This paper reports results obtained with two-dimensional numerical simulations of viscous incompressible flow in a symmetric channel with a sudden expansion and contraction, creating two facing cavities; a so-called double cavity. Based on time series recorded at discrete probe points inside the double cavity, different flow regimes are identified when the Reynolds number and the intercavity distance are varied. The transition from steady to chaotic flow behaviour can in general be summarized as follows: steady (fixed) point, period-1 limit cycle, intermediate regime (including quasi-periodicity) and torus breakdown leading to toroidal chaos. The analysis of the intracavity vorticity reveals a ‘carousel’ pattern, creating a feedback mechanism, that influences the shear-layer oscillations and makes it possible to identify in which regime the flow resides. A relation was found between the ratio of the shear-layer frequency peaks and the number of small intracavity structures observed in the flow field of a given regime. The properties of each regime are determined by the interplay of three characteristic time scales: the turnover time of the large intracavity vortex, the lifetime of the small intracavity vortex structures and the period of the dominant shear-layer oscillations.

The oscillatory motion of an electrically charged non-spherical colloidal particle in an oscillating electric field is investigated. The particle is immersed in an incompressible viscous fluid and assumed to have a thin electric double layer. For moderate-aspect-ration spheroids and cylinders, a simple algebraic expression is derived that accurately describes oscillatory electrophoretic particle motion in terms of the steady Stokes resistance, added mass, and Basset force. The effects of double-layer conduction and displacement currents within dielectric particles are included. The results indicate that electroacoustic measurements may be able to determine the ζ-potential, dielectric constant, surface conductivity (and microstructural information contained therein), size, density, volume fraction, and possibly shape of non-spherical particles in a dilute suspension. A simple formula is obtained for the high-frequency electrical conductivity of a dilute suspension of colloidal spheroids with arbitrary charge and dielectric constant; only the added mass and Basset force are required and the requisite parameters are given. The result is needed for electroacoustic measurements but it may also be independently useful for determining the dielectric constant, surface conductivity, volume fraction, and possibly the shape of non-spherical particles in a dilute suspension. Electroacoustic energy dissipation is described for a dilute colloidal suspension. It is shown that resistive electrical heating and viscous dissipation occur independently. Electrical and viscous dissipation coefficients that characterize the order volume fraction contributions of the suspended particles are calculated; the electrical dissipation coefficient is O(1) for all oscillation frequencies, whereas the latter vanishes at low- and high-frequencies. The fluid motion is shown to be a superposition of unsteady, viscous and potential flows past an oscillating particle with no applied electric field. The electro-osmotic flow field is insensitive to particle geometry and qualitatively different from the flow past an oscillating particle with no applied field.

A lubrication theory has been developed for the electro-osmotic flow of non-uniform buffers in narrow rectilinear channels. The analysis applies to systems in which the transverse dimensions of the channel are large compared with the Debye screening length of the electrolyte. In contrast with related theories of electrokinetic lubrication, here the streamwise variations of the velocity field stem from, and are nonlinearly coupled to, spatiotemporal variations in the electrolyte composition. Spatially non-uniform buffers are commonly employed in electrophoretic separation and transport schemes, including iso-electric focusing (IEF), isotachophoresis (ITP), field-amplified sample stacking (FASS), and high-ionic-strength electro-osmotic pumping. The fluid dynamics of these systems is controlled by a complex nonlinear coupling to the ion transport, driven by an applied electric field. Electrical conductivity gradients, attendent to the buffer non-uniformities, result in a variable electro-osmotic slip velocity and, in electric fields approaching 1 kV cm−1, Maxwell stresses drive the electrohydrodynamic circulation. Explicit semi-analytic expressions are derived for the fluid velocity, stream function, and electric field. The resulting approximations are found to be in good agreement with full numerical solutions for a prototype buffer, over a range of conditions typical of microfluidic systems. The approximations greatly simplify the computational analysis, reduce computation times by a factor 4–5, and, for the first time, provide general insight on the dominant fluid physics of two-dimensional electrically driven transport.

