Granular flows such as landslides, debris flows and avalanches are systems of particles with a large range of particle sizes that typically segregate while flowing. The physical mechanisms responsible for this process, however, are still poorly understood, and there is no predictive framework for ascertaining the segregation behaviour of a given system of particles. Here, we provide experimental evidence of individual large intruder particles being attracted to a fixed point in a dry two-dimensional flow of particles of otherwise uniform size. A continuum theory is proposed which captures this effect using only a single fitting parameter that describes the rate of segregation, given knowledge of the bulk flow field. Predictions of the continuum theory are compared with the experimental findings, both for the typical location and velocity field of a range of intruder sizes. For large intruder particle sizes, the continuum model successfully predicts that a fixed point attractor will form, where intruders are drawn to a single location.

We examine the fracture of a quasi-two-dimensional surfactant-laden aqueous foam under an applied driving pressure, using a network modelling approach developed for metallic foams by Stewart & Davis (J. Rheol., vol. 56, 2012, p. 543). In agreement with experiments, we observe two distinct mechanisms of failure analogous to those observed in a crystalline solid: a slow ductile mode when the driving pressure is applied slowly, where the void propagates as bubbles interchange neighbours through the T1 process; and a rapid brittle mode for faster application of pressures, where the void advances by successive rupture of liquid films driven by Rayleigh–Taylor instability. The simulations allow detailed insight into the mechanics of the fracturing medium and the role of its microstructure. In particular, we examine the stress distribution around the crack tip and investigate how brittle fracture localizes into a single line of breakages. We also confirm that pre-existing microstructural defects can alter the course of fracture.

Buoyant displacement flows of two miscible fluids in rectangular channels are studied, theoretically and experimentally. The scenario considered involves the displacement of a fluid by a slightly heavier one at nearly horizontal channel inclinations, where inertial effects are weak and laminar stratified flows may be expected. In the theoretical part, a lubrication approximation model is developed to simplify the displacement flow governing equations and furnish a semi-analytical solution for the heavy and light fluid flux functions. Three key dimensionless parameters govern the fluid flow motion, i.e. a buoyancy number, the viscosity ratio and the channel cross-section aspect ratio. When these parameters are specified, the reduced model can deliver the interface propagation in time, leading and trailing front heights, shapes and speeds, cross-sectional velocity fields, etc. In addition, the model can be exploited to provide various classifications such as single or multiple fronts as well as main displacement flow regimes at long times such as no sustained backflows, stationary interface flows and sustained backflows. Focusing on the variation of the buoyancy number, a large number of iso-viscous displacement experiments are performed in a square duct and the results are compared with those of the lubrication model. Qualitative displacement flow features observed in the theory and experiments are in good agreement, in particular, in terms of the main displacement flow regimes. The quantitative comparisons are also reasonable for small and moderate imposed displacement flow velocities. However, at large flow rates, a deviation of the experimental results from the model results is observed, which may be due to the presence of non-negligible inertial effects.

In a process involving electromagnetic shaping, a high-frequency electromagnetic field is used to deform a liquid conductor into a required shape. This is particularly relevant to processes such as levitation melting. In this paper the stability of such configurations are investigated. The second variation of an appropriate energy functional is derived whose minimum states correspond to stable configurations, thus providing a stability criterion. As an example, this is applied to the shaping of a levitated cylinder of circular cross-section and to an almost spherical axisymmetric shape. In both cases we find that these shapes are unstable. We then consider enclosing the entire shaping device in a metal shield, thus preventing the escape of the magnetic field. It is then shown that in general the shield has a stabilizing effect, whose exact nature depends on the topology of the liquid shape and on the field structure on its surface. This differing behaviour is discussed for two-dimensional spherical and toroidal shapes.

