Electrowetting on dielectric (EWOD) is a powerful tool in many droplet-manipulation applications with a notorious weakness caused by contact-angle saturation (CAS), a phenomenon limiting the equilibrium contact angle of an EWOD-actuated droplet at high applied voltage. In this paper, we study the spreading behaviours of droplets on EWOD substrates with the range of applied voltage exceeding the saturation limit. We experimentally find that at the initial stage of spreading, the driving force at the contact line still follows the Young–Lippmann law even if the applied voltage is higher than the CAS voltage. We then theoretically establish the relation between the initial contact-line velocity and the applied voltage using the force balance at the contact line. We also find that the amplitude of capillary waves on the droplet surface generated by the contact line's initial motion increases with the applied voltage. We provide a working framework utilising EWOD with voltages beyond CAS by characterising the capillary waves formed on the droplet surface and their self-similar behaviours. We finally propose a theoretical model of the wave profiles taking into account the viscous effects and verify this model experimentally. Our results provide avenues to utilise the EWOD effect with voltages beyond the CAS threshold, and have strong bearing on emerging applications such as digital microfluidic and ink-jet printing.

The control of stationary convective instabilities in the rotating disk boundary layer via a time-periodic modulation of the disk rotation rate is investigated. The configuration provides an archetypal example of a three-dimensional temporally periodic boundary layer, encompassing both the von Kármán and Stokes boundary layers. A velocity–vorticity formulation of the governing perturbation equations is deployed, together with a numerical procedure that utilises the Chebyshev-tau method. Floquet theory is used to determine the linear stability properties of these time-periodic flows. The addition of a time-periodic modulation to the otherwise steady disk rotation rate establishes a stabilising effect. In particular, for a broad range of modulation frequencies, the growth of the stationary convective instabilities is suppressed and the critical Reynolds number for the onset of both the cross-flow and Coriolis instabilities is raised to larger values than that found for the steady disk without modulation. An energy analysis is undertaken, where it is demonstrated that time-periodic modulation induces a reduction in the Reynolds stress energy production and an increase in the viscous dissipation across the boundary layer. Comparisons are made with other control techniques, including distributed surface roughness and compliant walls.

Wind-tunnel experiments have been conducted on cylindrical models with canted fins. The fins introduced a swirling motion into the wake downstream of a blunt-based afterbody aligned with a Mach 2 flow. Measurements of the velocity field downstream of the models and the pressure distribution at the model base show evidence of two wake flow patterns distinctively differing from the classical supersonic wake, depending on the degree of rotation introduced. For a fin-cant angle of 16$^\circ$, a rotating wake flow with a central, downstream-directed vortex tube and a concentric, counter-rotating, toric vortex pair forms. A higher fin-cant angle of 32$^\circ$, in turn, results in a swirling flow surrounding a region of low-momentum flow at the axis. Near the central axis of the flow field an upstream flow establishes, extending from the far wake up to the model base. Numerical simulations have been performed to explain the fluid-dynamic processes and the origins of the experimentally observed structural changes of the rotating wakes. The results of the large-scale-turbulence-resolving simulations agree qualitatively well with the measured flow fields. The numerical results show that the centrifugal forces decrease the base pressure and cause the experimentally observed structural changes in the wake.

Mass-spring models are essential for the description of sloshing resonances in engineering. By experimentally measuring the liquid's centre of mass in a horizontally oscillated rectangular tank, we show that low-amplitude sloshing obeys the Duffing equation. A bending of the response curve in analogy to a softening spring is observed, with growing hysteresis as the driving amplitude increases. At large amplitudes, complex wave patterns emerge (including wave breaking and run up at the tank walls), competition between flow states is observed and the dynamics departs progressively from Duffing. We also provide a quantitative comparison of wave shapes and response curves to the predictions of a multimodal model based on potential flow theory (Faltinsen & Timokha, Sloshing. Cambridge University Press, 2009) and show that it systematically overestimates the sloshing amplitudes and the hysteresis. We find that the phase lag between the liquid's centre of mass and the forcing is the key predictor of the nonlinear response maxima. The phase lag reflects precisely the onset of deviations from Duffing dynamics and – most importantly – at resonance the sloshing motion always lags the driving by $90^{\circ }$ (independently of the wave pattern). This confirms the theoretical $90^{\circ }$-phase-lag criterion (Cenedese & Haller, Proc. R. Soc. A, vol. 476, no. 2234, 2020, 20190494).

