This article examines a reduced form of the ‘purely dissipative’ model proposed several years ago as a general continuum model for the rheology of non-colloidal particle dispersions, ranging from Stokesian suspensions to non-cohesive granular media. Essential to the model is a positive-definite viscosity tensor $\boldsymbol{\eta}$, depending on the history of deformation and providing a crucial restriction on related models for anisotropic fluids and suspensions. In the present treatment, $\boldsymbol{\eta}$ is assumed to be as an isotropic function of a history-dependent second-rank ‘texture’ or ‘fabric’ tensor ${\textsfbi A}$. A formula for $\boldsymbol{\eta}({\textsfbi A})$ borrowed from the analogous theory of linear elasticity, and its subsequent expansion for weak anisotropy provides an explicit expression for the stress tensor in terms of fabric, strain-rate and eight material constants.

Detailed consideration is given to the special case of Stokesian suspensions, which represent an intriguing subset of memory materials without characteristic time. For this idealized fluid one finds linear dependence of all stresses, including viscometric normal stress, on present deformation rate, with the provision for an arbitrary fabric evolution (‘thixotropy’) in unsteady deformations. As a concrete example, a co-rotational memory integral is adopted for ${\textsfbi A}$ in terms of strain-rate history, and a memory kernel with two-mode exponential relaxation gives close agreement with the rather sparse experimental data on transient shear experiments. In the proposed model, an extremely rapid mode of relaxation is required to mimic the incomplete reversal of stress observed in experiments involving abrupt reversal of steady shearing, supporting the conclusion of others that non-hydrodynamic effects, with breaking of Stokesian symmetry, may be implicated in such experiments.

Qualitative comparisons are made to a closely related model, derived from a micro-mechanical analysis of Stokesian suspensions, but also involving non-Stokesian effects.

The present analysis may point the way to improved micro-mechanical analysis and to further experiments. Possible extensions of the model to the viscoplasticity of dry and liquid-saturated granular media also are discussed briefly.

The evolution of a stratified shear layer with mean shear in the horizontal direction, orthogonal to gravity, is numerically investigated with focus on the structural organization of the vorticity and density fields. Although the Reynolds number of the flow increases with time, facilitating instabilities and turbulence, the bulk Richardson number signifying the level of stratification also increases. Remarkably rich dynamics is found: turbulence; the emergence of coherent core/braid regions from turbulence; formation of a lattice of dislocated vortex cores connected by thin horizontal sheets of collapsed density and vorticity; density-driven intrusions at the edges of the shear layer; and internal wave generation and propagation. Stratification introduces significant vertical variability although it inhibits the vertical velocity. The molecular dissipation of turbulent kinetic energy and of turbulent potential energy are both found to be substantial even in the case with highest stratification, and primarily concentrated in thin horizontal sheets. The simulation data are used to help explain how buoyancy induces the emergence of columnar vortex cores from turbulence and then dislocates these cores to eventually form a lattice of ‘pancake’ eddies connected by thin sheets with large vertical shear (horizontal vorticity) and density gradient.

The present effort documents the population trends of prograde and retrograde spanwise vortex cores in wall turbulence outside the buffer layer. Large ensembles of instantaneous velocity fields are acquired by particle-image velocimetry in the streamwise–wall-normal plane of both turbulent channel flow at $\hbox{\it Re}_\tau\equiv u_*\delta/\nu=570$, 1185 and 1760 and a zero-pressure-gradient turbulent boundary layer at $\hbox{\it Re}_\tau=1400$, 2350 and 3450. Substantial numbers of prograde spanwise vortices are found to populate the inner boundary of the log layer of both flows and most of these vortices have structural signatures consistent with the heads of hairpin vortices. In contrast, retrograde vortices are most prominent at the outer edge of the log layer, often nesting near clusters of prograde vortices. Appropriate Reynolds-number scalings for outer- and inner-scaled population densities of prograde and retrograde vortices are determined. However, the Re$_\tau=570$ channel-flow case deviates from these scalings, indicating that it suffers from low-Re effects. When the population densities are recast in terms of fractions of resolved prograde and retrograde spanwise vortices, similarity is observed for $100\,{<}\,y^+\,{<}\,0.8\delta^+$ in channel flow and in both flows for $100\,{<}\,y^+\,{<}\,0.3\delta^+$ over the Re$_\tau$ range studied. The fraction of retrograde vortices increases slightly with $Re_\tau$ beyond the log layer in both flows, suggesting that they may play an increasingly important role at higher Reynolds numbers. Finally, while the overall prograde and retrograde population trends of channel flow and the boundary layer show little difference for $y\,{<}\,0.45\delta$, the retrograde populations differ considerably beyond this point, highlighting the influence of the opposing wall in channel flow.

