A three-dimensional linear stability analysis has been carried out to understand the origin of vortices and related density patterns in bounded uniform-shear flow of granular materials, using a kinetic-theory constitutive model. This flow is found to be unstable to pure spanwise stationary perturbations ($k_z\,{\neq}\, 0$, $k_x\,{=}\,0$ and $\partial/\partial y(.)\,{=}\,0$, where $k_i$ is the wavenumber for the $i$th direction) if the solid fraction is below some critical value $\nu\,{<}\, \nu_{3D}$. The growth rates of these spanwise instabilities are an order of magnitude larger than those of the two-dimensional ($k_z\,{=}\,0$) streamwise-independent ($k_x\,{=}\,0$) instabilities that occur if the solid fraction is above some critical value $\nu\,{>}\,\nu_{2D}$ (${>}\nu_{3D}$). The spanwise instabilities give birth to new three-dimensional travelling wave instabilities at non-zero values of the streamwise wavenumber ($k_x\,{\neq}\, 0$) in dilute flows ($\nu \,{<}\, \nu_{3D}$). For moderate-to-large densities with $k_x\,{\neq}\, 0$, there are additional three-dimensional instability modes in the form of both stationary and travelling waves, whose origin is tied to the corresponding two-dimensional instabilities.

While the two-dimensional streamwise-independent modes lead to the formation of stationary streamwise vortices for moderately dense flows ($\nu\,{>}\,\nu_{2D}$), the pure spanwise modes are responsible for the origin of such vortices in the dilute limit ($\nu\,{<}\,\nu_{3D}$). For more general kinds of perturbations ($k_x\,{\neq}\, 0$ and $k_z\,{\neq}\, 0$), ‘modulated’ streamwise vortices are born which could be either stationary or travelling depending on control parameters. The rolling motion of vortices will lead to a major redistribution of the streamwise velocity and hence such vortices can act as potential progenitors for the mixing of particles. The effect of non-zero wall slip has been investigated, and it is shown that some dilute-flow instabilities can disappear with the inclusion of the wall slip. Even though the streamwise granular vortices have similarities to the well-known stationary Taylor–Couette vortices (which are ‘hydrodynamic’ in origin), their origin is, however, tied to ‘constitutive’ instabilities, and hence they belong to a different class.

Consider the three-dimensional flow of a viscous Newtonian fluid upon a curved two-dimensional substrate when the fluid film is thin, as occurs in many draining, coating and biological flows. We derive a comprehensive model of the dynamics of the film, the model being expressed in terms of the film thickness $\eta$ and the average lateral velocity $\bar{\bm u}$. Centre manifold theory assures us that the model accurately and systematically includes the effects of the curvature of substrate, gravitational body force, fluid inertia and dissipation. The model resolves wavelike phenomena in the dynamics of viscous fluid flows over arbitrarily curved substrates such as cylinders, tubes and spheres. We briefly illustrate its use in simulating drop formation on cylindrical fibres, wave transitions, three-dimensional instabilities, Faraday waves, viscous hydraulic jumps, flow vortices in a compound channel and flow down and up a step. These models are the most complete models for thin-film flow of a Newtonian fluid; many other thin-film models can be obtained by different restrictions and truncations of the model derived here.

We study the jetting–dripping (J–D) transition of a flow-focused viscous liquid jet surrounded by a co-flowing immiscible, lower viscosity liquid. A theoretical model describing wave propagation in open cylindrical flows has been adapted to our problem and further expanded to incorporate spatio-temporal stability considerations (global modes). The J–D transition sets the minimum liquid flow rate issuing as a steady jet and breaking up into droplets whose size is commensurate with the jet diameter. At the onset of dripping, droplets become considerably larger than jetting droplets, under comparable flow parameters. A linear theory accounting for convective and absolute instability is provided, along with a detailed interpretation of the parametrical space, under realistic viscosity and density restrictions. The experimental part sums up a collection of laboratory data illustrating the J–D transition with good agreement with the theory.