As inkjet technology develops to produce smaller droplets, substrate features such as accidental scratches or manufacturing defects can potentially affect the outcome of printing, particularly for printed electronics where continuous tracks are required. Here, the deposition of micro-droplets onto a scratch of commensurate size is studied. The scratch is considered as a groove of rectangular cross-section, with rectangular side ridges representing material displaced from the substrate, and seven equilibrium morphologies are identified as a result of inertial spreading, contact-line pinning, imbibition into the scratch and capillary flow. A regime map is constructed in terms of scratch depth and width, and theoretical estimates of the regime boundaries are developed by adapting droplet spreading laws for flat surfaces to account for liquid entering the scratches. Good agreement is seen with numerical results obtained using a graphical processing unit-accelerated three-dimensional multiphase lattice Boltzmann model validated against published experiments, and the influences of Reynolds number, Weber number and advancing and receding contact angles are explored. Negative and positive implications of the results for printing applications are discussed and illustrated via multiple-droplet simulations of printing across and along scratches.

The dynamics of granular media within a silo in which the grain velocities are controlled by a conveyor belt has been experimentally investigated. To this end, the building of coarse-grained field maps of different magnitudes has allowed a deep analysis of the flow properties as a function of two parameters: the orifice size and the belt velocity. First, the internal dynamics of the particles within the silo has been fully characterized by the solid fraction, the velocity of the particles and the kinetic stress. Then, the analysis of the vertical profiles of the same magnitude (plus the acceleration) has allowed connection of the internal dynamics with the flow rate. In particular, we show that the gamma parameter – which accounts for the integration of the normalized acceleration along the vertical direction – can successfully discriminate the kind of flow established within the silo (from the quasistatic regime to the free discharge) depending on the outlet size and belt velocity.

In this paper, the kinetic energy cascade in stably stratified open-channel flows is investigated. A mathematical framework to incorporate vertical scales into the conventional kinetic energy spectrum and its budget is introduced. This framework defines kinetic energy density in horizontal spectral and vertical scale space. The energy cascade is studied by analysing the evolution of kinetic energy density. It is shown that energetic streamwise scales ($\lambda _x$) become larger with increasing vertical scale. For the strongest stratification, for which the turbulence becomes intermittent, the energetic streamwise scales are suppressed, and energy density resides in $\lambda _x$ of the size of the domain. It is shown that, in an unstratified case, vertical scales of the size comparable to the height of the logarithmic layer connect viscous regions to the outer layer. By contrast, in stratified cases, such a connection is not observed. Moreover, it is shown that nonlinear transfer for streamwise scales is dominated by in-plane triad interactions and inter-plane transfer is more active in transferring energy density among small vertical scales of the size comparable to the height of viscous sublayer. The vertical scales of size comparable to the height of the viscous sublayer and buffer layer are the most active scales in the viscous term and the production term in the energy density budget, respectively.

The behaviour of a viscous drop squeezed between two horizontal planes (a squeezed Hele-Shaw cell) is treated by both theory and experiment. When the squeezing force $F$ is constant and surface tension is neglected, the theory predicts ultimate growth of the radius $a\sim t^{1/8}$ with time $t$. This theory is first reviewed and found to be in excellent agreement with experiment. Surface tension at the drop boundary reduces the interior pressure, and this effect is included in the analysis, although it is negligibly small in the squeezing experiments. An initially elliptic drop tends to become circular as $t$ increases. More generally, the circular evolution is found to be stable under small perturbations. If, on the other hand, the force is reversed ($F<0$), so that the plates are drawn apart (the ‘contraction’, or ‘lifting plate’, problem), the boundary of the drop is subject to a fingering instability on a scale determined by surface tension. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a ‘one-eighth’ power law, but this is unstable, the instability originating at the boundary of the air bubble, i.e. the inner boundary of the annulus. The air bubble is realised experimentally in two ways: first by simply starting with the drop in the form of an annulus, as nearly circular as possible; and second by forcing four initially separate drops to expand and merge, a process that involves the resolution of ‘contact singularities’ by surface tension. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, before ultimate rupture of the film and complete separation of the plates, fingering spreads also from the boundary of any interior trapped air bubble, and small cavitation bubbles appear in the very low-pressure region, far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis.

