Density segregation of multi-component granular mixtures in a dense, gravity-driven flow over a rough and bumpy periodic chute surface is studied using theory and simulations. An existing theoretical model for predicting the steady-state concentration field of each species in a binary mixture using the forces acting on the particles is generalised for multi-component mixtures in this work. In addition, the rheological model for binary mixtures is also extended to multi-component mixtures. In contrast to the percolation velocity-based empirical segregation models that do not account for the rheology and need prior knowledge of the velocity field, the present approach accounts for the inter-coupling of rheology with segregation. The momentum balance equations are solved, along with the mixture rheological model as well as the multi-component density segregation model, to obtain concentration fields using an iterative numerical method. The theoretical predictions are compared with discrete element method (DEM) simulations for ternary, quaternary and quinary granular mixtures differing in density. The steady-state profiles for the concentration of different species as well as other flow properties predicted from the theory are in excellent agreement with the DEM simulation results for a variety of compositions over a range of inclination angles for different density ratios. Through this work, we take the first essential step towards proposing a generalised particle force-based segregation theory for multi-component mixtures differing in size and/or density.

Different types of neural networks have been used to solve the flow sensing problem in turbulent flows, namely to estimate velocity in wall-parallel planes from wall measurements. Generative adversarial networks (GANs) are among the most promising methodologies, due to their more accurate estimations and better perceptual quality. This work tackles this flow sensing problem in the vicinity of the wall, addressing for the first time the reconstruction of the entire three-dimensional (3-D) field with a single network, i.e. a 3-D GAN. With this methodology, a single training and prediction process overcomes the limitation presented by the former approaches based on the independent estimation of wall-parallel planes. The network is capable of estimating the 3-D flow field with a level of error at each wall-normal distance comparable to that reported from wall-parallel plane estimations and at a lower training cost in terms of computational resources. The direct full 3-D reconstruction also unveils a direct interpretation in terms of coherent structures. It is shown that the accuracy of the network depends directly on the wall footprint of each individual turbulent structure. It is observed that wall-attached structures are predicted more accurately than wall-detached ones, especially at larger distances from the wall. Among wall-attached structures, smaller sweeps are reconstructed better than small ejections, while large ejections are reconstructed better than large sweeps as a consequence of their more intense footprint.

This work investigates the time-averaged shape dynamics of a soft elastic capsule in two phase-shifted orthogonal ultrasonic standing waves. The capsule consists of an elastic membrane that encloses a viscous fluid and is immersed in another viscous fluid. Combining the acoustic perturbation theory of fluid dynamics with the thin-shell mechanics of capsule membrane deformation, two sets of equations are established to govern the ultrasonic propagation and the time-averaged response of the fluid–capsule system, respectively. These governing equations are solved numerically based on the finite element method. Numerical simulations show that the ultrasonic standing waves have pure elongation and pure rotation effects on the initially circular capsule when the phase difference is 0 and ${\rm \pi}/2$, respectively. By setting the phase difference between 0 and ${\rm \pi}/2$, it is found that the initially circular capsule exhibits a tank-treading motion due to the combined effect of the elongation and rotation. The capsule membrane elasticity and internal fluid viscosity have significant effects on the tank-treading behaviour of the initially circular capsule, including elongation deformation, inclination angle and tank-treading velocity. For the initially non-circular capsule, three types of dynamical states, including stable orientation, swinging and tumbling, are predicted by varying the phase difference and intensity of the ultrasonic standing waves, as well as the initial shape, membrane elasticity and internal fluid viscosity of the initially non-circular capsule. This work enriches the cell manipulation capabilities of ultrasonic standing wave micro-acoustofluidics and may inspire new biological applications.

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.

We consider the motion at large Reynolds number of an incompressible fluid around a thin circular disk of finite radius rotating in its plane. The disk is placed in a large tank filled with an initially stagnant fluid. Then it is brought into rotation about its centre with constant angular velocity. Due to viscosity, a layer of fluid adjacent to the disk gets involved in the circumferential motion. This activates centrifugal forces; the fluid particles start to deviate in the radial direction. When they cross the edge of the disk, a thin jet is formed. Firstly, we solve the classical boundary-layer equations for the flow in the boundary layer in the direct neighbourhood of the disk surface and in the jet. The solution is found to develop a discontinuity at the ‘head’ of the jet where the radial and circumferential velocity components experience a jump. This type of discontinuity, called a pseudo-shock, was previously observed by Ruban & Vonatsos (J. Fluid Mech., vol. 614, 2008, pp. 407–424). Then, we investigate the internal structure of the pseudo-shock. We find that the fluid motion is described by the Euler equations in the leading-order approximation. Their solution shows that, as the jet penetrates the stagnant fluid, it ejects the fluid from the boundary layer into the surrounding area. Analysis of the inviscid region outside the boundary layer reveals that the ejected fluid returns back to the boundary layer through the ‘entrainment process’. Finally, we conclude this paper with the study of the wake in the vicinity of the disk rim.

