We present a theoretical investigation of the effects of low-frequency vibrations on the motion of two-dimensional droplets on heterogeneous substrates in the presence of gravity and substrate heterogeneities, both chemical and topographical. A combined analytical and numerical approach is undertaken, extending the work of Savva & Kalliadasis (J. Fluid Mech., vol. 725, 2013, pp. 462–491) on inclined heterogeneous substrates to include the effects of substrate vibrations. Via a matching procedure and under the quasi-static assumption, we obtain evolution equations for the moving fronts. These equations are then invoked in a wide variety of case studies. It is demonstrated that vertically vibrated horizontal ratcheted substrates can induce unidirectional motion. For inclined substrates, we focus on a number of qualitative aspects of the peculiar vibration-induced climbing of droplets reported in experiments by Brunet, Eggers & Deegan (Phys. Rev. Lett., vol. 99, 2007, art. 144501). We examine the effects of weak inertia on the dynamics, deduce analytical criteria for the uphill motion in the limit of weak gravitational and vibrational effects, and demonstrate that substrate heterogeneities may be utilised to enhance droplet transport.

An approximate expression is given for the thickness of weak fully dispersed shock waves. Using available data on the thermodynamic properties of air, it is shown that shocks of the strength expected in sonic bangs are fully dispersed. Estimated relaxation times for dry and humid air lead to wide variations in possible thickness, varying from millimetres to metres.

The paper contains a detailed study of the inertial instability of a barotropic Bickley jet on the $f$-plane in the continuously stratified primitive equations model, and a comparison of this essentially ageostrophic instability with the classical baroclinic one. Analytical and numerical investigation of the linear stability of the jet in the long-wave sector is performed for a range of Rossby and Burger numbers. The major results are that: (1) the standard symmetric inertial instability, appearing at high enough Rossby numbers, turns out to be the infinite-wavelength limit of an asymmetric inertial instability, this latter having the highest growth rate for a large range of vertical wavenumbers; (2) inertial instability coexists with the standard baroclinic instability, which becomes dominant at small Rossby numbers. Nonlinear saturation of the inertial instability of the jet with a superimposed random small-amplitude perturbation is then studied, using the Weather Research and Forecast model. It is shown that at first stages the inertial instability dominates. It is localized near the maximum of the anticyclonic shear and is associated with the highest attainable value of the vertical wavenumber. The saturation of the inertial instability leads to the homogenization of the geostrophic momentum in the unstable region. At later stages, another baroclinic instability develops, characterized by lower values of the vertical wavenumber. This instability saturates by forming large-scale vortices downstream. It is identified as the leading instability of a marginally inertially stable jet resulting from the initial one through homogenization of the geostrophic momentum. The rough scenario of the evolution of essentially ageostrophic jets is, thus, as follows: the inertial instability rapidly saturates and baroclinic instability takes over. It is shown that reorganization of the flow due to developing instabilities is an efficient source of inertia-gravity waves.

Stationary axi-symmetrical pressurized bells formed by the impact of a liquid jet on a solid disc (so-called Savart water bells) can exhibit uncommon sharp and pointed shapes. They are characterized by two successive inflections of the two-dimensional bell generator profile, corresponding to partially concave bells. We show that this shape is incompatible with the usual assumption that the detail of the flow across the liquid sheet constitutive of the bell is unimportant. We consider the equilibrium of a curved liquid sheet of finite thickness sustaining a pressure difference between both sides, and show that several curvatures of the interface may be a solution under given flow conditions. The inflection of the bell profile is then explained in terms of a spontaneous transition from a ‘negative’ to a ‘positive’ curvature which conserves mass flow, linear and angular momenta. That inflection is also a transition from a super to a subcritical flow (with respect to capillary waves), having the status of a capillary hydraulic jump on a freely suspended sheet, a novel object in fluid mechanics. The azimuthal wrinkles forming at the jump result from the inertial destabilization of the sheet due to the centripetal acceleration fluid particles experience as they flow along the highly curved bell profile in the vicinity of the fold. This finding also explains the singular shape at the edge of freely flapping sheets.