This paper deals with the Soret separation of a binary mixture in a cylinder subjected to an axial temperature gradient. The study is connected to an experiment designed to measure the Soret coefficient of an AgI-KI mixture corresponding to a moderate Prandtl number (Pr = 0.6) and a high Schmidt number (Sc = 60). In such an experiment the species separation is often hidden by a mixing effect due to the buoyancy-driven convection generated by a horizontal temperature gradient induced by some defect of the heating system. Here, such a defect is simulated by a slight misorientation of the cell with respect to the vertical; a small inclination (γ = 1°) of the cell has been considered, but the results can be generalized for any other small γ. For situations corresponding to a top heating and a positive Soret parameter, S, two quite different regimes have been exhibited depending on the value of S. For moderate S, the induced solutal buoyancy balances the imposed thermal buoyancy, slowing down the flow and giving a good separation rate. For small S this balance does not exist (except in the centre), leading to a remixing of the species and thus to poor separation (the separation would be still worse for negative S). The smaller the (positive) Soret parameter is, the smaller the cell misorientation γ has to be to allow a good separation rate.

In this paper, we consider nearshore rotational currents directly forced by unsteady multidirectional wave breaking. Scaling relationships, simplified analytical solutions, and asymptotic limits are developed for the maximum forced cross-shore and longshore velocities. In all cases, forced longshore velocities are considerably larger than cross-shore velocities. On longshore-uniform beaches, strong fluctuating velocities are found for (i) large waves; (ii) strong directional spreading; and (iii) short peak wave periods. When topographic inhomogeneities control longshore scales of wave breaking, overall scaling changes and the largest fluctuating velocities are found for ($a$) large waves; ($b$) long wave periods; and ($c$) topographic features that vary quickly in the longshore direction. The ratio of fluctuating rotational velocities to mean longshore current does not depend on the wave height or period, but instead on the bottom friction, slope, deep water wave angle, and details of the wave spectrum.

The geometric properties are quantified for concentration iso-surfaces of a high-Schmidt-number passive scalar field produced by an iso-kinetic source with an initial finite characteristic length scale released into the inertial layer of fully developed open-channel-flow turbulent boundary layers. The coverage dimension and other measures of two-dimensional transects of the passive scalar iso-surfaces are found to be scale dependent. The coverage dimension is around 1.0 at the order of the Batchelor length scale and based on our data increases in a universal manner to reach a local maximum at a length scale around the Kolmogorov scale. We introduce a new parameter called the coverage length underestimate, which demonstrates universal behaviour in the viscous–convective regime for these data and hence is a potentially useful practical tool for many mixing applications. At larger scales (in the inertial–convective regime), the fractal geometry measures are dependent on the Reynolds number, injection length scale, and concentration threshold of the iso-surfaces. Finally, the lacunarity of the iso-surface structure shows that the instantaneous scalar field is most inhomogenous around the length scale corresponding to the Kolmogorov scale.

Solutions for inviscid rotating flow over a right circular cylinder of finite height are studied, and comparisons are made to quasi-geostrophic solutions. To study the combined effects of finite topography and the variation of the Coriolis parameter with latitude a steady inviscid model is used. The analytical solution consists of one part which is similar to the quasi-geostrophic solution that is driven by the potential vorticity anomaly over the topography, and another, similar to the solution of potential flow around a cylinder, that is driven by the matching conditions on the edge of the topography. When the characteristic Rossby wave speed is much larger than the background flow velocity, the transport over the topography is enhanced as the streamlines follow lines of constant background potential vorticity. For eastward flow, the Rossby wave drag can be very much larger than that predicted by quasi-geostrophic theory. The combined effects of finite height topography and time-dependence are studied in the inviscid initial value problem on the f-plane using the method of contour dynamics. The method is modified to handle finite topography. When the topography takes up most of the layer depth, a stable oscillation exists with all of the fluid which originates over the topography rotating around the topography. When the Rossby number is order one, a steady trapped vortex solution similar to the one described by Johnson (1978) may be reached.

Axisymmetric non-isothermal spreading/dewetting of droplets on a substrate is studied, wherein the surface tension is a function of temperature, resulting in Marangoni stresses. A lubrication theory is first extended to determine the drop shape for spreading/dewetting limited by slip. It is demonstrated that an apparent angle inferred from a fitted spherical cap shape does not relate to the contact-line speed as it would under isothermal conditions. Also, a power law for the thermocapillary spreading rate versus time is derived. Results obtained with direct numerical simulations (DNS), using a slip length down to $O(10^{-4})$ times the drop diameter, confirm predictions from lubrication theory. The DNS results further show that the behaviour predicted by the lubrication theory – that a cold wall promotes spreading, and a hot wall promotes dewetting – is reversed at sufficiently large contact angles and/or viscosity of the surrounding fluid. This behaviour is summarized in a phase diagram, and a simple model that supports this finding is presented. Although the key results are found to be robust when accounting for heat conduction in the substrate, a critical thickness of the substrate is identified above which wall conduction significantly modifies wetting behaviour.

