In 1954 R. A. Bagnold published his seminal findings on the rheological properties of a liquid–solid suspension. Although this work has been cited extensively over the last fifty years, there has not been a critical review of the experiments. The purpose of this study is to examine the work and to suggest an alternative reason for the experimental findings. The concentric cylinder rheometer was designed to measure simultaneously the shear and normal forces for a wide range of solid concentrations, fluid viscosities and shear rates. As presented by Bagnold, the analysis and experiments demonstrated that the shear and normal forces depended linearly on the shear rate in the ‘macro-viscous’ regime; as the grain-to-grain interactions increased in the ‘grain-inertia’ regime, the stresses depended on the square of the shear rate and were independent of the fluid viscosity. These results, however, appear to be dictated by the design of the experimental facility. In Bagnold’s experiments, the height (h) of the rheometer was relatively short compared to the spacing (t) between the rotating outer and stationary inner cylinder (h/t = 4.6). Since the top and bottom end plates rotated with the outer cylinder, the flow contained two axisymmetric counter-rotating cells in which flow moved outward along the end plates and inward through the central region of the annulus. At higher Reynolds numbers, these cells contributed significantly to the measured torque, as demonstrated by comparing Bagnold's pure-fluid measurements with studies on laminar-to-turbulent transitions that pre-date the 1954 study. By accounting for the torque along the end walls, Bagnold’s shear stress measurements can be estimated by modelling the liquid–solid mixture as a Newtonian fluid with a corrected viscosity that depends on the solids concentration. An analysis of the normal stress measurements was problematic because the gross measurements were not reported and could not be obtained.

This is a laboratory study of salt fingers at low Rayleigh numbers. We report on the stability boundary in the (RS, RT)-plane (where RS and RT are the salt and heat Rayleigh numbers respectively), the wavenumber of the observed fingers, and the planform. In this low RS, RT range, fingers have width comparable to their height, as predicted by linear stability theory. The planform appears to be close-packed polygonal cells when they are formed on curved profiles of temperature and salinity. However, the planform is distinctly rolls when care is taken to approximate linear profiles.

SO(3) and SO(2) decompositions of numerical channel flow turbulence are performed. The decompositions are used to probe, characterize, and quantify anisotropic structures in the flow. Close to the wall, the anisotropic modes are dominant and reveal the flow structures. The dominance of the (j, m) = (2, 1) mode of the SO(3) decomposition in the buffer layer is associated with hairpin vortices. The SO(2) decomposition in planes parallel to the walls allows us also to access the regions very close to the wall. In those regions we have found that the strong enhancement of intermittency can be explained in terms of streaklike structures and their signatures in the m = 2 and m = 4 modes of the SO(2) decomposition.

We present closures for the drag and virtual mass force terms appearing in a two-fluid model for flow of a mixture consisting of uniformly sized gas bubbles dispersed in a liquid. These closures were deduced through computational experiments performed using an implicit formulation of the lattice Boltzmann method with a BGK collision model. Unlike the explicit schemes described in the literature, this implicit implementation requires iterative calculations, which, however, are local in nature. While the computational cost per time step is modestly increased, the implicit scheme dramatically expands the parameter space in multiphase flow calculations which can be simulated economically. The closure relations obtained in our study are limited to a regular array of uniformly sized bubbles and were obtained by simulating the rise behaviour of a single bubble in a periodic box. The effect of volume fraction on the rise characteristics was probed by changing the size of the box relative to that of the bubble. While spherical bubbles exhibited the expected hindered rise behaviour, highly distorted bubbles tended to rise cooperatively. The closure for the drag force, obtained in our study through computational experiments, captured both hindered and cooperative rise. A simple model for the virtual mass coefficient, applicable to both spherical and distorted bubbles, was also obtained by fitting simulation results. The virtual mass coefficient for isolated bubbles could be correlated with the aspect ratio of the bubbles.