Adaptive DNS/LES (direct numerical simulation/large-eddy simulation) is used to compute the drag coefficient $c_D$ for the flow past a sphere at Reynolds number $\hbox{\it Re}\,{=}\,10^4$. Using less than $10^5$ mesh points, $c_D$ is computed to an accuracy of a few percent, corresponding to experimental precision, which is at least an order of magnitude cheaper than standard non-adaptive LES computations in the literature. Adaptive DNS/LES is a General Galerkin G2 method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate solutions to the Navier–Stokes equations, with the mesh being adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, in this paper $c_D$. Both the stopping criterion and the mesh refinement strategy are based on a posteriori error estimates, in the form of a space–time integral of residuals multiplied by derivatives of the solution of an associated dual problem, linearized at the approximate solution, and with data coupling to the output of interest. There is no filtering of the equations, and thus no Reynolds stresses are introduced that need modelling. The stabilization in the numerical method is acting as a simple turbulence model.

We present numerical simulations of forced rotating stratified turbulence dominated by vortical motion (i.e. with potential vorticity). Strong stratification and various rotation rates are considered, corresponding to a small Froude number and a wide range of Rossby numbers $\hbox{\it Ro}$ spanning the regimes of stratified turbulence ($\hbox{\it Ro}\,{=}\,\infty$) to quasi-geostrophic turbulence ($\hbox{\it Ro}\,{\ll}\,1$). We examine how the energy spectra and characteristic vertical scale of the turbulence vary with Rossby number between these two regimes. The separate dependence on $N/f$, where $N$ is the Brunt–Väisälä frequency and $f$ is the Coriolis parameter, is found to be of secondary importance. As the macroscale $\hbox{\it Ro}$ decreases below 0.4 and the microscale $\hbox{\it Ro}$ (at our resolution) decreases below 3, the horizontal wavenumber energy spectrum steepens and the flat range in the vertical wavenumber spectrum increases in amplitude and decreases in length. At large Rossby numbers, the vertical scale $H$ is proportional to the stratified turbulence value $U/N$, where $U$ is the root mean square velocity. At small $\hbox{\it Ro}$, $H$ takes the quasi-geostrophic form $(f/N)L$, where $L$ is the horizontal scale of the flow. Implications of these findings for numerical atmosphere and ocean modelling are discussed.

Microstreaming from oscillating bubbles is known to induce vigorous vortex flow. Here we show how to harness the power of bubble streaming in an experiment to achieve directed transport flow of high velocity, allowing design and manufacture of microfluidic MEMS devices. By combining oscillating bubbles with solid protrusions positioned on a patterned substrate, solid beads and lipid vesicles are guided in desired directions without microchannels. Simultaneously, the flow exerts controlled localized forces capable of opening and reclosing lipid membranes.

In order to understand the rheological and transport properties of a suspension of swimming micro-organisms, it is necessary to analyse the fluid-dynamical interaction of pairs of such swimming cells. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). The effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The interaction of two squirmers is calculated analytically for the limits of small and large separations and is also calculated numerically using a boundary-element method. The analytical and the numerical results for the translational–rotational velocities and for the stresslet of two squirmers correspond very well. We sought to generate a database for an interacting pair of squirmers from which one can easily predict the motion of a collection of squirmers. The behaviour of two interacting squirmers is discussed phenomenologically, too. The results for the trajectories of two squirmers show that first the squirmers attract each other, then they change their orientation dramatically when they are in near contact and finally they separate from each other. The effect of bottom-heaviness is considerable. Restricting the trajectories to two dimensions is shown to give misleading results. Some movies of interacting squirmers are available with the online version of the paper.

