Hydrodynamic diffusion in the absence of Brownian motion is studied via active microrheology in the ‘pure-hydrodynamic’ limit, with a view towards elucidating the transition from colloidal microrheology to the non-colloidal limit, falling-ball rheometry. The phenomenon of non-Brownian force-induced diffusion in falling-ball rheometry is strictly hydrodynamic in nature; in contrast, analogous force-induced diffusion in colloids is deeply connected to the presence of a diffusive boundary layer even when Brownian motion is very weak compared with the external force driving the ‘probe’ particle. To connect these two limits, we derive an expression for the force-induced diffusion in active microrheology of hydrodynamically interacting particles via the Smoluchowski equation, where thermal fluctuations play no role. While it is well known that the microstructure is spherically symmetric about the probe in this limit, fluctuations in the microstructure need not be – and indeed lead to a diffusive spread of the probe trajectory. The force-induced diffusion is anisotropic, with components along and transverse to the line of external force. The latter is identically zero owing to the fore–aft symmetry of pair trajectories in Stokes flow. In a naïve first approach, the vanishing relative hydrodynamic mobility at contact between the probe and an interacting bath particle was assumed to eliminate all physical contribution from interparticle forces, whereby advection alone drove structural evolution in pair density and microstructural fluctuations. With such an approach, longitudinal force-induced diffusion vanishes in the absence of Brownian motion, a result that contradicts well-known experimental measurements of such diffusion in falling-ball rheometry. To resolve this contradiction, the probe–bath-particle interaction at contact was carefully modelled via an excluded annulus. We find that interparticle forces play a crucial role in encounters between particles in the hydrodynamic limit – as they must, to balance the advective flux. Accounting for this force results in a longitudinal force-induced diffusion $D_{\Vert }=1.26aU_{S}{\it\phi}$, where $a$ is the probe size, $U_{S}$ is the Stokes velocity and ${\it\phi}$ is the volume fraction of bath particles, in excellent qualitative and quantitative agreement with experimental measurements in, and theoretical predictions for, macroscopic falling-ball rheometry. This new model thus provides a continuous connection between micro- and macroscale rheology, as well as providing important insight into the role of interparticle forces for diffusion and rheology even in the limit of pure hydrodynamics: interparticle forces give rise to non-Newtonian rheology in strongly forced suspensions. A connection is made between the flow-induced diffusivity and the intrinsic hydrodynamic microviscosity which recovers a precise balance between fluctuation and dissipation in far from equilibrium suspensions; that is, diffusion and drag arise from a common microstructural origin even far from equilibrium.

This paper considers the flow at high Mach number behind the curved shock formed when a supersonic stream impinges on an axisymmetrical body with a rounded nose.

The solution is obtained as a double expansion in $\delta = (\gamma -1)|(\gamma +1),$ where γ is the adiabatic index, and M−2; the expansion is developed to within terms of order (δ+M−2)3.

Expressions are obtained for the distance between the body and the shock, the radius of curvature of the shock compared with that of the body, and the pressure distribution on the body.

The development of the boundary layer accompanying the formation of a free surface at y′ = 0, from the two-dimensional uniform shear flow u′ = ωyω, is discussed. The analysis shows that the surface velocity and surface position vary as the cube root of the distance downstream, while the mass-transfer coefficient varies inversely as the cube root of this distance. It is shown how these may be applied to the formation of capillary jets.

A numerical study is conducted to investigate the influence of a uniform axial magnetic field on the global linear stability of the rotating-disc boundary layer. Simulation results obtained using a radially homogenized base flow were found to be in excellent agreement with an earlier linear stability analysis, which indicated that an axial magnetic field can locally suppress both convective and absolute instabilities. However, the numerical results obtained for the genuine, radially inhomogeneous, flow indicate that a global form of instability develops for sufficiently large magnetic fields. The qualitative nature of the global instability is similar to that which was observed in a previous study, where mass suction was applied at the rotating disc surface. It is shown that, just as for the case with mass suction, it is possible to explain the promotion of global instability by considering a model that includes detuning effects, which are associated with the radial variation of locally defined absolute temporal frequencies. The recurrence of the same type of instability behaviour when two distinct flow control strategies are implemented, one using suction and the other an axial magnetic field, indicates that the phenomena described by the model may be considered generic.