Parametrically excited solitary waves emerge as localized structures in high-aspect-ratio free surfaces subject to vertical vibrations. Herein, we provide the first experimental characterization of the hydrodynamics of these waves using particle image velocimetry. We show that the underlying velocity field of parametrically excited solitary waves is primarily composed of a subharmonic oscillatory component. Our results confirm the accuracy of Hamiltonian models with added dissipation in describing this field. Remarkably, our measurements also uncover the onset of a streaming velocity field which we show to be as important as other crucial nonlinear terms in the current theory. Using vorticity equations, we show that the streaming pattern arises from the coupling of the potential bulk flow with the oscillating boundary layers on the vertical walls. Numerical simulations provide good agreement between this model and experiments.

In Crowe & Taylor (J. Fluid Mech., vol. 850, 2018, pp. 179–211) we described a theory for the evolution of density fronts in a rotating reference frame subject to strong vertical mixing using an asymptotic expansion in small Rossby number, $Ro$. We found that the front reaches a balanced state where vertical diffusion is balanced by horizontal advection in the buoyancy equation. The depth-averaged buoyancy obeys a nonlinear diffusion equation which admits a similarity solution corresponding to horizontal spreading of the front. Here we use numerical simulations of the full momentum and buoyancy equations to investigate this problem for a wide range of Rossby and Ekman numbers. We examine the accuracy of our asymptotic solution and find that many aspects of the solution are valid for $Ro=O(1)$. However, the asymptotic solution departs from the numerical simulations for small Ekman numbers where the dominant balance in the momentum equation changes. We trace the source of this discrepancy to a depth-independent geostrophic flow that develops on both sides of the front and we develop a modification to the theory described in Crowe & Taylor (2018) to account for this geostrophic flow. The refined theory closely matches the numerical simulations, even for $Ro=O(1)$. Finally, we develop a new scaling for the intense vertical velocity that can develop in thin bands at the edges of the front.

The problem of interaction between disturbances and shock waves was solved by a theoretical approach called linear interaction analysis in the mid-twentieth century. More recently, great progress has been made in analysing shock–turbulence interactions by direct numerical simulation. However, an unsolved theoretical problem remains: What happens when no acoustic waves are stimulated behind the shock wave? The concept of a damped wave is introduced, which is a type of excited plane wave. Based on this, the dispersion and amplitude relationships between any incident plane wave and resulting stimulated waves are constructed analytically, systematically and comprehensively. The physical essence of damped waves and the existence of critical angles are clarified. It is demonstrated that a damped wave is a complex number space solution to the acoustic dispersion relationship under certain conditions. It acts as a bridge connecting fast and slow acoustic waves at the position where the $x$ component of the group velocity is zero. There are two critical angles that can excite fast and slow acoustic waves, which determine the conditions that stimulate a damped wave. Our results show good agreement with theoretical and simulation results. The contribution of each excited wave to the transmission coefficient is evaluated, the distribution of the transmission coefficient is analysed and application to an engineering wedge model is performed.

The dynamics of spherical particles in a fluid flow is governed by the well-accepted Maxey–Riley equation. This equation of motion simply represents Newton’s second law, equating the rate of change of the linear momentum with all forces acting on the particle. One of these forces, the Basset–Boussinesq memory term, however, is notoriously difficult to handle, which prompts most studies to ignore this term despite ample numerical and experimental evidence of its significance. This practice may well change now due to a clever reformulation of the particle equation of motion by Prasath, Vasan & Govindarajan (J. Fluid Mech., vol. 868, 2019, pp. 428–460), who convert the Maxey–Riley equation into a one-dimensional heat equation with non-trivial boundary conditions. Remarkably, this reformulation confirms earlier estimates on particle asymptotics, yields previously unknown analytic solutions and leads to an efficient numerical scheme for more complex flow fields.