The vertical flow of a fluid, under the influence of gravity, down the exterior of a rigid fibre is a flow accompanied by rich dynamics manifested via the formation of droplets, or beads, driven by a Rayleigh mechanism modulated by the presence of gravity. These droplets propagate down the fibre and undergo coalescence with preceding droplets. Different flow regimes are possible depending on system parameters such as the fibre radius, liquid flow rate and physical properties. We derive an evolution equation for the interface in the long-wavelength approximation, which captures the flow characteristics of the system; this model is similar to those previously used to investigate the dynamics of slender viscous threads in the absence of the fibre. Analytical and numerical solutions of the evolution equation yield information regarding the shape and propagation speeds of the droplets, which is in good agreement with available experimental data as well as those obtained as part of the present work. Connections with models already available in the literature are also established.

The generation and propagation of a packet of small-amplitude inertia–gravity waves (IGWs) in a rotating stratified balanced flow is described. The initially balanced geophysical flow is an unstable baroclinic jet which breaks up into a street of cyclonic and anticyclonic vortices. The small-amplitude unbalanced component of the flow is extracted from the large-amplitude mesoscale balanced flow using the optimal potential vorticity balance approach. This analysis reveals that during the instability the balanced flow spontaneously emits bursts of IGWs. The emission occurs along two directions, into and out of the anticyclonic vortices. The inward-waves remain trapped inside the vortices while the outward-waves propagate away from them as a packet of small-amplitude IGWs with a three-dimensional helical structure. The wave packet emission is confirmed for different spatial resolutions ($128^3$, $160^3$, $192^3$ and $256^3$ grid points). The ratio between the balanced vertical and horizontal velocity components is of the order of $10^{-3}$, as is typical of mesoscale geophysical flows. The ratio between the unbalanced vertical and horizontal components is about 0.1. Since the unbalanced horizontal and the balanced vertical velocity components are of similar magnitude, the vertical velocity of the IGWs is about $10^{-4}$ times the balanced horizontal velocity. The IGWs are dominated by frequencies close to the inertial frequency and have a clockwise-rotating horizontal velocity, similar to plane wave solutions.

Wakes and vortex streets such as those occurring behind towed or self-propelled bodies are generated by moving localized forces in a viscous fluid at moderate values of the Reynolds number, $\hbox{\it Re}\,{\sim}\,10^{2}$. The forcing is provided by an electromagnetic method and allows us to create a ‘virtual’ body without introducing any solid objects into the fluid. Characteristics of stable and unstable wakes, in particular the shedding frequency, are measured in the space of control parameters, namely the magnitude of the forcing and the speed of translational motion of the forcing. The results for a single force presented in the dimensionless form of the Strouhal number demonstrate quantitative similarity to those for the classical flow around a cylinder. The problem considered here has an extra degree of freedom compared to the problem of the flow around a cylinder and exhibit a wider array of different regimes. These regimes are documented in both our visualization experiments and particle image velocimetry measurements.

In high-Reynolds-number two-dimensional turbulence with a −5/3 power-law energy spectrum, the clustering of inertial particles reflects the clustering of acceleration stagnation points for all particle relaxation times smaller than the integral time scale $T$ of the turbulence. Acceleration stagnation points and small inertial particles on these points are swept together by large-scale motions. In synthetic turbulence where there is no sweeping and acceleration stagnation points do not cluster, inertial particles do nevertheless cluster as a result of the repelling action of persistent velocity stagnation-point clusters. This repelling action has a negligible effect on the clustering of inertial particles in the presence of acceleration stagnation points clustering.

