A numerical investigation of steady states, their stability, onset of oscillatory instability, and slightly supercritical unsteady regimes of an axisymmetric swirling flow of a Newtonian incompressible fluid in a closed circular cylinder with a rotating lid is presented for aspect ratio (height/radius) 1 [les ] γ [les ] 3.5. Various criteria for the appearance of vortex breakdown are discussed. It is shown that vortex breakdown takes place in this system not as a result of instability but as a continuous evolution of the stationary meridional flow with increasing Reynolds number. The dependence of the critical Reynolds number Recr and frequency of oscillations ωcr on the aspect ratio of the cylinder γ is obtained. It is found that the neutral curve Recr(γ) and the curve ωcr(γ) consist of three successive continuous segments corresponding to different modes of the dominant perturbation. The calculated critical parameters are in good agreement with the available experimental and numerical data for γ < 3. It is shown that the onset of the oscillatory instability does not depend on the existence of a separation bubble in the subcritical steady state. By means of a weakly nonlinear analysis it is shown that the axisymmetric oscillatory instability sets in as a result of a supercritical Hopf bifurcation for each segment of the neutral curve. A weakly nonlinear asymptotic approximation of slightly supercritical flows is carried out. The results of the weakly nonlinear analysis are verified by direct numerical solution of the unsteady Navier-Stokes equation using the finite volume method. The analysis of the supercritical flow field for aspect ratio less than 1.75, for which no steady vortex breakdown is found, shows the existence of an oscillatory vortex breakdown which develops as a result of the oscillatory instability.

Experimental studies of a forced transitional jet at moderate Reynolds numbers reveal interesting transitions between low-dimensional states, namely tangent bifurcations, intermittency and hysteresis. The experiments were carried out in an axisymmetric air jet with a laminar top-hat exit profile in the low-noise ambient of a large anechoic chamber, using forcing amplitude and frequency as control parameters. Tangent bifurcations are seen to occur in two different transitions from periodicity to chaos: (i) from stable pairing to nearly periodic modulations of pairing and (ii) from stable double pairing to a quarter-harmonic chaotic attractor. In case (i), an empirically derived mapping closely reproduces the temporal dynamics of one chaotic attractor at a point just after the tangent bifurcation. In case (ii), the intermittency was characterized by estimating the scaling exponent of the p.d.f. of the periodic durations, which was found to be close to the characteristic value for type-II intermittency. Hysteresis is seen at higher Strouhal numbers in the transitions between aperiodic modulations and the periodic/chaotic double pairing states. Based on simultaneous flow visualization and velocimetry, the hysteresis appears to be associated with intermittent tilting of (otherwise) axisymmetric vortices. These transitions are explained in terms of feedback-driven dynamics from vortex roll-ups and pairings, which can be phase-locked (periodic) or unlocked (nearly quasi-periodic and chaotic). The observed transitions connect large regions of deterministic behaviour in the phase diagram, confirming the existence of a low-dimensional dynamical system in transitional jets-an open flow of technological relevance.

Two-point coherence measurements indicate that spatial coupling (and, hence, coherent motion from pairing dynamics) extends for five to eight diameters from the exit, well beyond the locations of pairing and double pairing. This justifies the use of single-point measurements and confirms our hypothesis that the dynamics in this convectively unstable flow are primarily temporal rather than spatio-temporal.

An experimental study involving flow visualization and two-component LDV measurements has been undertaken to elucidate the deformation of an axisymmetric jet (ReD ≈ 1950 and 4160) caused by tabs placed at the nozzle exit. Previous studies have shown the profound distortion of high-speed jets with tabs and have demonstrated that a pair of counter-rotating streamwise vortices generated by each tab are responsible for the deformation of the jet core. This work illustrates the distortion as well as some of the more subtle features of the tab effect. Extensive visualizations taken simultaneously from two perspectives reveal the real-time evolution of complex three-dimensional flow structures. Velocity data show the expected overall distortion, and the existence and strength of the streamwise vortices responsible for this deformation. Furthermore, a second set of weaker streamwise vortices was detected near each tab, the size and location of which was consistent with a horseshoe vortex system. The data showed a widespread increase in both Reynolds normal and shear stresses and generally indicated the accelerated development of the mixing layer when tabs were inserted. A brief analysis employing vortex dynamics - an alternative to previous work which utilized pressure gradient arguments-is used to explain the tab effect, resulting in similar conclusions.