Buoyancy-driven surface currents were generated in the laboratory by releasing buoyant fluid from a source adjacent to a vertical boundary in a rotating container. Different bottom topographies that simulate both a continental slope and a continental ridge were introduced in the container. The topography modified the flow in comparison with the at bottom case where the current grew in width and depth until it became unstable once to non-axisymmetric disturbances. However, when topography was introduced a second instability of the buoyancy-driven current was observed. The most important parameter describing the flow is the ratio of continental shelf width W to the width L* of the current at the onset of the instability. The values of L* for the first instability, and L*−W for the second instability were not influenced by the topography and were 2–6 times the Rossby radius. Thus, the parameter describing the flow can be expressed as the ratio of the width of the continental shelf to the Rossby radius. When this ratio is larger than 2–6 the second instability was observed on the current front. A continental ridge allowed the disturbance to grow to larger amplitude with formation of eddies and fronts, while a gentle continental slope reduced the growth rate and amplitude of the most unstable mode, when compared to the continental ridge topography. When present, eddies did not separate from the main current, and remained near the shelf break. On the other hand, for the largest values of the Rossby radius the first instability was suppressed and the flow was observed to remain stable. A small but significant variation was found in the wavelength of the first instability, which was smaller for a current over topography than over a flat bottom.

The stability of steep gravity–capillary solitary waves in deep water is numerically investigated using the full nonlinear water-wave equations with surface tension. Out of the two solution branches that bifurcate at the minimum gravity–capillary phase speed, solitary waves of depression are found to be stable both in the small-amplitude limit when they are in the form of wavepackets and at finite steepness when they consist of a single trough, consistent with observations. The elevation-wave solution branch, on the other hand, is unstable close to the bifurcation point but becomes stable at finite steepness as a limit point is passed and the wave profile features two well-separated troughs. Motivated by the experiments of Longuet-Higgins & Zhang (1997), we also consider the forced problem of a localized pressure distribution applied to the free surface of a stream with speed below the minimum gravity–capillary phase speed. We find that the finite-amplitude forced solitary-wave solution branch computed by Vanden-Broeck & Dias (1992) is unstable but the branch corresponding to Rayleigh’s linearized solution is stable, in agreement also with a weakly nonlinear analysis based on a forced nonlinear Schrödinger equation. The significance of viscous effects is assessed using the approach proposed by Longuet-Higgins (1997): while for free elevation waves the instability predicted on the basis of potential-flow theory is relatively weak compared with viscous damping, the opposite turns out to be the case in the forced problem when the forcing is strong. In this régime, which is relevant to the experiments of Longuet-Higgins & Zhang (1997), the effects of instability can easily dominate viscous effects, and the results of the stability analysis are used to propose a theoretical explanation for the persistent unsteadiness of the forced wave profiles observed in the experiments.

When a bubble collapses mildly the interior pressure field is spatially uniform; this is an assumption often made to close the Rayleigh–Plesset equation of bubble dynamics. The present work is a study of the self-consistency of this assumption, particularly in the case of violent collapses. To begin, an approximation is developed for a spatially non-uniform pressure field, which in a violent collapse is inertially driven. Comparisons of this approximation show good agreement with direct numerical solutions of the compressible Navier–Stokes equations with heat and mass transfer. With knowledge of the departures from pressure uniformity in strongly forced bubbles, one is in a position to develop criteria to assess when pressure uniformity is a physically valid assumption, as well as the significance of wave motion in the gas. An examination of the Rayleigh–Plesset equation reveals that its solutions are quite accurate even in the case of significant inertially driven spatial inhomogeneity in the pressure field, and even when wave-like motions in the gas are present. This extends the range of utility of the Rayleigh–Plesset equation well into the regime where the Mach number is no longer small; at the same time the theory sheds light on the interior of a strongly forced bubble.

A liquid bridge is a column of liquid, pinned at each end. Here we analyse the stability of a bridge pinned between planar electrodes held at different potentials and surrounded by a non-conducting, dielectric gas. In the absence of electric fields, surface tension destabilizes bridges with aspect ratios (length/diameter) greater than π. Here we describe how electrical forces counteract surface tension, using a linearized model. When the liquid is treated as an Ohmic conductor, the specific conductivity level is irrelevant and only the dielectric properties of the bridge and the surrounding gas are involved. Fourier series and a biharmonic, biorthogonal set of Papkovich–Fadle functions are used to formulate an eigenvalue problem. Numerical solutions disclose that the most unstable axisymmetric deformation is antisymmetric with respect to the bridge’s midplane. It is shown that whilst a bridge whose length exceeds its circumference may be unstable, a sufficiently strong axial field provides stability if the dielectric constant of the bridge exceeds that of the surrounding fluid. Conversely, a field destabilizes a bridge whose dielectric constant is lower than that of its surroundings, even when its aspect ratio is less than π. Bridge behaviour is sensitive to the presence of conduction along the surface and much higher fields are required for stability when surface transport is present. The theoretical results are compared with experimental work (Burcham & Saville 2000) that demonstrated how a field stabilizes an otherwise unstable configuration. According to the experiments, the bridge undergoes two asymmetric transitions (cylinder-to-amphora and pinch-off) as the field is reduced. Agreement between theory and experiment for the field strength at the pinch-off transition is excellent, but less so for the change from cylinder to amphora. Using surface conductivity as an adjustable parameter brings theory and experiment into agreement.