The flipping motion of a blood platelet convected under the action of a simple shear flow over a substrate is discussed. The platelet is modelled as a rigid oblate spheroid with aspect ratio equal to 0.25 whose axis of revolution is perpendicular to the vorticity of the simple shear flow. The particle motion from a given initial position is computed using a boundary element method for Stokes flow based on the double-layer representation. When the platelet is far from the wall, the motion is described by Jeffery's exact solution. As the platelet approaches the wall, the rate of rotation is reduced significantly when the platelet mid-plane is parallel to wall, and mildly when the mid-plane is perpendicular to the wall. Comparison with laboratory data indicates that wall effects alone do not explain the observed slow rate of rotation. It is proposed that a distributed adhesion force imparts to the particle an effective external force and torque at the nominal point of contact, and these slow down the rate of rotation. The process is demonstrated by computing the motion of an adhering platelet whose lowest point is immobilized under the action of a suitable force and torque.

Long gas bubbles rising in a vertical tube are observed to lose axial symmetry and become unstable in a downward liquid flow. In this paper an approximate linear stability analysis of this phenomenon is presented. It is found that, under the combined effect of gravity and the pressure gradient which drives the liquid flow, the relative velocity between the bubble and the liquid decreases with increasing downflow, which diminishes the stabilizing effect of convection. The decrease of the relative velocity is accompanied by a flattening of the bubble nose, which also has a destabilizing effect by strengthening the Rayleigh–Taylor instability at the bubble nose.

The speed of a fluid intrusion propagating along a sharp density interface is predicted using conservation of mass, momentum and energy. For the special case in which the intrusion density equals the depth-weighted mean density of the upper and lower layers, the theory of Holyer & Huppert (J. Fluid Mech., vol. 100, 1980, p. 739) predicts that the intrusion occupies one-half the total depth, its speed is one-half the interfacial long-wave speed and the interface ahead of the intrusion remains undisturbed. For all other intrusion densities, the interface is deflected vertically by a long wave that travels ahead of the intrusion and thereby changes the local upstream conditions. In these cases, the conservation equations must be matched to an exact solution of the two-layer shallow water equations, which describe the spatial evolution of the nonlinear wave. We obtain predictions for the intrusion speed that match closely with experiments and numerical simulations, and with a global energy balance analysis by Cheong, Keunen & Linden (J. Fluid Mech., vol. 552, 2006, p. 1). Since the latter does not explicitly include the energetics of the upstream wave, it is inferred that the energy carried by the wave is a small fraction of the intrusion energy. However, the new more detailed model also shows that the kinematic influence of the upstream wave in changing the level of the interface is a critical component of the flow that has previously been ignored.

Numerical computations and order of magnitude estimates are presented for the periodic generation and coalescence of bubbles due to the injection of a constant flow rate of a gas through a circular orifice at the bottom wall of an inviscid dielectric or very polar liquid that is at rest and subject to a uniform vertical electric field far from the orifice. The problem depends on five dimensionless parameters: a Bond number based on the radius of the orifice; Weber and electric Bond numbers whose square roots are dimensionless measures of the flow rate of gas and the applied electric field; the dielectric constant of the liquid; and the contact angle of the liquid with the bottom wall. The bubbles that grow quasi-statically at the orifice for small values of the Weber number are always elongated vertically by the electric stress that acts on their surface when an electric field is applied. The volume of these bubbles at detachment may reach a maximum at a certain value of the electric Bond number, if the Bond number is sufficiently small, or decrease monotonically with the electric Bond number if the Bond number is larger. In both cases the bubbling ceases to be periodic beyond a certain value of the electric Bond number, apparently giving way to more complex bubbling regimes, which are not investigated here. Bubble interaction and eventually coalescence occur when the Weber number is increased keeping the electric Bond number in the range of periodic bubbling. Different periodic regimes are described. It is shown that a moderate electric field may increase the value of the Weber number above which coalescence occurs without changing the shape of the bubbles much. A large electric field may suppress coalescence but it also favours the development of upward and downward jets that cross the bubbles and may cause their breakdown.

This paper reports on a lattice Boltzmann simulation of laminar mixing in a coaxial microreactor at low Reynolds numbers. The main focus of the study was to compare the effectiveness of a multi-holed baffle plate, a set of $2 \,{\times}\, 2$ square bars, and a flat-plate arrangement on the enhancement of reagent mixing inside the reactor. It was found that all three mixers increased mixing through the mechanism of generating coherent structures, which in turn increased interface contact between the fluid streams. However, for this particular microreactor, the efficiency of each mixer depended on its ability to generate coherent structures with high helicity and thereby inducing a swirl motion to the flow.