The flow structures giving rise to the force on a circular cylinder in uniform, circular, orbital flows have been investigated for keulegan—Carpenter numbers (K) less than 2 and a Stokes parameter (β) of 483 using the random-vortex method. Comparisons with analysis using the method of inner and outer expansions are made and good agreement is found for K = 0.1. For higher K-values, the viscous force (the difference between the total force and the potential-flow force) acts mainly in opposition to the potential-flow force causing a substantial reduction in total force, in keeping with experimental measurements. Significant separation does not occur at K ≤ 1.5 and vorticity organizes itself asymmetrically about the line through the cylinder centre parallel to the incident velocity vector. Vorticity of one sense of rotation remains close to the half-surface lagging the velocity vector, while an area of vorticity of the opposite sense wraps itself around the cylinder. The net circulation in the flow (the circulation within a path encircling the cylinder at a large radius) is zero. Vortex shedding occurs at K > 1.5. Viscous forces due to non-uniform, orbital flows around a horizontal cylinder beneath waves were similar although vortex shedding tended to occur at lower K-values.

A Mach-Zehnder interferometer was used to study the interaction between convection currents in the sea water and the ice-water interface during freezing. The temperature at the ice-water interface during the early stages of freezing was approximately 0 °C, not the equilibrium freezing temperature. Salt plumes were observed to appear first at specific sites on the ice-water interface. This is contrary to earlier results reported by Farhadieh & Tankin (1972). The salinity profiles in the salt plumes were measured using interferograms in conjunction with Abel inversion. The maximum salinity, along the centre-line of the salt plume, is an indication of the salinity of the brine trapped within the ice.

We explore the transport of heavy particles through an upward displacement-ventilated space. The space incorporates a localised source of buoyancy which generates a turbulent buoyant plume. The plume fluid is contaminated with a small concentration of particles, which are subject to gravitational settling. A constant flow of uncontaminated fluid is supplied at a low level into the space, while an equal amount of fluid is vented from the space at a high level. At steady state, a two-layer density stratification develops associated with the source of buoyancy. New laboratory experiments are conducted to explore how particles are transported by this flow. The experiments identify that the upper layer may either become well-mixed in particles or it may develop a vertical stratification in particle concentration, with the particle concentration decreasing with height. We develop a quantitative model which identifies that such stratification develops for larger particle setting speeds, or smaller ventilation rates. In accord with our experiments, the model predicts that the number of particles extracted from the space through the high-level vent is controlled by the magnitude of the particle stratification in the upper layer, and this in turn depends on the particle settling speed relative to the ventilation speed and also the cross-sectional area and height of the space. We compare the predictions of the model with measurements of the flux of particles vented from the space for a range of operating conditions. We explore the relevance of the model for the removal of airborne contaminants by displacement ventilation in hospital rooms, and we discuss how contamination is propagated in the room as a result of lateral mixing of pathogens in the upper layer.

The time-dependent settling of a dilute monodisperse suspension in a centrifugal force field is considered. The settling takes place between two axisymmetric narrowly spaced conical disks that are rotating rapidly. Clear fluid and suspension are assumed to behave as Newtonian fluids of different densities. The viscosities are, for simplicity, assumed to be the same. All effects of the sediment are neglected. The fluid motion is assumed to be almost parallel with the disks and is computed by using lubrication theory. This leads to a nonlinear hyperbolic equation of first order for the location of the interface between clear fluid and suspension. Local multivaluedness of the solution is removed by inserting shocks. Such a shock is a model for a small region where the slope of the interface, as scaled in the lubrication approximation, is large. Two problems are solved: batch settling and a case where the suspension is pumped into a conical channel that is initially filled with clear fluid. In the batch-settling case, the solution is quite similar to that computed by Herbolzheimer & Acrivos for settling due to a constant gravity field in a narrow tilted channel. For large values of the Taylor number, it is found that the blocking of radial flow outside the Ekman layers leads to a somewhat slower separation process than expected. In the filling problem, the character of the solution is distinctly different for Taylor numbers of order unity and large values of this parameter.