Mass transfer through the flat shear-free surface of a turbulent open-channel flow is investigated over a wide range of Schmidt number (1 $ \le $Sc$ \le $ 200) by means of large-eddy simulations using a dynamic subgrid-scale model. In contrast with situations previously analysed using direct numerical simulation, the turbulent Reynolds number Re is high enough for the near-surface turbulence to be fairly close to isotropy and almost independent of the structure of the flow in the bottom region (the statistics of the velocity field are identical to those described by I. Calmet & J. Magnaudet J. Fluid Mech. vol. 474, 2003, p. 355). The main statistical features of the concentration field are analysed in connection with the structure of the turbulent motion below the free surface, characterized by a velocity macroscale $u$ and an integral length scale $L$. All near-surface statistical profiles are found to be Sc-independent when plotted vs. the dimensionless coordinate Sc$^{1 / 2}yu$/$\nu $ ($y$ is the distance to the surface and $\nu $ is the kinematic viscosity). Mean concentration profiles are observed to be linear throughout an inner diffusive sublayer whose thickness is about one Batchelor microscale, i.e. LSc$^{ - 1 / 2 }$Re$^{ - 3 / 4}$. In contrast, the concentration fluctuations are found to reach their maximum near the edge of the outer diffusive layer which scales as LSc$^{ - 1 / 2}$Re$^{ - 1 / 2}$. Instantaneous views of the near-surface isovalues of the concentration and vertical velocity are used to reveal the influence of the Schmidt number. In particular, it is observed that at high Schmidt number, the tiny concentration fluctuations that subsist in the diffusive sublayer just mirror the divergence of the two-component surface velocity field. Co-spectra of concentration and vertical velocity fluctuations indicate that the main contribution to the turbulent mass flux is provided by eddies whose horizontal size is close to $L$, which strongly supports the view that the mass transfer is governed by large-scale structures. The dimensionless mass transfer rate is observed to be proportional to Sc$^{ - 1 / 2}$ over the whole range of Schmidt number. Based on a frequency analysis of the concentration equation and on the Sc$^{ - 1 / 2}$Re$^{ - 3 / 4 }$scaling of the diffusive sublayer, it is shown that the mass transfer rate at a given Sc is proportional to $\langle {\beta ^2}\rangle ^{1 / 4}$, $\langle {\beta ^2}\rangle $ being the variance of the divergence of the surface velocity field. This yields dimensionless mass transfer rates of the form $\alpha$Sc$^{ - 1 / 2}$Re$^{ - 1 / 4}$, where the value of $\alpha$ is shown to result from both the kinematic blocking of the vertical velocity and the viscous damping of the horizontal vorticity components induced by the free surface.

A novel boundary-integral algorithm is used to study the general, three-dimensional motion of neutrally buoyant prolate and oblate spheroids in a low-Reynolds-number Poiseuille flow between parallel plates. Adaptive meshing of the spheroid surface assists in obtaining accurate numerical results for particle–wall gaps as small as 1.3% of the spheroid's major axis. The resistance formulation and lubrication asymptotic forms are then used to obtain results for arbitrarily small particle–wall separations. Spheroids with their major axes shorter than the channel spacing experience oscillating motion when the spheroid's centre is initially located in or near the midplane of the channel. For both two-dimensional and three-dimensional oscillations, the period length decreases with an increase in the initial inclination of the spheroid's major axis with respect to the lower wall. These spheroids experience tumbling motions for centre locations further from the midplane of the channel, with a period length that decreases as the spheroid is located closer to a wall. The transition from two-dimensional oscillating motion to two-dimensional tumbling motion occurs for an initial centre location closer to a wall as the initial inclination of the major axis is increased. For these spheroids, the average translational velocity along the channel length for two-dimensional oscillating motion decreases for an increase in the initial inclination of the major axis, and the average translational velocity for two-dimensional tumbling motion decreases for a decrease in the initial centre location. A prolate spheroid with its major axis 50% longer than the channel spacing and confined to the ($x_2$, $x_3$)-plane (where $x_2$ is the primary flow direction and $x_3$ is normal to the walls) cannot experience two-dimensional tumbling; instead, the spheroid becomes wedged between the walls for initial centre locations near the midplane of the channel when the initial inclination of the large spheroid's major axis is steep, and experiences two-dimensional oscillations for initial centre locations near a wall. When this spheroid's major axis is not confined to the ($x_2$, $x_3$)-plane, it experiences three-dimensional oscillations for initial centre locations in or near the midplane of the channel, and three-dimensional tumbling for initial centre locations near a wall.