Liquid helium at 4.2 K has a viscosity that is about 40 times smaller than that of water at room temperature, and about 600 times smaller than that of air at atmospheric pressure. It is therefore a convenient fluid for generating in a table-top apparatus turbulent flows at high Reynolds numbers that require large air and water facilities. Here, we produce turbulence behind towed grids in a liquid helium chamber that is 5 cm2 in cross-section at mesh Reynolds numbers of up to 7×105. Liquid nitrogen is intermediate in its viscosity as well as refrigeration demands, and so we also exploit its use to generate towed-grid turbulence up to mesh Reynolds number of about 2×104. In both instances, we map two-dimensional fields of velocity vectors using particle image velocimetry, and compare the data with those in water and air.

By experiments and supporting computations we investigate two methods of transport enhancement in two-dimensional open cellular flows with inertia. First, we introduce a spatial dependence in the velocity field by periodic modulation of the shape of the wall driving the flow; this perturbs the steady-state streamlines in the direction perpendicular to the main flow. Second, we introduce a time dependence through transient acceleration–deceleration of a flat wall driving the flow; surprisingly, even though the streamline portrait changes very little during the transient, there is still significant transport enhancement. The range of Reynolds and Reynolds–Strouhal numbers studied is 7.7[les ]Re[les ]46.5 and 0.52[les ]ReSr[les ]12.55 in the spatially dependent mode and 12[les ]Re[les ]93 and 0.26[les ]ReSr[les ]5.02 in the time-dependent mode. The transport is described theoretically via lobe dynamics. For both modifications, a curve with one maximum characterizes the various transport enhancement measures when plotted as a function of the forcing frequency. A qualitative analysis suggests that the exchange first increases linearly with the forcing frequency and then decreases as 1/Sr for large frequencies.

An improvement of the Stokesian Dynamics method for many-particle systems is presented. A direct calculation of the hydrodynamic interaction is used rather than imposing periodic boundary conditions. The two major difficulties concern the accuracy and the speed of calculations. The accuracy discussed in this work is not concerned with the lubrication correction but, rather, focuses on the multipole expansion which until now has only been formulated up to the so-called FTS version or the first order of force moments. This is improved systematically by a real-space multipole expansion with force moments and velocity moments evaluated at the centre of the particles, where the velocity moments are calculated through the velocity derivatives; the introduction of the velocity derivatives makes the formulation and its extensions straightforward. The reduction of the moments into irreducible form is achieved by the Cartesian irreducible tensor. The reduction is essential to form a well-defined linear set of equations as a generalized mobility problem. The order of truncation is not limited in principle, and explicit calculations of two-body problems are shown with order up to 7. The calculating speed is improved by a conjugate-gradient-type iterative method which consists of a dot-product between the generalized mobility matrix and the force moments as a trial value in each iteration. This provides an O(N2) scheme where N is the number of particles in the system. Further improvement is achieved by the fast multipole method for the calculation of the generalized mobility problem in each iteration, and an O(N) scheme for the non-adaptive version is obtained. Real problems are studied on systems with N = 400 000 particles. For mobility problems the number of iterations is constant and an O(N) performance is achieved; however for resistance problems the number of iterations increases as almost N1/2 with a high accuracy of 10−6 and the total cost seems to be O(N3/2).