The linear stability of the flow of an incompressible viscous fluid through a curved pipe of circular cross-section is considered. There is a sinusoidal pressure gradient, with zero mean, acting down the pipe. The flow is shown to be unstable to a Taylor-Görtler mode of instability, with vortices aligned with the basic flow first appearing at the outer bend of the pipe when a critical value of the Taylor number is exceeded. A WKBJ perturbation solution is constructed and the form of the vortex amplitude is determined. This solution is found to break down in the vicinity of the pipe's outer bend, and an inner solution is presented to overcome this. The solution is determined by identifying a saddle point in the complex plane of the cross-sectional angle coordinate. This leads to an eigenvalue problem for the Taylor number, for fixed wavenumber and cross-sectional angle coordinate, which in turn leads to the determination of the critical Taylor number above which instability sets in.

This paper examines an often overlooked point in the theory of dispersion of passive contaminants in parallel flows – the behaviour of a cloud of solute which has been injected into the flow over a period of time. The problem is linear if the solute has neutral buoyancy and the diffusivity is independent of concentration, and so it can be treated by a Fourier transform in time. For such a Fourier-transform method to be successful, a non-standard eigenvalue problem has to be solved to determine the concentration pattern, the speed at which it is convected downstream and its decay distance downstream. The eigenvalue problem is not self-adjoint, the eigenvalue enters it nonlinearly if longitudinal diffusion is included, and it involves both regular and singular perturbation aspects. The eigenvalue problem is examined generally and the conclusions of Chatwin (1973a) are re-assessed. The eigenvalue problem is then solved numerically for three cases of interest. Two of these cases (dispersion from a harmonically varying source in Poiseuille and plane Couette flow) reveal most unusual eigenvalue structure, whilst the third (dispersion from a harmonically varying source in turbulent channel flow) is not exceptional. The effect of weak longitudinal diffusion is examined theoretically (for general applications) and numerically in one instance (the application to Poiseuille flow). Longitudinal diffusion, even if weak, has a marked effect on the eigenvalue structure. The paper concludes with a suggestion for an alternative attack on the original problem of dispersion from time-dependent sources.

The nonlinear dynamics of nonlinear modulated cross-waves of resonant frequency ω1 and carrier frequency ω ≈ ω1 is investigated. In a long channel of width b, that contains fluid of depth d and which is subjected to a vertical oscillation of frequency 2ω, the wave can appear in solitary form. As has been shown previously, the solitary wave is only stable in a certain parameter regime; depending on damping and driving amplitudes the wave becomes unstable. The nonlinear development of the instabilities of solitary waves is the central problem of this paper. It is shown how instabilities are saturated following generic routes to chaos in time with spatially coherent structures. Finally, the case of time-modulated driving amplitudes is also considered. In most cases it appears that nonlinear waves of simple spatial structures take part in the nonlinear dynamics, but a few cases of spatial chaos are also reported.

A bifurcation study is made of laminar flow in curved ducts. The problem is formulated in a curvilinear coordinate system, and the governing equations, after orthogonal mapping is applied, are solved numerically by an iterative finite-difference method. Many computer runs were made with various duct cross-sections ranging from a circle to a square, to learn the transition of bifurcation structure with this change in cross-section and to reconcile the differences between them. In addition, a simpler technique is proposed to generate symmetric four-cell solutions in a circular pipe and a means is put forward to stabilize four-vortex structures in a complete cross-section.