Supersonic laminar boundary-layer equations near the plane of symmetry of a cone at incidence are treated by the similarity method. Numerical integration of differential equations governing such a flow is performed, taking into consideration the temperature dependence of the Prandtl number Pr and viscosity μ throughout the boundary layer. On the leeward side, a detailed consideration of the solutions shows the existence of two solutions up to a critical incidence beyond which it appears that no solution may be found. Calculations carried out for a set of values of the external flow Mach number show up a significant effect of this parameter on the behaviour of the boundary layer.

The design of vortex promoters in a heated-wall duct is often limited by the considerations of practicality, especially in complex systems such as fusion blankets. The present study investigates the use of current injection to invoke a street of vortices in quasi-two-dimensional high transverse magnetic field magnetohydrodynamic duct flows to enhance instability behind a cylinder. The intent is to generate intensive flow vorticity parallel to a magnetic field downstream of a field-aligned cylinder. Electric current enters the flow through an electrode embedded in one of the Hartmann walls, radiates outward, imparting a rotational forcing around the electrode due to the Lorentz force. The quasi-two-dimensional nature of these flows then promotes a vortical rotation across the interior of the duct with axis aligned to the magnetic field. The hot and cold walls are parallel to the magnetic field. Electric current amplitude and pulse width, excitation frequency and electrode position are systematically varied to explore their influences on the convective heat transport phenomenon. This investigation builds on a recommendation from previous work of Bühler (J. Fluid Mech., vol. 326, 1996, pp. 125–150) dedicated to understanding of the flow stability in a similar configuration. This study provides supportive evidence for the use of current injection as an alternative to the conventional mechanically actuated turbuliser, with heat transfer almost doubled for negligible additional pumping power requirements.

Addition of particles or droplets to turbulent liquid flows or addition of droplets to turbulent gas flows can lead to modulation of turbulence characteristics. Corresponding observations have been reported for very small particle or droplet volume loadings ${\Phi }_{v} $ and therefore may be important when simulating such flows. In this work, a modelling framework that accounts for preferential concentration and reproduces isotropic and anisotropic turbulence attenuation effects is presented. The framework is outlined for both Reynolds-averaged Navier–Stokes (RANS) and joint probability density function (p.d.f.) methods. Validations are performed involving a range of particle and flow-field parameters and are based on the direct numerical simulation (DNS) study of Boivin, Simonin & Squires (J. Fluid Mech., vol. 375, 1998, pp. 235–263) dealing with heavy particles suspended in homogeneous isotropic turbulence (Stokes number $\mathit{St}= O(1{\unicode{x2013}} 10)$, particle/fluid density ratio ${\rho }_{p} / \rho = 2000$, ${\Phi }_{v} = O(1{0}^{- 4} )$) and the experimental investigation of Poelma, Westerweel & Ooms (J. Fluid Mech., vol. 589, 2007, pp. 315–351) involving light particles ($\mathit{St}= O(0. 1)$, ${\rho }_{p} / \rho \gtrsim 1$, ${\Phi }_{v} = O(1{0}^{- 3} )$) settling in grid turbulence. The development in this work is restricted to volume loadings where particle or droplet collisions are negligible.

Blocked, continuously stratified, crest-controlled flows have hydraulically supercritical downslope flow in the lee of a ridge-like obstacle. The downslope flow separates from the obstacle and, depending on conditions further downstream, transitions to a subcritical state. A controlled, stratified overflow and its transition to a subcritical state are investigated here in a set of three-dimensional numerical experiments in which the height of a second, downstream ridge is varied. The downslope flow is associated with an isopycnal and streamline bifurcation, which acts to form a nearly-uniform-density isolating layer and a sharp pycnocline that separates deeper blocked and stratified fluid between the ridges from the flow above. The height of the downstream obstacle is communicated upstream via gravity waves that propagate along the density interface and set the separation depth of the downslope flow. The penetration depth of the downslope flow, its susceptibility to shear instabilities, and the amount of energy dissipated in the turbulent outflow all increase as the height of a downstream ridge, which effectively sets the downstream boundary conditions, is reduced.