We consider axisymmetric magnetohydrodynamic motion in a spherical shell driven by rotating the inner boundary relative to the stationary outer boundary – spherical Couette flow. The inner solid sphere is rigid with the same electrical conductivity as the surrounding fluid; the outer rigid boundary is an insulator. A force-free dipole magnetic field is maintained by a dipole source at the centre. For strong imposed fields (as measured by the Hartmann number M), the numerical simulations of Dormy et al. (1998) showed that a super-rotating shear layer (with angular velocity about 50% above the angular velocity of the inner core) is attached to the magnetic field line [Cscr ] tangent to the outer boundary at the equatorial plane of symmetry. At large M, we obtain analytically the mainstream solution valid outside all boundary layers by application of Hartmann jump conditions across the inner- and outer-sphere boundary layers. We formulate the large-M boundary layer problem for the free shear layer of width M−1/2 containing [Cscr ] and solve it numerically. The super-rotation can be understood in terms of the nature of the meridional electric current flow in the shear layer, which is fed by the outer-sphere Hartmann layer. Importantly, a large fraction of the current entering the shear layer is tightly focused and effectively released from a point source at the equator triggered by the tangency of the [Cscr ]-line. The current injected by the source follows the [Cscr ]-line closely but spreads laterally due to diffusion. In consequence, a strong azimuthal Lorentz force is produced, which takes opposite signs either side of the [Cscr ]-line; order-unity super-rotation results on the equatorial side. In fact, the point source is the small equatorial Hartmann layer of radial width M−2/3 ([Lt ]M−1/2) and latitudinal extent M−1/3. We construct its analytic solution and so determine an inward displacement width O(M−2/3) of the free shear layer. We compare our numerical solution of the free shear layer problem with our numerical solution of the full governing equations for M in excess of 104. We obtain excellent agreement. Some of our more testing comparisons are significantly improved by incorporating the shear layer displacement caused by the equatorial Hartmann layer.

We study the formation of shocks on the surface of a granular material draining through an orifice at the bottom of a quasi-two-dimensional silo. At high flow rates, the surface is observed to deviate strongly from a smooth linear inclined profile, giving way to a sharp discontinuity in the height of the surface near the bottom of the incline, the typical response of a choking flow such as encountered in a hydraulic jump in a Newtonian fluid like water. We present experimental results that characterize the conditions for the existence of such a jump, describe its structure and give an explanation for its occurrence.

The instability of a bed of particles sheared by a viscous fluid is investigated theoretically. The viscous flow over the wavy bed is first calculated, and the bed shear stress is derived. The particle transport rate induced by this bed shear stress is calculated from the viscous resuspension theory of Leighton & Acrivos (1986). Mass conservation of the particles then gives explicit expressions for the wave velocity and growth rate, which depend on four dimensionless parameters: the wavenumber, the fluid thickness, a viscous length and the shear stress. The mechanism of the instability is given. It appears that for high enough fluid-layer thickness, long-wave instability arises as soon as grains move, while short waves are stabilized by gravity. For smaller fluid thickness, the destabilizing effect of fluid inertia is reduced, so that the moving at bed is stable for small shear stress, and unstable for high shear stress. The most amplified wavelength scales with the viscous length, in agreement with the few available experiments for small particle Reynolds numbers. The results are also compared with related studies for turbulent flow.

This study quantifies the dynamics of actuation for the temporally forced, round gas jet injected transversely into a crossflow, and incorporates these dynamics in developing a methodology for open loop jet control. A linear model for the dynamics of the forced jet actuation is used to develop a dynamic compensator for the actuator. When the compensator is applied, it allows the jet to be forced in a manner which results in a more precisely prescribed, temporally varying exit velocity, the RMS amplitude of perturbation of which can be made independent of the forcing frequency. Use of the compensator allows straightforward comparisons among different conditions for jet excitation. Clear identification can be made of specific excitation frequencies and characteristic temporal pulse widths which optimize transverse jet penetration and spread through the formation of distinct, deeply penetrating vortex structures.

The paper is devoted to the theoretical investigation of the possible existence of stationary mixing layers and of their structure in nearly perfectly conducting, nearly inviscid fluids with a longitudinal magnetic field. A system of two equations is used, which generalizes the well-known Blasius equation (for flow around a semi-infinite plate) to the case under consideration. The system depends on the magnetic Prandtl number, Pm=ν/νm, where ν and νm are the usual and the magnetic viscosities, respectively.