We examine the rotor–oscillator flow, a slow viscous flow between long parallel plates driven by the rotation of a slender cylinder (the rotor) and the longitudinal oscillation of one of the plates (the oscillator). For rotor locations of interest to us, this flow exhibits a hyperbolic mixing region, characterized by homoclinic tangling associated with a hyperbolic fixed point, and a degenerate mixing region, characterized by heteroclinic tangling associated with two degenerate fixed points on one of the boundary plates (normally the oscillator). These mixing regions are investigated both theoretically, by applying various dynamical tools to a mathematical model of the flow, and experimentally, by observing the advection of a passive tracer in a specially constructed apparatus. Although degenerate mixing regions have been largely ignored or undervalued in previous research on chaotic mixing, our results demonstrate that more mixing is associated with the degenerate mixing region than the hyperbolic one in many cases. We have also discovered a peculiar phenomenon, which we call Melnikov resonance, involving a rapid fluctuation in the size of the hyperbolic mixing region as the frequency of the oscillator is varied.

Axisymmetric Rayleigh–Bénard convection in a cylinder with vertical axis is studied. The nonlinear behaviour is investigated near the onset of convection using an eigenfunction expansion. It is found that initially a steady target pattern develops; the number of rolls depends only on the aspect ratio of the box. At about 5% beyond onset, an oscillatory pattern develops, in which the number of rolls oscillates between two adjacent values. The transitions between the initial steady state and this oscillatory pattern are also investigated, and fall into two main categories. As the Rayleigh number is reduced to the transition point, either the period of the travelling wave tends to infinity, whilst its amplitude stays finite, or there is a sudden transition to a vascillating pattern, the amplitude of which becomes smaller and finally vanishes, whilst the period remains finite. The results are compared with experimental work.

We study numerically the scale-to-scale transfers of enstrophy and passive-tracer variance in two-dimensional turbulence, and show that these transfers display significant differences in the inertial range of the enstrophy cascade. While passive-tracer variance always cascades towards small scales, enstrophy is characterized by the simultaneous presence of a direct cascade in hyperbolic regions and of an inverse cascade in elliptic regions. The inverse enstrophy cascade is particularly intense in clusters of small-scales elliptic patches and vorticity filaments in the turbulent background, and it is associated with gradient-decreasing processes. The inversion of the enstrophy cascade, already noticed by Ohkitani (Phys. Fluids A, vol. 3, 1991, p. 1598), appears to be the main difference between vorticity and passive-tracer dynamics in incompressible two-dimensional turbulence.

This experimental study investigates the effect of imposed rotary oscillation on the flow-induced vibration of a sphere that is elastically mounted in the cross-flow direction, employing simultaneous displacement, force and vorticity measurements. The response is studied over a wide range of forcing parameters, including the frequency ratio $f_{R}$ and velocity ratio $\unicode[STIX]{x1D6FC}_{R}$ of the oscillatory forcing, which vary between $0\leqslant f_{R}\leqslant 5$ and $0\leqslant \unicode[STIX]{x1D6FC}_{R}\leqslant 2$. The effect of another important flow parameter, the reduced velocity, $U^{\ast }$, is also investigated by varying it in small increments between $0\leqslant U^{\ast }\leqslant 20$, corresponding to the Reynolds number range of $5000\lesssim Re\lesssim 30\,000$. It has been found that when the forcing frequency of the imposed rotary oscillations, $f_{r}$, is close to the natural frequency of the system, $f_{nw}$, (so that $f_{R}=f_{r}/f_{nw}\sim 1$), the sphere vibrations lock on to $f_{r}$ instead of $f_{nw}$. This inhibits the normal resonance or lock-in leading to a highly reduced vibration response amplitude. This phenomenon has been termed ‘rotary lock-on’, and occurs for only a narrow range of $f_{R}$ in the vicinity of $f_{R}=1$. When rotary lock-on occurs, the phase difference between the total transverse force coefficient and the sphere displacement, $\unicode[STIX]{x1D719}_{total}$, jumps from $0^{\circ }$ (in phase) to $180^{\circ }$ (out of phase). A corresponding dip in the total transverse force coefficient $C_{y\,(rms)}$ is also observed. Outside the lock-on boundaries, a highly modulated amplitude response is observed. Higher velocity ratios ($\unicode[STIX]{x1D6FC}_{R}\geqslant 0.5$) are more effective in reducing the vibration response of a sphere to much lower values. The mode I sphere vortex-induced vibration (VIV) response is found to resist suppression, requiring very high velocity ratios ($\unicode[STIX]{x1D6FC}_{R}>1.5$) to significantly suppress vibrations for the entire range of $f_{R}$ tested. On the other hand, mode II and mode III are suppressed for $\unicode[STIX]{x1D6FC}_{R}\geqslant 1$. The width of the lock-on region increases with an increase in $\unicode[STIX]{x1D6FC}_{R}$. Interestingly, a reduction of VIV is also observed in non-lock-on regions for high $f_{R}$ and $\unicode[STIX]{x1D6FC}_{R}$ values. For a fixed $\unicode[STIX]{x1D6FC}_{R}$, when $U^{\ast }$ is progressively increased, the response of the sphere is very rich, exhibiting characteristically different vibration responses for different $f_{R}$ values. The phase difference between the imposed rotary oscillation and the sphere displacement $\unicode[STIX]{x1D719}_{rot}$ is found to be crucial in determining the response. For selected $f_{R}$ values, the vibration amplitude increases monotonically with an increase in flow velocity, reaching magnitudes much higher than the peak VIV response for a non-rotating sphere. For these cases, the vibrations are always locked to the forcing frequency, and there is a linear decrease in $\unicode[STIX]{x1D719}_{rot}$. Such vibrations have been termed ‘rotary-induced vibrations’. The wake measurements in the cross-plane $1.5D$ downstream of the sphere position reveal that the sphere wake consists of vortex loops, similar to the wake of a sphere without any imposed rotation; however, there is a change in the timing of vortex formation. On the other hand, for high $f_{R}$ values, there is a reduction in the streamwise vorticity, presumably leading to a decreased total transverse force acting on the sphere and resulting in a reduced response.