This paper analyses numerically the quasi-two-dimensional flow of an incompressible electrically conducting viscous fluid past a localized zone of applied magnetic field, denominated a magnetic obstacle. The applied field is produced by the superposition of two parallel magnetized square surfaces, uniformly polarized in the normal direction, embedded in the insulating walls that contain the flow. The area of these surfaces is only a small fraction of the total flow domain. By considering inertial effects in the analysis under the low magnetic Reynolds number approximation, it is shown that the flow past a magnetic obstacle may develop vortical structures and eventually instabilities similar to those observed in flows interacting with bluff bodies. In the small zone where the oncoming uniform flow encounters the non-negligible magnetic field, the induced electric currents interact with the field, producing a non-uniform Lorentz force that brakes the fluid and creates vorticity. The effect of boundary layers is introduced through a friction term. Due to the localization of the applied magnetic field, this term models either the Hartmann braking within the zone of high magnetic field strength or a Rayleigh friction in zones where the magnetic field is negligible. Finite difference numerical computations have been conducted for Reynolds numbers $\hbox{\it Re}\,{=}\,100$ and 200, and Hartmann numbers in the range $1 \le \hbox{\it Ha} \le 100$ ($\hbox{\it Re}$ and $\hbox{\it Ha}$ are based on the side length of the magnetized square surfaces). Under these conditions, a wake is formed behind the obstacle. It may display two elongated streamwise vortices that remain steady as long as the Hartmann number does not exceed a critical value. Once this value is reached, the wake becomes unstable and a vortex shedding process similar to the one observed in the flow past bluff bodies is established. Similarities and differences with the flow around solid obstacles are discussed.

We present a three-dimensional linear stability analysis of a couple of co-rotating vertical vortices in a stratified fluid. When the fluid is non-stratified, the two vortices are unstable to the elliptic instability owing to the elliptic deformation of their core. These elliptic instability modes persist for weakly stratified flow: $F_h \,{>}\, 10$, where $F_h$ is the horizontal Froude number ($F_h\,{=}\,\Gamma_b/\pi a_b^2 N$ where $\Gamma_b$ is the circulation of the vortices, $a_b$ their core radius and $N$ the Brunt–Väisälä frequency). For strong stratification ($F_h \,{<}\, 2.85$), a new zigzag instability is found that bends each vortex symmetrically with almost no internal deformation of the basic vortices. This instability may modify the vortex merging since at every half-wavelength along the vertical, the vortices are alternatively brought closer, accelerating the merging, and moved apart, delaying the merging. The most unstable vertical wavelength $\lambda_m$ of this new instability is shown to be proportional to $F_h b_b$, where $b_b$ is the distance between the vortices, implying that $\lambda_m$ decreases with increasing stratification. The maximum growth rate, however, is independent of the stratification and proportional to the strain $S\,{=}\,\Gamma_b/2 \pi b_b^2 $. These scaling laws and the bending motion induced by this instability are similar to those of the zigzag instability of a counter-rotating vortex pair in a stratified fluid.

