Flow of a thin viscous film down a flat inclined plane becomes unstable to long-wave interfacial fluctuations when the Reynolds number based on the mean film thickness becomes larger than a critical value (this value decreases as the angle of inclination to the horizontal increases, and in particular becomes zero when the plate is vertical). Control of these interfacial instabilities is relevant to a wide range of industrial applications including coating processes and heat or mass transfer systems. This study considers the effect of blowing and suction through the substrate in order to construct from first principles physically realistic models that can be used for detailed passive and active control studies of direct relevance to possible experiments. Two different long-wave, thin-film equations are derived to describe this system; these include the imposed blowing/suction as well as inertia, surface tension, gravity and viscosity. The case of spatially periodic blowing and suction is considered in detail and the bifurcation structure of forced steady states is explored numerically to predict that steady states cease to exist for sufficiently large suction speeds since the film locally thins to zero thickness, giving way to dry patches on the substrate. The linear stability of the resulting non-uniform steady states is investigated for perturbations of arbitrary wavelength, and any instabilities are followed into the fully nonlinear regime using time-dependent computations. The case of small amplitude blowing/suction is studied analytically both for steady states and their stability. Finally, the transition between travelling waves and non-uniform steady states is explored as the amplitude of blowing and suction is increased.

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.

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.

A data-driven model-free framework is introduced for the calculation of reduced-order models (ROMs) capable of accurately predicting time-mean responses to external forcings, or forcings needed for specified responses, e.g. for control, in fully turbulent flows. The framework is based on using the fluctuation–dissipation theorem (FDT) in the space of a limited number of modes obtained from dynamic mode decomposition (DMD). Use of the DMD modes as the basis functions, rather than the commonly used proper orthogonal decomposition modes, resolves a previously identified problem in applying FDT to high-dimensional non-normal turbulent flows. Employing this DMD-enhanced FDT method ($\text{FDT}_{DMD}$), a linear ROM with horizontally averaged temperature as state vector is calculated for a 3D Rayleigh–Bénard convection system at a Rayleigh number of $10^{6}$ using data obtained from direct numerical simulation. The calculated ROM performs well in various tests for this turbulent flow, suggesting $\text{FDT}_{DMD}$ as a promising method for developing ROMs for high-dimensional turbulent systems.

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).

Asymptotic solutions are obtained for the rise of an axisymmetric hot plume from a localized source at the base of a half-space filled with very viscous fluid. We consider an effectively point source, generating a prescribed buoyancy flux $B$, and show that the length scale of the plume base is $z_0 \,{=}\, (32\upi \kappa^2\nu/B)$, where $\nu$ and $\kappa$ are the kinematic viscosity and thermal diffusivity. The internal structure of the plume for $z \,{\gg}\, z_0$ is found using stretched coordinates, and this is matched to a slender-body expansion for the external Stokes flow. Solutions are presented for both rigid (no-slip) and free-slip (no tangential stress) conditions on the lower boundary. In both cases we find that the typical vertical velocity in the plume increases slowly with height as $(B/\nu)^{1/2} [\ln (z/z_0)]^{1/2}$, and the plume radius increases as $(zz_0)^{1/2} [\ln (z/z_0)]^{-1/4}$.

In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section – a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure–flux and pressure–pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

A method is established for the calculation of the trajectories of shocks moving upward in the atmosphere, on the basis of the assumption that they are of the self-propagating type. The results of calculations for self-similar motions are given, and these are used to establish a propagation law based upon the concepts of the Chisnell, Chester and Whitham (CCW) approximation. This propagation law enters a characteristics law based upon that proposed by Whitham, but reformulated for the computation of axisymmetric shocks with varying density.

An asymptotic self-preserving shock shape is investigated, and is computed for the case γ = 1·4. A parabolic approximation scheme suggested by the self-preserving solution is developed, in which the solution near the axis is reduced to the solution of a system of ordinary differential equations. Finally, the governing equation for the general case without axial symmetry (but without winds) is presented.

In a companion paper (Part 1, Jameson et al. J. Fluid Mech. vol. 649, 2010, 19–43), the discovery of a new type of water bell was reported. When a vertical liquid jet impacts on the underside of a large horizontal plate, the resulting thin film spreads radially along the plate to an unspecified abrupt departure point, from whence it falls away from the plate of its own accord. The departure radius of the fluid from the plate is seen to depend strongly on the volumetric flow rate. The falling liquid may then coalesce to form a water bell. Here we present a theoretical analysis and explanation of this phenomenon. A force balance determining the maximum radial extension of the thin film flow along the plate is considered as a mechanism for fluid departure from the plate, for which an analytical model is developed. This model gives good predictions of the measured radius of departure. When a water bell has been formed, and the flow rate is altered, many interesting shapes are produced that depend on the shapes at previous flow rates. We discuss the origin of this hysteresis, and also present a leading order theory for the bell shape under a regime of changing flow rate. The models are compared with experimental results spanning two orders of magnitude in viscosity.