For the existence of stationary flows the ratio between the flow velocity vx and the Alfvén velocity cA=Hx/(4πρ)1/2 (ρ being the fluid density) plays a critical role. Super-Alfvén (vx>cA) flows are possible at any value of Pm and for any values of vx and Hx on the layer boundaries. Sub-Alfvén (vx<cA) stationary flows are impossible at any value of Pm and for any values of the differences in vx and Hx across the layer, except for two cases: Pm=0 and Pm=1. When Pm=0, i.e. when the fluid is strictly inviscid, ν=0, flow is possible in both the super- and sub-Alfvén regimes; however, the magnetic field must be uniform, Hx=const, Hy=0 in this case. For Pm=1 both flow regimes are also possible; however, the sub-Alfvén flow is possible only for a definite relationship between the magnetic field and velocity differences: ΔHx=−δvx (in corresponding units). For the case where the relative differences in vx and Hx across the layer are small, Δvx[Lt ]vx, ΔHx[Lt ]Hx, solutions are obtained in explicit form for arbitrary Pm (here vx and Hx are averaged over the layer). For the specific case Pm=1, exact analytical solutions of basic system are found and studied in detail.

The variances of the fluid-particle acceleration and of the pressure-gradient and viscous force are given. The scaling parameters for these variances are velocity statistics measureable with a single-wire anemometer. For both high and low Reynolds numbers, asymptotic scaling formulas are given; these agree quantitatively with DNS data. Thus, the scaling can be presumed known for all Reynolds numbers. Fluid-particle acceleration variance does not obey K41 scaling at any Reynolds number; this is consistent with recent experimental data. The non-dimensional pressure-gradient variance named λT/λP is shown to be obsolete.

The objective of the present work is to test experimentally the two-phase modelling approach which is widely used in fluidization. A difficulty of this way of modelling fluidized beds is the use of empirical relations in order to close the system of equations describing the fluidized bed as a two-phase continuum, especially concerning the description of the solid phase. We performed an experimental investigation of the primary wavy instability of liquid-fluidized beds. Experiments demonstrate that the wave amplitude saturates up the bed and we were able to measure the precise shape of this voidage wave. We then related this shape to the unknown solid phase viscosity and pressure functions of a simple two-phase model with a Newtonian stress-tensor for the solid phase. We found the scaling laws and the particle concentration dependence for these two quantities. It appears that this simplest model is quite satisfactory to describe the one-dimensional voidage waves in the limited range of parameters that we have studied. In our experimental conditions, the drag on the particles nearly balances their weight corrected for buoyancy, the small imbalance being mostly accounted for by solid phase viscous stress with a much smaller contribution from the solid phase pressure.

Scale model experiments on axially symmetric bodies exhibiting fore–aft asymmetry are described. Body shapes are specified by a three parameter equation: two of the parameters (a and b) describe the length and breadth of the body and the third (c) the degree of asymmetry. Objects of this shape orientate as they sediment downwards under gravity until the narrower end lies uppermost, after which they fall vertically downward with no further change in orientation. For the range of parameters investigated the sedimentation velocities, both when vertical and horizontal, are governed principally by a and b, while the rate of orientation is determined by c. The sedimentation characteristics of bodies which cannot be described exactly by the equation can be predicted approximately using best-fit values for a, b and c. These results are applied to consider the role of front–rear asymmetry in ciliated free-swimming micro-organisms. The shape asymmetry is probably sufficient to account for the observed orientation rates in the ciliated protozoan Paramecium. It is suggested that these results may be used to deduce the sedimentation behaviour of ciliates from microscope images of individual cells. In small flagellates such as Chlamydomonas the orientating effects of the protruding flagella are much larger than the effects of cell body asymmetry. The extreme sensitivity of the orientation rate to slight changes in body shape and flagellar beat patterns may explain why experiments to distinguish between various orientational mechanisms involved in gravitaxis have in the past produced equivocal results.

We consider viscous shear flow of a monolayer of solid spheres and discuss the effect that microscopic particle surface roughness has on the stress in the suspension. We consider effects both within and outside the dilute régime. Away from jamming concentrations, the viscosity is lowered by surface roughness, and for dilute suspensions it is insensitive to friction between the particles. Outside the dilute region, the viscosity increases with increasing friction coefficient. For a dilute system, roughness causes a negative first normal stress difference (N1) at order c2 in particle area concentration. The magnitude of N1 increases with increasing roughness height in the dilute limit but the trend reverses for more concentrated systems. N1 is largely insensitive to interparticle friction. The dilute results are in accord with the three-dimensional results of our earlier work (Wilson & Davis 2000), but with a correction to the sign of the tangential friction force.