This paper investigates the three-dimensional stability of a pair of co-rotating vertical vortices in a rotating strongly stratified fluid. In a companion paper (Otheguy, Chomaz & Billant 2006), we have shown that such a basic flow in a strongly stratified fluid is affected by a zigzag instability which bends the two vortices symmetrically. In the non-rotating flow, the most unstable wavelength of this instability scales as the buoyancy length and its growth rate scales as the external strain that each vortex induces on the other one. Here, we show that the zigzag instability remains active whatever the magnitude of the planetary rotation and is therefore connected to the tall-column instability in quasi-geostrophic fluids. Its growth rate is almost independent of the Rossby number. The most amplified wavelength follows the universal scaling $\lambda\,{=}\, 2 \pi F_h b \sqrt{{\gamma_1}/{\hbox{\it Ro}^2} +{\gamma_2}/{\hbox{\it Ro}} +\gamma_3 }$, where $b$ is the separation distance between the two vortices, ($\gamma_1$, $\gamma_2$, $\gamma_3$) are constants, $F_h$ is the horizontal Froude number and $\hbox{\it Ro}$ the Rossby number ($F_h=\Gamma / \pi a^2 N$, $\hbox{\it Ro}= \Gamma / \pi a^2 f$, where $\Gamma$ is the circulation of each vortex, $a$ the vortex radius, $N$ the Brunt–Väisälä frequency and $f$ the Coriolis parameter). When $\hbox{\it Ro}\,{=}\,\infty$, the scaling $\lambda \,{\propto}\, F_h b$ found in the companion paper Otheguy et al. (2006) is recovered. When $\hbox{\it Ro} \,{\rightarrow}\, 0$, $\lambda \,{\propto}\,b f/N$ in agreement with the quasi-geostrophic theory. In contrast to previous results, the wavelength is found to depend on the separation distance between the two vortices $b$, and not on the vortex radius $a$.

The evolution of a counter-rotating vortex pair in a stably stratified fluid is investigated using direct numerical simulations. The study focuses on the short-wavelength elliptic instability occurring in this flow and the subsequent decay of the vortices. Depending on the level of stratification, as characterized by the Froude number which indicates the time scale of buoyancy to that of the instability, and the stage of evolution, stratification effects may significantly alter the behaviour of the flow. In the case of weak to moderate stratification, the elliptic instability develops qualitatively in the same manner as in unstratified fluid. The primary effect of stratification is to reduce the vortex separation distance which enhances the mutually induced strain. Consequently, the instability has an earlier onset and higher growth rate with increasing stratification. The behaviour is essentially described by linear stability theory for unstratified flow if the varying separation distance is taken into account. On the other hand, the final breakdown and decay of the flow may be greatly modified by stratification since buoyancy effects eventually emerge after sufficient time has elapsed. The decay is enhanced owing to additional mechanisms not present in unstratified flow. Secondary vertical vortex structures form between the primary vortices promoting fluid exchange in the transverse direction. Detrainment of fluid from the primary vortices by the generated baroclinic torque also contributes to the more rapid breakdown of the flow. In the case of strong stratification, in which the time scale of buoyancy is comparable to that of the instability, the flow is significantly altered. As a result of strong baroclinic torque, the primary vortices are brought together and detrainment occurs earlier. The associated reduction in radii of the vortices results in a higher axial wave mode and a more complex radial structure of the instability. Detrainment and mixing accelerate their decay. Late time evolution is dominated by the successive generation of alternate signed baroclinic torque which results in an oscillation of the total flow circulation at the buoyancy frequency.

The transitions of the flow in an enclosed cylinder driven by the constant rotation of an endwall, from steady axisymmetric flow to aperiodic flow characterized by intermittent bursting dynamics where all the spatial and spatio-temporal symmetries have been broken, is studied numerically. The problem is controlled by two parameters, the Reynolds number and the cylinder aspect ratio. We vary the Reynolds number, fixing the aspect ratio at a value where the primary bifurcation of the axisymmetric steady state is to an axisymmetric periodic flow. The final transition to weak turbulence, however, is governed by a non-axisymmetric branch of rotating waves, which is the primary mode at lower aspect ratios, and the various branches of modulated rotating waves associated with subsequent bifurcations from the rotating wave. We study in detail the spatio-temporal characteristics of the various states encountered along the way, and how the symmetry of the problem impacts on the transition dynamics.

