instability modes ($m$ is the azimuthal wavenumber). This preference for one mode over the other makes the AMRI wave appear as a ‘one-winged butterfly’.

This study reports the first time-resolved particle image velocimetry characterization of a planar two-phase mixing layer flow, whose velocity field is measured simultaneously in gas and liquid streams. Two parallel air and water flows meet downstream of a splitter plate, giving rise to an initially spanwise invariant configuration. The aim is to elucidate further the mechanisms leading to the flow breakup in gas-assisted atomization. The complete experimental characterization of the velocity field represents a database that could be used in data-driven reduced-order models to investigate the global behaviour of the flow system. After the analysis of a selected reference case, a parametric study of the flow behaviour is performed by varying the liquid ($Re_l$) and gas ($Re_g$) Reynolds numbers, and as a consequence also the gas-to-liquid dynamic pressure ratio ($M$), shedding light on both time-averaged (mean) and unsteady velocity fields. In the reference case, it is shown that the mean flow exhibits a wake region just downstream of the splitter plate, followed by the development of a mixing layer. By increasing both $Re_l$ and $Re_g$, the streamwise extent of the wake decreases and eventually vanishes, the flow resulting in a pure mixing layer regime. The spectral analysis of the normal-to-flow velocity fluctuations outlines different flow regimes by variation of the governing parameters, giving more insights into the global characteristics of the flow field. As a major result, it is found that at high $Re_g$ and $M$ values, the velocity fluctuations are characterized by low-frequency temporal oscillations synchronized in several locations within the flow field, which suggest the presence of a global mode of instability. The proper orthogonal decomposition of velocity fluctuations, performed in both gas and liquid phases, reveals finally that the synchronized oscillations are associated with a low-frequency dominant flapping mode of the gas–liquid interface. Higher-order modes correspond to interfacial wave structures travelling with the so-called Dimotakis velocity. For lower gas Reynolds numbers, the leading modes describe higher frequency fingers shedding at the interface.

A hydrodynamic theory of premixed flame propagation within closed vessels is developed assuming the flame is much thinner than all other fluid dynamic lengths. In this limit, the flame is confined to a surface separating the unburned mixture from burned combustion products, and propagates at a speed determined from the analysis of its internal structure. Unlike freely propagating flames that propagate under nearly isobaric conditions, combustion in a closed vessel results in continuous increases in pressure, burning rate and flame temperature, and a progressive decrease in flame thickness. The flame speed is shown to depend on the voluminal stretch rate, which measures the deformation of a volume element of the flame zone, and on the rate of pressure rise. Both effects are modulated by pressure-dependent Markstein numbers that depend on heat release and mixture properties while capturing the effects of temperature-dependent transport and stoichiometry. The model applies to flames of arbitrary shape propagating in general flows, laminar or turbulent, within vessels of general configurations. The main limitation of hydrodynamic flame theories is the assumption that variations inside the flame zone due to chemistry or turbulence, which could potentially alter its internal structure, are physically unresolved. Nonetheless, the theory, deduced from physical first principles, identifies the various mechanisms involved in the combustion process as demonstrated in detailed discussions of planar flames propagating in rectangular channels and spherically expanding flames in spherical vessels. It also enables the construction of instructive models to numerically simulate the evolution of multi-dimensional and corrugated flames under confinement.

Flow between axially rotating concentric cylinders is well known to exhibit rich dynamics. Hence, Taylor instabilities have been studied, both experimentally and theoretically, for many years. Although usually studied in the abstract, such geometries arise in a range of practical situations including drilling, when a drilling fluid flow enters a well via a pipe that is the centre body and returns via the annulus between the pipe and the borehole wall. In drilling, the centre body rotates and the annular flow contains rock cuttings. Here, we report the development of an Eulerian-Eulerian solver, based on OpenFOAM, that solves for this cuttings transport problem in the presence of both gravity and Taylor vortices. To check the reliability of the solver, we conduct a set of experiments spanning a wide range of complex flow regimes. We show that the model successfully predicts, in all regimes, the observed complex redistribution of particulates. However, for suspension flows under viscously dominated conditions, high particle concentrations and in rectilinear flow, particle pressure and normal stress differences are sufficient to capture particle migration. Results show that in more complex flows exemplified by the Taylor–Couette flow studied here, more realistic predictions of non-Brownian particle migration require inclusion of forces arising through the relative velocity of the two phases including lift forces originating both from inertia and particle rotation.