Transient growth of small disturbances may lead to the initiation of the laminar–turbulent transition process. Such growth in a two-dimensional laminar flow in a channel with a corrugated wall is analysed. The corrugation has a wavy form that is completely characterized by its wavenumber and amplitude. The maximum possible growth and the form of the initial disturbance that leads to such growth have been identified for each form of the corrugation. The form that leads to the largest growth for a given corrugation amplitude, i.e. the optimal corrugation, has been found. It is shown that the corrugation acts as an amplifier for disturbances that are approximately optimal in the smooth channel case but has little effect in the other cases. The interplay between the modal (asymptotic) instability and the transient growth, and the use of the variable corrugation for modulation of the growth are discussed.

The work concerns the nonlinear dynamics of oceanic internal waves in resonance with a surface shear current. The resonance occurs when the celerity of the wave matches the mean flow speed at the surface. The evolution of weakly nonlinear waves long compared to the thickness of the upper mixed layer is found to be described by two linearly coupled equations (a linearized intermediate long wave equation and the Riemann wave equation). The presence of a pseudodifferential operator leads to qualitatively new features of the wave dynamics compared to the previously studied case of shallow water. The system is investigated primarily by means of numerical analysis. It possesses a variety of both periodic and solitary wave stationary solutions, including ‘delocalized solitons’ with a localized core and very small non-decaying oscillatory tails (throughout the paper we use the term ‘soliton’ as synonymous with ‘solitary wave’ and do not imply any integrability of the system). These ‘solitons’ are in linear resonance with infinitesimal waves, which in the evolutionary problem normally results in radiative damping. However, the rate of the energy losses proves to be so small, that these delocalized radiating solitons can be treated as quasi-stationary, that is, effectively, as true solitons at the characteristic time scales of the system. Moreover, they represent a very important class of intermediate asymptotics in the evolution of initial localized pulses. A typical pulse evolves into a sequence of solitary waves of all kinds, including the ‘delocalized’ ones, plus a decaying train of periodic waves. The remarkable feature of this evolution is that of all the products of the pulse fission (in a wide range of parameters of the initial pulse) the radiating solitons have by far the largest amplitudes. We argue that the radiating solitons acting as intermediate asymptotics of initial-value problems are a generic phenomenon not confined to the particular model under consideration.

A new mechanism for the instability of steady mountain waves is found through analysis of the linear stability problem. Steady flow of a hydrostatic stratified fluid is known to be unstable when the streamlines are at, or very close to, overturning. When the topography has multiple peaks, it is shown that this criterion can be superseded by an instability owing to a resonant triad interaction. For flow over two peaks, the threshold heights for instability are roughly half those which produce overturning streamlines. The mechanism behind the instability is the parametric amplification of counter-propagating gravity waves. The resonant nature of the instability is further illustrated by the existence of discrete peak-to-peak separation distances where the growth rate is a maximum.

We analyse the topology of vortex breakdown in a closed cylindrical container in the steady domain under variation of three parameters, the aspect ratio of the cylinder, the Reynolds number, and the ratio of the angular velocities of the covers. We develop a general post-processing method to obtain topological bifurcation diagrams from a database of simulations of two-dimensional flows and apply the method to axisymmetric simulations of the flow in the cylinder. Interpreting the diagrams with the aid of bifurcation theory, we obtain complete topological bifurcation diagrams for the rotation ratio in the interval [−0.04, 0.075]. In this narrow range we identify three codimension-3 bifurcation points which act as organising centres for the entire bifurcation diagram.