A cylinder rotating about its vertical axis is filled with homogeneous liquid and subjected to oscillatory mechanical forcing. Depending on the ratio, τ, of the forcing period to the spin-down time, the flow adjacent to the sidewall resembles either a classic Stokes layer (small τ), or a modulated Stewartson layer (large τ). Laboratory experiments show that the flow becomes unstable to columnar disturbances that are aligned with the axis of rotation. This azimuthally wavy instability can lead to the formation of strong vertical vortices which penetrate into the interior. Quasigeostrophic depth-invariant linear instability theory is compared with the experiments. The theory is much too stable and grossly overestimates the experimental critical points. An inertial adjustment model, which in a crude way takes account of observed small-scale (ageostrophic) instability and turbulence in the near-wall region, is in much better agreement with the laboratory measurements of the onset of the columnar vortices. Thus, the origin of the vertically coherent structures appears to be crucially related to alterations of the laminar Stokes-Stewartson profiles by fine structure in the boundary layer.

Recent developments in solving the problem of the elastohydrodynamic collision between two solid elastic bodies involved elaborate numerical procedures in order to simultaneously account for the elastic deformation of the solid surfaces and viscous fluid pressure. This paper describes a simple analytical approximation based upon a Hertzian-like profile for the elastic deformation of the two solid elastic spheres. By introducing a scaling coefficient, a closed-form solution has been developed which is capable of predicting the evolution of the relative particle velocity, force and restitution coefficient to an accuracy that is comparable with the exact numerical solutions.

This paper describes experiments and theoretical modelling of a naturally ventilated enclosure containing two point sources of buoyancy. Previous work on the flow and stratification that develop in a space owing to the presence of a single source of buoyancy has been extended to cover two sources of positive or negative buoyancy in an attempt to produce more realistic models of practical problems. Examples include naturally ventilated buildings, and industrial or medical processes involving dispersal of contaminants by air currents. The experimental technique involves the use of a water-filled enclosure with salt solution injected at two points. These sources of negative buoyancy form turbulent plumes within the enclosure. Two sources of unequal strength are found to produce a vertical density profile consisting of three distinct, fully mixed layers. The paper describes a theoretical model that successfully predicts the depths and densities of these layers. The positions of the interfaces between the three layers were found to be a function only of the effective area A* of the enclosure openings, the height of the enclosure H and the ratio of the strengths of the two sources of buoyancy B1/B2. The behaviour of a system with one source of positive buoyancy at the lower boundary and one of negative buoyancy at the upper boundary of the enclosure is also examined. This case has particular relevance to the prediction of thermal stratification in rooms with both cold and warm surfaces present. Again the height of the interfaces is dependent only on the geometry of the enclosure and the source strength ratio. Three distinct types of flow pattern are predicted and observed.

This paper presents an approximate model of the flow and stratification within a naturally ventilated enclosure containing multiple sources of buoyancy. The sources are assumed to produce plumes which rise without interaction throughout the enclosure. Buoyant fluid accumulates at the top of the enclosure and flows out through upper level openings, and ambient air flows in at low levels to generate an upward displacement ventilation. It is shown that the sources produce a multiple layered stratification with each plume terminating in a given layer. The weakest plume, with the lowest buoyancy flux, produces the lowest interface, and stronger plumes rise higher up within the space before discharging their buoyant fluid into the environment. The model is approximate in that it ignores the stratification within the space when calculating the properties of each plume, and it is shown that this approximation is satisfactory over a wide range of conditions. As a result it is possible to calculate the stratification and ventilation rate for any number of unequal, and equal, sources of buoyancy within a space.