This article uses triglobal stability analysis to address the question of shock-buffet unsteadiness, and associated modal dominance, on infinite wings at high Reynolds number, expanding upon recent biglobal work, aspiring to elucidate the flow phenomenon's origin and characteristics. Infinite wings are modelled by extruding an aerofoil to finite aspect ratios and imposing a periodic boundary condition without assumptions on spanwise homogeneity. Two distinct steady base flows, spanwise uniform and non-uniform, are analysed herein on straight and swept wings. Stability analysis of straight-wing uniform flow identifies both the oscillatory aerofoil mode, linked to the chordwise shock motion synchronised with a pulsation of its downstream shear layer, and several monotone (non-oscillatory), spatially periodic shock-distortion modes. Those monotone modes become outboard travelling on the swept wing with their respective frequencies and phase speeds correlated with the sweep angle. In the limiting case of very small wavenumbers approaching zero, the effect of sweep creates branches of outboard and inboard travelling modes. Overall, triglobal results for such quasi-three-dimensional base flows agree with previous biglobal studies. On the contrary, cellular patterns form in proper three-dimensional base flow on straight wings, and we present the first triglobal study of such an equilibrium solution to the governing equations. Spanwise-irregular modes are found to be sensitive to the chosen aspect ratio. Nonlinear time-marching simulations reveal the flow evolution and distinct events to confirm the insights gained through dominant modes from routine triglobal stability analysis.

The lowest swirling wave mode arising in upright circular cylinders as a response to circular orbital excitation has been widely studied in the last decade, largely due to its high practical relevance for orbitally shaken bioreactors. Our recent theoretical study (Horstmann et al., J. Fluid Mech., vol. 891, 2020, A22) revealed a damping-induced symmetry breaking mechanism that can cause spiral wave structures manifested in the so far widely disregarded higher rotating wave modes. Building on this work, we develop a linear criterion describing the degree of spiralisation and classify different spiral regimes as a function of the most relevant dimensionless groups. The analysis suggests that high Bond numbers and shallow liquid layers favour the formation of coherent spiral waves. This result paved the way to find the predicted wave structures in our interfacial sloshing experiment. We present two sets of experiments, with different characteristic damping rates, verifying the formation of both coherent and overdamped spiral waves in conformity with the theoretical predictions.

Particle segregation is common in natural and industrial processes involving flowing granular materials. Complex, and seemingly contradictory, segregation phenomena have been observed for different boundary conditions and forcing. Using discrete element method simulations, we show that segregation of a single particle intruder can be described in a unified manner across different flow configurations. A scaling relation for the net segregation force is obtained by measuring forces on an intruder particle in controlled-velocity flows where gravity and flow kinematics are varied independently. The scaling law consists of two additive terms: a buoyancy-like gravity-induced pressure gradient term and a shear rate gradient term, both of which depend on the particle size ratio. The shear rate gradient term reflects a kinematics-driven mechanism whereby larger (smaller) intruders are pushed toward higher (lower) shear rate regions. The scaling is validated, without refitting, in wall-driven flows, inclined wall-driven flows, vertical silo flows, and free-surface flows down inclines. Comparing the segregation force with the intruder weight results in predictions of the segregation direction that match experimental and computational results for various flow configurations.

The efficiency of tidal stream turbines in a large array depends on the balance between negative effects of turbine-wake interactions and positive effects of bypass-flow acceleration due to local blockage, both of which are functions of the layout of turbines. In this study we investigate the hydrodynamics of turbines in an infinitely large array with aligned or staggered layouts for a range of streamwise and lateral turbine spacing. First, we present a theoretical analysis based on an extension of the linear momentum actuator disc theory for perfectly aligned and staggered layouts, employing a hybrid inviscid-viscous approach to account for the local blockage effect within each row of turbines and the viscous (turbulent) wake mixing behind each row in a coupled manner. We then perform large-eddy simulation (LES) of open-channel flow for 28 layouts of tidal turbines using an actuator line method with doubly periodic boundary conditions. Both theoretical and LES results show that the efficiency of turbines (or the power of turbines for a given bulk velocity) in an aligned array decreases as we reduce the streamwise turbine spacing, whereas that in a staggered array remains high and may even increase due to the positive local blockage effect (causing the local flow velocity upstream of each turbine to exceed the bulk velocity) if the lateral turbine spacing is sufficiently small. The LES results further reveal that the amplitude of wake meandering tends to decrease as we reduce the lateral turbine spacing, which leads to a lower wake recovery rate in the near-wake region. These results will help to understand and improve the efficiency of tidal turbines in future large arrays, even though the performance of real tidal arrays may depend not only on turbine-to-turbine interactions within the array but also on macro-scale interactions between the array and natural tidal currents, the latter of which are outside the scope of this study.