We present a broad range of measurements of the angular orientation $\theta_0(t)$ of the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection as a function of time. We used two cylindrical samples of different overall sizes, but each with its diameter nearly equal to its height. The fluid was water with a Prandtl number of 4.38. The time series $\theta_0(t)$ consisted of meanderings similar to a diffusive process, but in addition contained large and irregular spontaneous reorientation events through angles $\uDelta \theta$. We found that reorientations can occur by two distinct mechanisms. One consists of a rotation of the circulation plane without any major reduction of the circulation strength. The other involves a cessation of the circulation, followed by a restart in a randomly chosen new direction. Rotations occurred an order of magnitude more frequently than cessations. Rotations occurred with a monotonically decreasing probability distribution $p(\uDelta \theta)$, i.e. there was no dominant value of $\uDelta \theta$ and small $\uDelta \theta$ were more common than large ones. For cessations, $p(\uDelta\theta)$ was uniform, suggesting that information of $\theta_0(t)$ is lost during cessations. Both rotations and cessations have Poissonian statistics in time, and can occur at any $\theta_0$. The average azimuthal rotation rate $|\skew4\dot\theta|$ increased as the circulation strength of the LSC decreased. Tilting the sample relative to gravity significantly reduced the frequency of occurrence of both rotations and cessations.

This paper presents a theoretical analysis of perfect gas flow over a convex corner of a rigid-body contour. It is assumed that the flow is subsonic before the corner. It accelerates around the corner to become supersonic, and then undergoes an additional acceleration in the expansion Prandtl–Meyer fan that forms in the supersonic part of the flow behind the corner. The entire process is described by a self-similar solution of the Kármán–Guderley equation. The latter shows that the boundary layer approaching the apex of the corner is exposed to a singular pressure gradient, ${\rm d} p / {\rm d} x \sim (-x)^{-3/5}$, where $x$ denotes the coordinate measured along the body surface from the corner apex. Under these conditions, the solution for the boundary layer also develops a singularity. In particular, the longitudinal velocity near the body surface behaves as $U \sim Y^{1/2}$. Here $Y$ is the normal coordinate scaled with the boundary-layer thickness $Re^{-1/2}$; $Re$ being the Reynolds number, assumed large in this theory.

As usual, the boundary layer splits up into two parts, a viscous near-wall sublayer and a locally inviscid main part of the boundary layer. The analysis of the displacement effect of the boundary layer shows that neither the viscous sublayer nor the main part determines the displacement thickness. Instead, the overlapping region situated between them proves to be responsible for the shape of the streamlines at the outer edge of the boundary layer. This leads to a significant simplification of the analysis of the flow behaviour in the viscous–inviscid interaction region that forms in a small vicinity of the corner. In order to describe the flow behaviour in this region, one has to solve the Kármán–Guderley equation for the inviscid part of the flow outside the boundary layer. The influence of the boundary layer is expressed through a boundary condition, that relates the streamline deflection angle $\vartheta $ at the outer edge of the boundary layer to the pressure gradient ${\rm d}p / {\rm d}x$ acting upon the boundary layer. The boundary-layer analysis leads to an analytical formula that relates $\vartheta $ and ${\rm d}p /{\rm d}x$ (unlike in previous studies of the viscous–inviscid interaction). The interaction problem was solved numerically to confirm that the solution develops a finite-distance singularity.

The interaction between wind dynamics and the waving of crop canopies is explored. On-site experiments with wheat and alfalfa fields have allowed us to quantify the motion of a large set of plants subject to wind, using an image-correlation technique. The coherent part of the waving motion is extracted by a bi-orthogonal decomposition of the spatio-temporal velocity field of the crop surface. It is shown that the corresponding space and time features cannot be explained using predictions from the mixing-layer analogy of wind above canopies, which is the most common model for perturbations in this environment. We show that the plant bending stiffness plays an important role in the frequency and wavelength selection for the coherent motion of the canopy. A fully coupled model, where the wind fluctuations and the plant dynamics interact through a drag term, is then proposed. This model allows us to demonstrate a lock-in mechanism, similar in principle to what is found in vortex-induced vibration, whereby the frequency of the instability deviates from its expected value when approaching the natural frequency of the oscillating medium. This finding is then compared with data from on-site experiments, and good agreement, in both the frequency and wavelength of the propagating patterns observed on the canopy surface, is found.

We derive an analytical bound on the resolvent of pipe Poiseuille flow in large parts of the unstable half-plane. We also consider the linearized equations, Fourier transformed in axial and azimuthal directions. For certain combinations of the wavenumbers and the Reynolds number, we derive an analytical bound on the resolvent of the Fourier transformed problem. In particular, this bound is valid for the perturbation which numerical computations indicate to be the perturbation that gives the largest transient growth. Our bound has the same dependence on the Reynolds number as given by the computations.