We treat the infiltration of an initially dry deformable porous medium by a pressurized liquid, taking into account the influence of variations in permeability within the deformed porous medium. Chief assumptions of our analysis are neglect of gravity, of inertial forces, and of partial saturation in the porous medium. We focus on unidirectional infiltration under constant liquid pressure, and present data from the infiltration of polyurethane sponge by ethylene glycol in a configuration of nearly unidirectional infiltration with reflief from friction effects along sample sides. We find excellent agreement between theory and experiment at longer infiltration times. We examine an additional assumption, namely the neglect of solid-phase velocity compared with average local liquid velicity at lower porous-medium strains. Agreement of this simplified model with experimental data, albeit less good, remains quite acceptable given the considerable computational simplicity it produces.

Fibres can be coated by passing them through a solution. At low velocity, the thickness of the entrained film is given by the Landau law. For liquids of low viscosity, we discuss the high-speed withdrawal regimes which are of technological interest. We focus on inertial effects and geometrical limitations. New experimental data are presented and discussed by using dimensional arguments. Finally, a classification is proposed.

The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinary-differential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by Hall, Malik & Poll (1984) showed that the two-dimensional disturbance is stable provided that the Reynolds number $\overline{R} < 583.1 $. In the present study, the restrictive assumptions on the disturbance field are relaxed to allow for more general solutions. Results of the present analysis indicate that unstable perturbations other than the special symmetric two-dimensional mode referred to above do exist in the attachment-line boundary layer provided $\overline{R} > 646 $. Both symmetric and antisymmetric two- and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric two-dimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.

This paper presents experiments on planar monopolar vortex structures generated in a non-rotating, stratified fluid. In order to study the dynamics of such planar vortices in the laboratory, angular momentum was generated in a specific horizontal layer of the stratified fluid, by using three different generation mechanisms. The lens-shaped monopolar vortices thus created were in some cases stable and conserved their circular symmetry, while in other cases they appeared to be unstable, leading to the formation of a multipoled vortex with a different topology. Characteristics such as cross-sectional profiles (angular velocity and vorticity) and vorticity-stream function scatter plots have been measured experimentally by using digital image processing techniques. The characteristics of the monopolar vortices are compared with analytical vortex models known from literature. Simple models, based on vertical diffusion of vorticity, are proposed to describe the monopolar vortex decay; they show reasonable agreement with the experimental results.

From the multipolar structures, the tripolar vortex and a specific case of a triangular vortex, neither having been observed before in a stratified fluid, are studied in detail. A comparison with point-vortex models yields good agreement. Although these multipolar vortices appear to persist for a long while, they are found eventually to be unstable and to transform into a monopolar vortex.

Axisymmetric steady flows driven by an electric field about a deformable fluid drop suspended in an immiscible fluid are studied within the framework of the leaky dielectric model. Deformations of the drop and the flow fields are determined by solving the nonlinear free-boundary problem composed of the Navier-Stokes system governing the flow field and Laplace's system governing the electric field. The solutions are obtained by using the Galerkin finite-element method with an elliptic mesh generation scheme. Under conditions of creeping flow and vanishingly small drop deformations, the results of finite-element computations recover the asymptotic results. When drop deformations become noticeable, the asymptotic results are often found to underestimate both the flow intensity and drop deformation. By tracking solution branches in parameter space with an arc-length continuation method, curves in parameter space of the drop deformation parameter D versus the square of the dimensionless field strength E usually exhibit a turning point when E reaches a critical value Ec. Along such a family of drop shapes, steady solutions do not exist for E > Ec. The nonlinear relationship revealed computationally between D and E2 appears to be capable of providing insight into discrepancies reported in the literature between experiments and predictions based on the asymptotic theory. In some special cases with fluid conductivities closely matched, however, drop deformations are found to grow with E2 indefinitely and no critical value Ec is encountered by the corresponding solution branches. For most cases with realistic values of physical properties, the overall electrohydrodynamic behaviour is relatively insensitive to effects of finite-Reynolds-number flow. However, under extreme conditions when fluids of very low viscosities are involved, computational results illustrate a remarkable shape turnaround phenomenon: a drop with oblate deformation at low field strength can evolve into a prolate-like drop shape as the field strength is increased.