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.

The interaction between near-wall turbulence and wall curvature is described for the incompressible flow in a plane channel with a small concave–convex–concave bump on the bottom wall, with height comparable to the wall-normal location of the main turbulent structures. The analysis starts from a database generated by a direct numerical simulation and hinges upon the anisotropic generalised Kolmogorov equations, i.e. the exact budget equations for the second-order structure function tensor. The influence of the bump on the wall cycle and on the energy production, redistribution and transfers is described in the physical and scale spaces. Over the upstream side of the bump, the energy drained from the mean flow to sustain the streamwise fluctuations decreases, and the streaks of high and low streamwise velocity weaken and are stretched spanwise. After the bump tip, instead, the production of streamwise fluctuations grows and the streaks intensify, progressively recovering their characteristic spanwise scale. The wall-normal fluctuations, and thus the quasi-streamwise vortices, are sustained by the mean flow over the upstream side of the bump, while energy flows from the vertical fluctuations to the mean field over the downstream side. On the concave portion of the upstream side, the near-wall fluctuations form structures of spanwise velocity which are consistent with Taylor–Görtler vortices at an early stage of development. Their evolution is described by analysing the scale-space pressure–strain term. A schematic description of the bump flow is presented, in which various regions are identified according to the signs of curvature and streamwise pressure gradient.

We investigate jumping of sessile droplets from a solid surface in ambient oil using modulated electrowetting actuation. We focus on the case in which the electrowetting effect is activated to cause droplet spreading and then deactivated exactly at the moment the droplet reaches its maximum deformation. By systematically varying the control parameters such as the droplet radius, liquid viscosity and applied voltage, we provide detailed characterisation of the resulting behaviours including a comprehensive phase diagram separating detachment from non-detachment behaviours, as well as how the detach velocity and detach time, i.e. duration leading to detachment, depend on the control parameters. We then construct a theoretical model predicting the detachment condition using energy conservation principles. We finally validate our theoretical analysis by experimental data obtained in the explored ranges of the control parameters.

Direct numerical simulations are performed to explore the evolution behaviour of the turbulent/non-turbulent interface (TNTI) in a temporally evolving turbulent plane jet, using the evolution equation for the TNTI surface area. A novel algorithm is used to calculate the surface area of the TNTI and entrainment flux. It is shown that the surface area remains relatively constant, which leads to the mean entrainment velocity being inversely proportional to the square root of time. On average, the effects of the stretching and curvature/viscous terms on the TNTI area roughly counterbalance each other, while the curvature/inviscid term associated with vortex stretching is virtually zero. More specifically, the stretching term contributes to the production of the surface area, while the curvature/viscous term is associated with a destruction in the surface area. The local effect of the curvature/viscous term exhibits high spatial intermittency with small-scale extreme/intense events, whereas the effect of the large-scale stretching term is more continuous. To shed light on the contribution of curvature/viscous term to the evolution of the surface area, we decompose it into three components. The effect of the curvature/normal diffusion term (the curvature/viscous dissipation term) in the bulging regions (the valley regions) mainly contributes to the production of the area. The continuous decrease of the average mean curvature is associated with the production of the bulging regions and the destruction of the valley regions. Finally, although the entrainment velocity is mainly dominated by the normal diffusion effect, all three components related to the viscous effect are indispensable to the production and destruction of the TNTI area. This numerical study contributes to a better understanding of the evolution of the TNTI area.