We consider the release of preferentially wetting fluid in a laterally extensive V-shaped channel initially filled with a second fluid, presenting solutions for the initial exchange flow and the late time spreading of the wetting fluid along the narrow part of the channel. We also show that, if there is a buoyancy force acting in the cross-channel direction, the early time exchange flow depends on the Bond number, and the intermediate time slumping flow may initially be dominated by buoyancy, but at long times becomes controlled by capillarity. Where there is an along-channel component of gravity we show that the flow spreads out downslope, with capillarity controlling the structure of the nose. We then consider the case where the channel is connected to a reservoir of wetting fluid at constant pressure. We show that, depending on this pressure, either a zero flux exchange flow develops, or a net inflow through the whole width of the channel develops, as in the classical Washburn, Lucas, Bell and Cameron capillary imbibition flow. We show these flows are analogous to the classical model for one-dimensional capillary driven flows in porous media, with the current width in the channel corresponding to the saturation in the pore space.

This paper explores an idealized model of the ocean surface in which widely separated surface-wave packets and point vortices interact in two horizontal dimensions. We start with a Lagrangian which, in its general form, depends on the fields of wave action, wave phase, stream function and two additional fields that label and track the vertical component of vorticity. By assuming that the wave action and vorticity are confined to infinitesimally small, widely separated regions of the flow, we obtain model equations that are analogous to, but significantly more general than, the familiar system consisting solely of point vortices. We analyse stable and unstable harmonic solutions, solutions in which wave packets eventually coincide with point vortices (violating our assumptions), and solutions in which the wave vector eventually blows up. Additionally, we show that a wave packet induces a net drift on a passive vortex in the direction of wave propagation which is equivalent to Darwin drift. Generalizing our analysis to many wave packets and vortices, we examine the influence of wave packets on an otherwise unstable vortex street and show analytically, according to linear stability analysis, that the wave-packet-induced drift can stabilize the vortex street. The system is then numerically integrated for long times and an example is shown in which the configuration remains stable, which may be particularly relevant for the upper ocean.

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.

We study experimentally the behaviour of a bubble injected into a horizontal liquid solid-body rotating flow, in a range of rotational velocities where the bubble is close to the axis of rotation. We first study the stretching of the bubble as a function of its size and of the rotation of the cell. We show that the bubble aspect ratio can be predicted as a function of the bubble Weber number by the model of Rosenthal (J. Fluid Mech., vol. 12, 1962, 358–366) provided an appropriate correction due to the impact of buoyancy is included. We next deduce the drag and lift coefficients from the mean bubble position. For large bubbles straddling the axis of rotation, we show that the drag coefficient $C_D$ is solely dependent on the Rossby number $Ro$, with $C_D \approx 1.5/Ro$. In the same limit of large bubbles, we show that the lift coefficient $C_L$ is controlled by the shear Reynolds number $Re_{shear}$ at the scale of the bubble. For $Re_{shear}$ larger than 3000 we observe a sharp transition, wherein large fluctuations in the bubble aspect ratio and mean position occur, and can lead to the break-up of the bubble. We interpret this regime as a resonance between the periodic forcing of the rotating cell and the eigenmodes of the stretched bubble.