Theory and numerical simulations are used to investigate the concentration fluctuations and the microstructure in dilute sedimenting suspensions of orientable and deformable particles at zero Reynolds number. The case of orientable particles is studied using prolate and oblate spheroids, while viscous droplets in the small deformation regime illustrate the effects of deformability. An efficient method based on a point-particle approximation and on smooth localized force representations is implemented to simulate full-scale suspensions with both periodic and slip boundaries, where the latter are used to qualitatively reproduce the effects of horizontal walls. The concentration instability predicted theoretically for suspensions of spheroids is captured in the simulations, and we find that including horizontal walls provides a mechanism for wavenumber selection, in contrast to periodic systems in which the longest wavelength set by the size of the container dominates. A theoretical model for the case of slightly deformable particles is developed, and a linear stability analysis shows that such suspensions are also unstable to concentration fluctuations under sedimentation. In the absence of diffusion, the model predicts that density fluctuations are equally unstable at all wavelengths, but we show that diffusion, whether Brownian or hydrodynamic, should damp high-wavenumber fluctuations. Simulations are also performed for deformable particles, and again an instability is observed that shows a similar mechanism for the wavenumber selection in finite containers. Our results demonstrate that all sedimentation processes of orientable or deformable particles are subject to spontaneous concentration inhomogeneities, which control the sedimentation rates in these systems.

We consider the problem of nonlinear oscillatory convection in a horizontal mushy layer rotating about a vertical axis. Under a near-eutectic approximation and the limit of large far-field temperature, we determine the stable and unstable oscillatory solutions of the weakly nonlinear problem by using perturbation and stability analyses. It was found that depending on the values of the parameters, supercritical simple travelling modes of convection in the form of hexagons, squares, rectangles or rolls can become stable and preferred, provided the value of the rotation parameter $\tau$ is not too small and is below some value, which can depend on the other parameter values. Each supercritical form of the oscillatory convection becomes subcritical as $\tau$ increases beyond some value, and each subcritical form of the oscillatory convection is unstable. In contrast to the non-rotating case, qualitative properties of the left-travelling modes of convection are different from those of the right-travelling modes, and such qualitative difference is found to be due to the interactions between the local solid fraction and the Coriolis term in the momentum-Darcy equation.

We study dynamo processes in a convective layer of Boussinesq fluid rotating about the vertical. Irrespective of rotation, if the magnetic Reynolds number is large enough, the convection acts as an efficient small-scale dynamo with a growth time comparable with the turnover time and capable of generating a substantial amount of magnetic energy. When the rotation is important (large Taylor number) the characteristic horizontal scale of the convection decreases and the flow develops a well-defined distribution of kinetic helicity antisymmetric about the mid-plane. We find no convincing evidence of large-scale dynamo action associated with this helicity distribution. Even when the rotation is strong, the magnetic energy at large scales remains small, and comparable with that in the non-rotating case. By externally imposing a uniform field, we measure the average electromotive force. We find this quantity to be extremely strongly fluctuating, and are able to compute the associated $\alpha$-effect only after very long time averaging. In those cases for which reasonable convergence is achieved, the $\alpha$-effect is small, and controlled by the magnetic diffusivity. Thus we demonstrate the existence of a system whose small-scale dynamo growth rate is turbulent, i.e. independent of diffusivity, but whose $\alpha$-effect is laminar, i.e. dependent on diffusivity. The implications of these results to the problem of the generation of strong mean fields are discussed.

We study the problem of nonlinear development of oscillatory convective instability in a two-dimensional mushy layer during solidification of a binary mixture. We adopt the near-eutectic limit, making the problem analytically tractable using standard perturbation techniques. We consider also a distinguished limit of large Stefan number, which allows a destabilization of the system to an oscillatory mode of convection. We find that either travelling waves or standing waves can be supercritically stable, depending strongly on the sensitivity of permeability of the mushy layer to variations in the local solid fraction: mushy-layer systems with relatively weak sensitivity are more likely to select travelling waves rather than standing waves in the nonlinear regime. Furthermore, the decrease in permeability is found to promote the subcritical, and hence more unstable, primary oscillatory states. In addition to mapping out the location of different stable oscillatory patterns in the available parameter space, we give the detailed spatio-temporal structure of the corresponding thermal, flow and solid-fraction fields within the mushy layer, as well as the local bulk composition in the resulting eutectic solid.