The dissolution of a single droplet, containing a mixture of oils, in water is experimentally studied. The oils in the droplet varied in terms of their solubility in water and their hydrophobicity. We demonstrate that the polarity of the droplet constituents strongly influences the dissolution dynamics. A binary-component droplet, containing two polar components (one soluble the other insoluble) exhibits a retarded dissolution as compared to a droplet containing only the soluble component. We argue that in this case the mixture in the droplet can be assumed homogeneous, leading to a smaller effective contact area of the soluble liquid in the droplet with the bulk water, and thus delayed dissolution. On the other hand, it is shown that this is not the case when a polar, soluble component is mixed with an insoluble non-polar component, in which case segregation between the different liquids inside the droplet occurs, leading to Marangoni flows and superspreading of the droplet. The segregation is confirmed by volumetric measurements and by the use of a solvatochromic dye in combination with confocal microscopy, which clearly showed that during dissolution local concentration differences inside the droplet developed.

The steady motion of a flat surfboard propelled by a solitary wave is considered. The shape of the free surface and the flow of the fluid are determined numerically by series truncation for flows without spray or splash. These flows all bifurcate from the uniform horizontal flow at the critical value of the Froude number. Various limiting cases of these special flows are described analytically. Flows past submerged hydrofoils are discussed also.

The coupled Kuramoto–Sivashinsky (CKS) equations for multilayer downflowing films are derived and explored. The CKS equations exhibit a wealth of dynamical behaviour, displaying travelling periodic waves, regular and chaotic-like patterns, coexistence of different attractors, and perfect and imperfect synchronization of the interfaces. New physical effects are found, such as suppression of the Rayleigh–Taylor instability for heavy-top stratified films, and new surface-tension-driven instability.

The near-resonant flow of a stratified fluid through a localized contraction is considered in the long-wavelength weakly nonlinear limit to investigate the transient development of nonlinear internal waves and whether these might lead to local steady hydraulic flows. It is shown that under these circumstances the response of the fluid will fall into one of three categories, the first governed by a forced Korteweg–de Vries equation and the latter two by a variable-coefficient form of this equation. The variable-coefficient equation is discussed using analytical approximations and numerical solutions when the forcing is of the same (positive) and of opposite (negative) polarity to that of free solitary waves in the fluid. For positive and negative forcing, strong and weak resonant regimes will occur near the critical point. In these resonant regimes for positive forcing the flow becomes locally steady within the contraction, while for negative forcing it remains unsteady within the contraction. The boundaries of these resonant regimes are identified in the limits of long and short contractions, and for a number of common stratifications.

Unsteady electrolysis of a dilute solution of a metal salt made up of two ions in a system with vertical electrodes is considered for large values of the Rayleigh and Schmidt numbers. The mass transfer at the electrodes is assumed to be related to the local charge transfer potential and concentration by a nonlinear Butler–Volmer law. Free convection of the electrolyte appears owing to the variation of the concentration field. After a short initial period, the electrolyte becomes strongly stratified and the motion takes place in boundary layers at the solid boundaries. An approximate model equation for the evolution of the stratification is derived by using perturbation theory. Predictions from the simplified model are found to be in good agreement with numerical solutions of the complete problem. Significant differences compared with earlier studies for linear kinetics, i.e. cases in which the electric current density at the electrodes is constant, are found. Among other things, for large values of the difference ΔV in electric potential between the electrodes, most of the dissolved salt eventually collects near the bottom of the cell. The concentration in the bulk of the electrolyte is, for large values of ΔV, approximately given by a ninth-order polynomial to be compared with a linear behaviour for linear kinetics.

A theoretical study is made of the onset of cellular convection in a shallow two-dimensional container of fluid when the temperature difference between the horizontal boundaries is a monotonic function of horizontal distance. The typical lengthscale of the horizontal variation in the temperature difference is taken to be of the same order of magnitude as the length of the container, and both are much longer than the depth of the container. The resulting flow may be regarded as consisting of two parts, a steady base flow and a disturbed flow. It is found that a weak disturbance taking the form of transverse rolls is first set up near the endwalls, but as the temperature difference between the horizontal boundaries is uniformly increased, an instability in the form of longitudinal rolls takes place near the hotter end of the container. This description is in good qualitative agreement with experiment.