Simple shearing flow of a dry soap foam composed of identical Kelvin cells is analysed. An undeformed Kelvin cell has six planar quadrilateral faces with curved edges and eight non-planar hexagonal faces with zero mean curvature. The elastic-plastic response of the foam is modelled by determining the bubble shape that minimizes total surface area at each value of strain. Computer simulations were performed with the Surface Evolver program developed by Brakke. The foam structure and macroscopic stress are piecewise continuous functions of strain. Each discontinuity corresponds to a topological change (Tl) that occurs when the film network is unstable. These instabilities involve shrinking films, but the surface area and edge lengths of a shrinking film do not necessarily vanish smoothly with strain. Each Tl reduces surface energy, results in cell-neighbour switching, and provides a film-level mechanism for plastic yield behaviour during foam flow. The foam structure is determined for all strains by choosing initial foam orientations that lead to strain-periodic behaviour. The average shear stress varies by an order of magnitude for different orientations. A Kelvin foam has cubic symmetry and exhibits anisotropic linear elastic behaviour; the two shear moduli and their average over all orientations are Gmin = 0.5706, Gmax = 0.9646, and $\overline{G} = 0.8070$, where stress is scaled by T/V1/3, T is surface tension, and V is bubble volume. An approximate solution for the microrheology is also determined by minimizing the total surface area of a Kelvin foam with flat films.

A multiple-timescale analysis is employed to analyse Taylor-dispersion-like convective-diffusive processes in converging and diverging flows. A long-time asymptotic equation governing the cross-sectionally averaged solute probability density is derived. The form of this equation is shown to be dependent upon the number of spatial dimensions characterizing the duct or ‘cone’. The two-dimensional case (non-parallel plates) is shown to be fundamentally different from that for three dimensions (circular cone) in that, in two dimensions, a Taylor dispersion description of the process is possible only for small Peclet numbers or angles of divergence. In contrast, in three dimensions, a Taylor dispersion description is always possible provided sufficient time has passed since the initial introduction of solute into the system. The convective Taylor dispersion coefficients $\overline{D}_c$ for the respective cases of low-Reynolds-number flow between non-parallel plates and in a circular cone are computed and their limiting values, $\overline{D}_c^0$, for zero apex angle are shown to be consistent with the known results for Taylor dispersion between parallel plates and in a circular cylinder. When plotted in the non-dimensional form of $\overline{D}_c/\overline{D}^0_c$ versus the half-vertex angle θ0, the respective dispersivity results for the two cases hardly differ from one another, increasing monotonically from 1.0 for θ0 = 0 to approximately 2.6 for a fully flared duct, θ0 = θ/2. Lastly, the techniques developed above for the case of rectilinear channel and duct boundaries are extended to the case of curvilinear boundaries, and an illustrative calculation performed for the case of axisymmetric flow in a flared Venturi tube.

Surfactant transport dynamics and the consequences for rectified diffusion of microbubbles are treated for bubbles undergoing arbitrarily large-amplitude periodic radial oscillations. A perturbation technique is used to reveal averaged equations for the slow convection-enhanced diffusive transport of surfactant molecules. These equations have a readily obtained asymptotic limit in the form of a single nonlinear integral equation — this may be interpreted as a dynamic equilibrium adsorption isotherm. For a lightly populated interface, an explicit solution for the surface excess population of surfactants may be obtained. Bubble oscillations are shown to drive an increased number of surfactant molecules to the interface, if it is lightly populated, but to reduce the maximum possible population of surfactants on the interface. These effects have important consequences for rectified diffusion, in which the interfacial resistance to gas transfer of a surfactant monolayer is a strong function of the surface excess population.