The formation and the evolution of axisymmetric vortex rings in a uniformly rotating fluid, with the rotation axis orthogonal to the ring vorticity, have been investigated by numerical and laboratory experiments. The flow dynamics turned out to be strongly affected by the presence of the rotation. In particular, as the background rotation increases, the translation velocity of the ring decreases, a structure with opposite circulation forms ahead of the ring and an intense axial vortex is generated on the axis of symmetry in the tail of the ring. The occurrence of these structures has been explained by the presence of a self-induced swirl flow and by inspection of the extra terms in the Navier–Stokes equations due to rotation. Although in the present case the swirl was generated by the vortex ring itself, these results are in agreement with those of Virk et al. (1994) for polarized vortex rings, in which the swirl flow was initially assigned as a ‘degree of polarization’.

If the rotation rate is further increased beyond a certain value, the flow starts to be dominated by Coriolis forces. In this flow regime, the impulse imparted to the fluid no longer generates a vortex ring, but rather it excites inertial waves allowing the flow to radiate energy. Evidence of this phenomenon is shown.

Finally, some three-dimensional numerical results are discussed in order to justify some asymmetries observed in flow visualizations.

The linear stability of two incompressible coaxial jets, separated by a thick duct wall, is investigated by means of both a modal and a non-modal approach within a global framework. The attention is focused on the range of unitary velocity ratios for which an alternate vortex shedding from the duct wall is known to dominate the flow. In spite of the inherent convective nature of jet flow instabilities, such behaviour is shown to originate from an unstable global mode of the dynamics linearised around the axisymmetric base flow. The corresponding wavemaker is located in the recirculating-flow region formed behind the duct wall. At the same time, the transient-growth analysis reveals that huge amplifications (up to $20$ orders of magnitude) of small flow perturbations at the nozzle exit can occur in the subcritical regime, especially for high ratios between the outer and the inner velocities.

The thin-Debye-layer model is utilized to analyse the electro-hydrodynamic flow about a colloidal particle which is exposed to a direct ionic current, emitted by a proximate reactive electrode. This flow is driven by electro-osmotic slip on the particle, as well as a comparable slip on the electrode itself. The small particle–electrode separation allows for the use of singular perturbation. Thus, the electro-neutral bulk-fluid domain is decomposed into an ‘inner’ gap region, where the electric field and shear rate are large, and an ‘outer’ region, consisting of the remaining bulk domain, where they are moderate. Matched asymptotic expansions in both regions provide the requisite flow field. The intensive shear rate in the narrow gap region is associated with a lubrication-type pressure build-up, which is responsible for the leading-order hydrodynamic force on the particle. This force acts to repel the particle away from the electrode, thereby supporting it against gravity. Its magnitude is inversely proportional to the gap width. At large distances from the particle the fluid velocity decays with the third power of distance, while near the electrode it decays with the fourth power. The inward pointing flow near the electrode tend to entrain neighbouring particles, thereby resulting in two-dimensional particle clusters. For equal values of particle and anode zeta potentials, this process is dominated by the particle-slip contribution.