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.

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.

This paper presents an analytical study of the Riemann problem for dilute granular gas using initial conditions that result in a shock–contact–shock wave structure. The Euler equations for molecular gas were perturbed with small granular energy dissipation, resulting in an approximate analytical solution that is valid for early evolution time. This approximate analytical solution shows good agreement with the numerical solution of the full problem obtained using a shock-capturing scheme. It is shown that the wave structure of the Riemann problem for dilute granular gas follows that of molecular gas. However, the variables in regions between the discontinuities are functions of both space and time. Our solution shows that the ‘density overshoot’ – reported by Reddy & Alam (J. Fluid Mech., vol. 779, 2015, R2) – is not part of the shock layer but a signature of the density variation in the star region between the left and right shocks, with the maximum density occurring at the contact discontinuity.

A gas volume rising in a vertical tube is called a Taylor bubble when its equivalent spherical diameter is greater than the diameter of the tube. Continuity dictates that the rising velocity of such bullet-shaped bubbles must be such that the drainage flow in the liquid film separating the gas from the tube wall equals the rate at which space is vacated by the bubble tail on its rise. The numerical simulations presented here show that, if the volume of the bubble is made to execute small-amplitude oscillations, this drainage flow can be markedly decreased so that the bubble slows down so much as to very nearly stop. The mechanism is a subtle nonlinear interplay between the upward flow imposed by the bubble expansion and the inertia of the falling liquid. Two manners of excitation are considered. In the first one the liquid column above the bubble is forced to oscillate sinusoidally, e.g. by a piston. In the second one, an oscillating pressure is imposed on the top liquid surface. With the first arrangement the bubble slows down during its entire ascent by an amount depending on the oscillation frequency and amplitude. With the second one, for a given frequency, the bubble slows down only in the neighbourhood of a certain resonant depth at which the volume oscillations are maximized, but is little affected at deeper and shallower depths.

Recently, the existence of so-called granular bubbles and droplets has been demonstrated experimentally. Granular bubbles and droplets are clusters of particles that respectively rise and sink if submerged in an aerated and vibrated bed of another granular material of different size and/or density. However, currently, there is no model that explains the coherent motion of these clusters and predicts the transition between a rising and sinking motion. Here, we propose an analytical model predicting accurately the neutral buoyancy limit of a granular bubble/droplet. This model allows the compilation of a regime map identifying five distinct regimes of granular bubble/droplet motion.

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.

The influence of an applied magnetic field on the collisional plasma Richtmyer–Meshkov instability (RMI) is investigated through numerical simulation. The instability is studied within the five-moment multifluid plasma model without any simplifying assumptions such as infinite speed of light, negligible electron inertia or quasineutrality. The plasma is composed of ion and electron fluids, and elastic collisions are modelled with the Braginskii transport coefficients. A collisional regime is investigated and the magnetic field is applied in the direction of shock propagation, which is perpendicular to the density interface. The primary instability is influenced by several terms affecting the evolution of circulation, the most significant of which are the baroclinic, magnetic field torque and intraspecies collisional terms. The applied magnetic field results in a reduction of interface perturbation growth, agreeing qualitatively with previous numerical simulations for the case of an ideal multifluid plasma RMI. The only major difference in the present case's instability mitigation by applied magnetic field, relative to the ideal case with applied magnetic field, is that the elastic collisions replace and obstruct the secondary vorticity suppression mechanism through collisional dissipation of vorticity. Additionally the collisions, influenced by the combination of self-generated and the applied magnetic field, introduce anisotropy to the problem. The primary suppression mechanism for the RMI is unchanged relative to the ideal case, i.e. the magnetic field torque resisting baroclinic deposition of vorticity in the ion fluid.

The electromagnetically driven anticyclonic flow in the wide gap of a rotating concentric spheres system is studied experimentally and numerically in the laminar regime. The working fluid is an electrolyte contained in the spherical gap. The outer sphere rotates at constant angular speed, and the inner sphere is at rest. The electromagnetic stirring is generated due to the interaction of a direct current, which is injected radially through ring-shaped electrodes located at the equatorial zone of each sphere, and a dipolar magnetic field produced by a permanent magnet located inside the inner sphere. Experimental velocity fields in the equatorial plane were obtained with particle image velocimetry. Additionally, full three-dimensional numerical simulations were performed. The high shearing at the equatorial region promotes an instability that can be perceived as a triggering mechanism of four tornado-like vortical structures tilted and entangled in the polar direction. The instability structure rotates in either the cyclonic or the anticyclonic direction, depending on the flow parameters, namely, the angular velocity of the outer sphere and the applied electric current. To the best knowledge of the authors, this is the first time the phenomenon under study is reported with electromagnetic forcing. The numerical results agree quantitatively with the experimental observations.