A computational method is presented for the analysis of two-dimensional flow through a non-uniform gauze. The method, based upon the linearized theory due to Elder (1959), permits solutions for most practical cases to be obtained using relatively simple numerical techniques. Comparison with experimental data shows that the computed solutions are satisfactory provided the restrictions inherent in the linearized theory are observed.

A finite-difference numerical method for the solution of the unsteady flow of a viscous incompressible fluid through axisymmetric circular ducts of variable axial geometry is developed and applied to the flow in a spherical-cavity geometry approximating the human aortic valve. The presence and motion of the valve leaflets are considered only as long as they can be assumed to present negligible impedance to the flow. The numerical solution is based on the vorticity/streamfunction approach, and is carried out for the systolic acceleration phase of the heart beat. A hybrid-mesh design consisting of a fine cell structure in the region close to the solid walls and a coarser grid in the core region is used. An experimental flow-visualization study in an acrylic model of the spherical cavity shows good agreement with the numerical simulation. An early separation of flow occurs at the entrance to the cavity, and an annular eddy grows in the wake until it occupies most of the cavity. The use of the hybrid mesh also makes possible the simulation of fine secondary-flow features in the cavity under peak-flow conditions.

A theoretical study is made of the global nonlinear growth or decay, in space and time, of an unsteady non-neutral disturbance/wavepacket when a time-dependent nonlinear viscous critical layer is present. The basic flow considered is a steady quasiparallel channel flow, boundary layer or liquid-layer flow at high Reynolds number. The unsteadiness with regard to the critical layer shows itself less in the internal dynamics than in the relatively slow movement of the layer across the flow, the temporal and spatial rate of movement discussed being sufficient to affect the nonlinear viscous balance in the layer. This greatly reduces the mean-flow distortions produced. The disturbance amplitude, in contrast, responds nonlinearly on faster time- and space-scales, both inside and outside the critical layer. These slower- and faster-scale properties inside the critical layer and outside, i.e. globally, are coupled together in general. The work addresses first the structure and nonlinear evolution equations for the growing or decaying free disturbance and the critical layer. But preliminary analysis in special cases suggests, among other things, the significant result that previous nonlinear studies based on quasineutral assumptions give unstable subcritical threshold amplitudes, above which increasingly fast disturbance growth takes place globally.

Measurements of the fluctuating turbulent stresses (uiuj)’ in the mixing layer of a two-dimensional jet are presented. These include the intensity, spatial correlation and the wave-number frequency spectra of the fluctuating stresses.

The acoustic spectrum perpendicular to the axis of a jet can be approximated with the aid of assumptions about jet similarity and a knowledge of the four-dimensional wave-number frequency spectrum at zero wave-number. In the present paper the four-dimensional spectrum was inferred from measured two-dimensional spectra and the self-noise spectrum perpendicular to a round jet estimated. Good agreement was obtained with measured acoustic spectra.

A configuration consisting of two superposed fluids bounded above by a free surface is considered. Steady three-dimensional potential solutions generated by a moving pressure distribution are computed. The pressure can be applied either on the interface or on the free surface. Solutions of the fully nonlinear equations are calculated by boundary-integral equation methods. The results generalize previous linear and weakly nonlinear results. Fully localized gravity–capillary interfacial solitary waves are also computed, when the free surface is replaced by a rigid lid.

We study time-modulated Taylor–Couette flow for the simple case in which the inner cylinder's angular velocity oscillates around zero mean at given amplitude and frequency and the outer cylinder is at rest. We find that, provided the amplitude of modulation is large enough, two classes of Taylor vortex flows are possible – reversing and non-reversing. In the latter, which takes place at relatively high frequency, the direction of the Taylor vortices does not change every half-cycle.