A general constitutive model is constructed and validated for highly concentrated monodisperse emulsions of deformable drops with insoluble surfactant through long-time, large-scale and high-resolution multidrop simulations. There is the same amount of surfactant on each drop, and the linear model is assumed for the surface tension versus the surfactant concentration. The surfactant surface transport is coupled to multidrop hydrodynamics through the convective–diffusive equation and the interfacial stress balance. Only the limit of small surfactant diffusivities is addressed, when this parameter does not affect the rheology. An Oldroyd constitutive equation is postulated, with five variable coefficients depending on one instantaneous flow invariant (chosen as the drop-phase contribution to the dissipation rate). These coefficients are found by fitting the model to five precise rheological functions from two steady base flows at arbitrary deformation rates. One base flow is planar extension (PE) ($\dot{\unicode[STIX]{x1D6E4}}x_{1},-\dot{\unicode[STIX]{x1D6E4}}x_{2},0$), the other one is planar mixed flow (PM) ($\dot{\unicode[STIX]{x1D6FE}}x_{2}$, $\dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D712}x_{1}$, 0) with $\unicode[STIX]{x1D712}=0.16$. A small but finite $\unicode[STIX]{x1D712}$ (a precise choice in the range $\unicode[STIX]{x1D712}\sim 0.1$ is unimportant) provides a necessarily perturbation to exclude severe ergodic difficulties and abnormal, kinked behaviour inherent in simple shear for high drop volume fractions $c$, especially at small capillary numbers $Ca$ and small drop-to-medium viscosity ratios $\unicode[STIX]{x1D706}$. The database rheological functions are obtained for $c=0.45{-}0.6$, $\unicode[STIX]{x1D706}=0.25{-}3$ and surfactant elasticities $\unicode[STIX]{x1D6FD}=0.05{-}0.2$ (based on the equilibrium surfactant concentration) from long-time simulations by a multipole-accelerated boundary-integral code with $N=100{-}200$ drops in a periodic cell and 2000–4000 boundary elements per drop. The code is an extension from Zinchenko & Davis (J. Fluid Mech., vol. 779, 2015, pp. 197–244) to account for surfactant transport and Marangoni stresses. Massive drop cusping or (sometimes) drop break-up limit the range of $Ca$ from above in the base flows, but there is no substantial lower limitation owing to the absence of phase transition difficulties. At small $\unicode[STIX]{x1D706}$, even minimal surface contamination may have a strong effect on the rheology. The simulations remain accurate for quite strong drop interactions, when the PE emulsion viscosity is nine times that for the carrier fluid. The model validation against a steady PM flow with a different $\unicode[STIX]{x1D712}=0.5$ shows a very good agreement for various $Ca$, $c$ and $\unicode[STIX]{x1D706}$. In the three PE and PM time-dependent flow tests, the quasi-steady approximation is found to predict stresses poorly. In contrast, the combination of the steady-state results for PE and PM used in the present method to generate the Oldroyd parameters gives a model with much better predictions for these time-dependent flows.

We describe new experiments in which a bubble plume, produced from a point source of bubbles, rises through an ambient fluid composed of two layers of fluid of different density. In the lower layer, the speed of the plume exceeds the bubble rise speed and the motion is well described using classical theory of turbulent buoyant plumes. As the mixture enters the upper layer, it is either buoyant and rises to the top of the layer, or is negatively buoyant and forms a fountain. In our experiments, in which a fountain forms in the upper layer, the bubble rise speed exceeds the characteristic speed of this fountain, and a separated flow develops. The bubbles rise to the top of the system, while the lower layer fluid in the fountain rises a finite distance into the upper layer, entrains some of the upper layer fluid, and then collapses. This mixture of fluids then feeds a growing layer of density which is intermediate between the upper and lower layer. The height of rise of the fountain scales with the square of the Froude number of the fountain and the rate of entrainment of upper layer fluid into the fountain is directly proportional to the height of the fountain. This is analogous to the scaling for single-phase fountains with Froude numbers in the same range, $1<Fr<7$, but the constants of proportionality are smaller. We illustrate the relevance of the work for the design of mixing and aeration systems in freshwater reservoirs.

The Lagrangian averaged Navier–Stokes–alpha (LANS-α) model of turbulence is found to possess a Kármán–Howarth (KH) theorem for the dynamics of its second-order autocorrelation functions in homogeneous isotropic turbulence. This KH result implies that alpha-filtering in the LANS-α model of turbulence does not affect the exact Navier–Stokes relation between second and third moments at separation distances large compared to the model's length scale α. Moreover, at separations r that are smaller than α, the KH scaling between energy dissipation rate and longitudinal third-order autocorrelation changes to match the scaling found in two-dimensional incompressible flow. This is consistent with the corresponding change in scaling of the kinetic energy spectrum from k−5/3 for larger scales with kα < 1, which switches to k−3 for smaller scales with kα > 1, as discovered in Foias, Holm & Titi (2001).