We present a computational study of the equilibrium dynamics of a polydisperse hard-sphere colloidal dispersion confined in a spherical cavity. We account for many-body hydrodynamic and lubrication interactions between particles and with the confining cavity utilizing our confined Stokesian dynamics model, expanded here for size polydispersity. We find that, even though the tendency of polydispersity to homogenize structure in a suspension is still present in confinement, strong correlations induced by the cavity resist homogenization. Although seemingly opposite, these two effects have a common driver, which is to maximize configurational entropy of particles in the cavity interior. These structural effects couple with the hydrodynamics to change the particle dynamics: polydispersity weakens lubrication effects near the cavity wall, allowing small (large) particles to diffuse faster (slower) than in a monodisperse suspension. As a small (large) particle gets farther from the wall, polydispersity weakens many-body hydrodynamic couplings, driving diffusivity up (down). While the local cage dynamics dominates short-time self-diffusion, long-time dynamics is also affected. In the concentrated regime, polydispersity and confinement combine to induce radial de-mixing into size-segregated populations. The cavity becomes the most influential ‘nearest neighbour’, setting the length scale of and dynamics within these radial domains. This intermediate length-scale caging makes the angular dynamics insensitive to polydispersity but leads to radial long-time mean-square displacement that changes qualitatively with volume composition. These results hold promise for explaining colloidal-scale physics implicated in the functioning of biological cells, and the engineering of non-living confined colloids where size de-mixing could be useful in the design of encapsulated micro-reactors and therapeutic vesicles.

Drops subjected to electric fields can deform into singular shapes exhibiting apparent sharp tips. At high field strengths, a perfectly conducting drop surrounded by a perfectly insulating exterior fluid deforms into a prolate-shaped drop with conical ends and can exist in hydrostatic equilibrium. On the conical ends, capillary stress, which is due to the out-of-plane curvature and is singular, balances electric normal stress which is also singular. If the two phases are not perfect conductors/insulators but are both leaky dielectrics and the drop is much more conducting and viscous than the exterior, electric tangential stress disrupts the hydrostatic force balance and leads to jet emission from the cone's apex. If, however, the physical situation is inverted so that a weakly conducting, slightly viscous drop is immersed in a highly conducting, more viscous exterior, the drop deforms into an oblate lens-like profile before eventually becoming unstable. In experiments, the equator of a lenticular drop superficially resembles a wedge prior to instability. Such a drop disintegrates by equatorial streaming by ejecting a thin liquid sheet from its equator. We show theoretically by performing a local analysis that a lenticular drop's equatorial profile can be a wedge only if an approximate form of the surface charge transport equation – continuity of normal current condition – is used. Moreover, we demonstrate via numerical simulation that such wedge-shaped drops do not become unstable and therefore cannot emit equatorial sheets. We then show by transient simulations how equatorial streaming can occur when charge transport along the interface is analysed without approximation.

Turbulent channel flow simulation of a shear-thinning fluid is considered – see Arosemena et al. (J. Fluid Mech., vol. 908, 2021, p. A43) – and compared with a Newtonian base case to reveal the effects of the shear-dependent rheology on the near-wall structures. Analyses of different flow statistics revealed that, for the shear-thinning fluid case, the streamwise vortices appear to grow in size, depart from the wall and present a lessening in their intensity. Information regarding variations in the quasi-longitudinal vortices is also obtained from three-dimensional structures identified through a normalized $Q$-criterion. With shear-thinning rheology, it is shown that the structures are comprised of wall-attached and -detached families which are taller than for a Newtonian fluid. Also, for a given height, the structures appear to be longer, with approximately the same width and overall larger volume for the shear-thinning fluid case; albeit their fractal dimension remains the same when compared to the Newtonian base case. Moreover, it is observed that the number density of vortical structures decreases with shear-thinning fluid behaviour. These observations, in conjunction with the known changes to the longitudinal velocity structures which appear to be less streaky, more spanwise separated and thickened with shear-thinning rheology, strongly suggest that the near-wall self-sustaining process has been disrupted. As we move slightly away from the wall and with shear-thinning behaviour, the local increase in viscosity seems to lead to less energetic vortices whereas the streaks are provided with an additional source of energy due to fluctuations in viscosity.

Large-scale intermittency is a widely observed phenomenon in convective surface layer turbulence that induces non-Gaussian temperature statistics, while such a signature is not observed for velocity signals. Although approaches based on probability density functions have been used so far, those are not able to explain to what extent the signals’ temporal structure impacts the statistical characteristics of the velocity and temperature fluctuations. To tackle this issue, a visibility network analysis is carried out on a field-experimental dataset from a convective atmospheric surface layer flow. Through surrogate data and network-based measures, we demonstrate that the temperature intermittency is related to strong nonlinear dependencies in the temperature signals. Conversely, a competition between linear and nonlinear effects tends to inhibit the temperature-like intermittency behaviour in streamwise and vertical velocities. Based on present findings, new research avenues are likely to be opened up in studying large-scale intermittency in convective turbulence.