We studied the transport and deposition behaviour of point particles in Rayleigh–Bénard convection cells subjected to Couette-type wall shear. Direct numerical simulations (DNSs) are performed for Rayleigh number ($Ra$) in the range $10^{7} \leq Ra \leq ~10^9$ with a fixed Prandtl number $Pr = 0.71$, while the wall-shear Reynolds number ($Re_w$) is in the range $0 \leq Re_w \leq ~12\,000$. With the increase of $Re_w$, the large-scale rolls expanded horizontally, evolving into zonal flow in two-dimensional simulations or streamwise-oriented rolls in three-dimensional simulations. We observed that, for particles with a small Stokes number ($St$), they either circulated within the large-scale rolls when buoyancy dominated or drifted near the walls when shear dominated. For medium $St$ particles, pronounced spatial inhomogeneity and preferential concentration were observed regardless of the prevailing flow state. For large $St$ particles, the turbulent flow structure had a minor influence on the particles’ motion; although clustering still occurred, wall shear had a negligible influence compared with that for medium $St$ particles. We then presented the settling curves to quantify the particle deposition ratio on the walls. Our DNS results aligned well with previous theoretical predictions, which state that small $St$ particles settle with an exponential deposition ratio and large $St$ particles settle with a linear deposition ratio. For medium $St$ particles, where complex particle–turbulence interaction emerges, we developed a new model describing the settling process with an initial linear stage followed by a nonlinear stage. Unknown parameters in our model can be determined either by fitting the settling curves or using empirical relations. Compared with DNS results, our model also accurately predicts the average residence time across a wide range of $St$ for various $Re_w$.

Turbulent shear flows driven by a combination of a pressure gradient and buoyancy forcing are investigated using direct numerical simulations. Specifically, we consider the set-up of a differentially heated vertical channel subject to a Poiseuille-like horizontal pressure gradient. We explore the response of the system to its three control parameters: the Grashof number $Gr$, the Prandtl number $Pr$, and the Reynolds number $Re$ of the pressure-driven flow. From these input parameters, the relative strength of buoyancy driving to the pressure gradient can be quantified by the Richardson number $Ri=Gr/Re^2$. We compare the response of the mixed vertical convection configuration to that of mixed Rayleigh–Bénard convection, and find a nearly identical behaviour, including an increase in wall friction at higher $Gr$, and a drop in the heat flux relative to natural convection for $Ri=O(1)$. This closely matched response is despite vastly different flow structures in the systems. No large-scale organisation is visible in visualisations of mixed vertical convection – an observation that is confirmed quantitatively by spectral analysis. This analysis, combined with a statistical description of the wall heat flux, highlights how moderate shear suppresses the growth of small-scale plumes and reduces the likelihood of extreme events in the local wall heat flux. Vice versa, starting from a pure shear flow, the addition of thermal driving enhances the drag due to the emission of thermal plumes.

The global linear stability analysis for the magnetohydrodynamic liquid metal flow past an insulated sphere subjected to a constant streamwise magnetic field is investigated in the range of the Reynolds number $Re\leq 400$ and the interaction number $N\leq 40$ coupled with direct numerical simulations, where $N$ stands for strength of the electromagnetic force. The stability of the steady axisymmetric base flow to independent time-azimuthal modes is discussed. Five critical curves associated with various wake transitions are obtained in the $\{Re, N\}$ phase diagram. These critical curves reveal the stabilising effect of a weak magnetic field, the destabilising effect of a strong magnetic field and re-stabilising effect of a much stronger magnetic field. To explore the impact of the magnetic field on flow instability, a sensitivity analysis utilizing an adjoint method is performed for the first regular bifurcation. Sensitivity functions of growth rate to base-flow modifications and Lorentz force are defined to identify the region that has the most significant influence on flow instability, such as the recirculation region responsible for the stabilising effect at a weak magnetic field and the shear layer region responsible for the destabilising effect at a strong magnetic field. Furthermore, a competition between the stabilising and shear destabilising effects of the magnetic field is discussed. This analysis provides valuable insights into the non-monotonic effect of the magnetic field on flow instability.