Mass transport under a progressive sea wave propagating over a rippled bed is investigated. Wave amplitudes a* of the same order of magnitude as that of the boundary layer thickness δ* and of the ripple wavelength l* are considered. All the above quantities are assumed to be much smaller than the wavelength L* of the sea wave and much larger than the amplitude 2ε* of the ripples. The analysis is carried out up to the second order in the wave slope a*/L* and in the parameter ε*/δ* which is a measure of ripple steepness. Because of these assumptions, the slow damping of wave amplitude in the direction of wave propagation is taken into account. Attention is focused on the bottom boundary layer where an order (ε*/δ*)2 correction of the steady velocity components described by Longuet-Higgins (1953) is found. This correction persists at the outer edge of the bottom boundary layer and affects the solution in the entire water column.

It seems that no treatment of pulsating flow in deformable tubes is complete without a reference to the work of Womersley (1955) which for an infinitely long tube deals with the waves of axismmetric transverse motion and of longitudinal motion of the walls. This theory has so far been subjected to experimental test only for tethered tubes in which longitudinal wall motion is absent.

A series of measurements of the longitudinal motion has been made on horizontal water-filled latex tubes suspended by an array of strings so that there is minimal longitudinal constraint except at the ends, which are fixed. One end of the tube is driven by oscillating flow produced by a piston; the other end is closed. Theory and experiment agree when the tube is long provided an entrance length greater than a wavelength is included. Tubes which are short enough for reflection from the closed end to be significant present a more complicated problem. It is found that in the entrance length the theory of Womersley cannot be applied. A more refined theory is required which takes into account a distributed end constraint more completely than as a simple boundary condition.

Experiments on tethered tubes in which longitudinal wall motion is absent are also presented. These serve to demonstrate that the theory for such tubes agrees with measurements without any appreciable end effect and also shows that the small viscoelasticity of the latex rubber is correctly included.

When dealing with systems showing a Hopf bifurcation as the first instability from a conductive state leading to travelling waves, the distinction between convective and absolute instability becomes significant. The convectively unstable regime is characterized by the fact that a homogeneous disturbance may have a positive growth rate, while a single localized perturbation cannot trigger the onset of nonlinear convection. In this paper the convective instability occurring in binary fluid mixtures for a negative separation ratio is utilized for amplifying intrinsic thermal fluctuations, which in this way become accessible to quantitative measurements. The experiments are performed in a quasi-one-dimensional convection channel which, by means of subcritical ramps, effectively prevents the reflection of the travelling waves from the sidewalls. Thus, that range of the convective instability within which linear waves can be observed is strongly enhanced. The temperature variations involved in the observed travelling-wave states are quantified by using the shadowgraph method. By resonantly stimulating the system with its linear Hopf frequency, the reflection ability and some coefficients of the amplitude equation appropriate for describing the convection features near onset can be determined. Without stimulation, travelling-wave states of very small amplitudes showing an erratic spatio-temporal behaviour occur spontaneously inside the convectively unstable regime. The temporal correlation function calculated from the measured light intensity caused by these states is compared with a theoretical expression obtained from a Ginzburg—Landau equation containing a noise term. A very good agreement is found for the amplitude if thermal noise is assumed to be the reason for these fluctuating convection rolls, thus supporting the idea that the response of the system to thermal fluctuations is observed.

The two-dimensional steady-state turbulent-diffusion equation is solved by a separation-of-variables process that leads to a Sturm–Liouville eigenvalue problem. The general solution, for arbitrary velocity and diffusivity distributions, is shown to be in the form of an eigenfunction expansion. For a steady uniform flow in a wide, open channel the velocity distribution in the vertical is well approximated by a logarithmic law and the diffusivity distribution is approximately parabolic. For these distributions the power-series solution technique for ordinary differential equations is used to determine the eigenfunctions and eigenvalues. The solution is compared with the standard solution that Holley, Siemans & Abraham (1972) obtained for a uniform velocity and diffusivity distribution. Experimental results are presented, and these show that

(1) the use of the correct velocity and diffusivity distribution results in a significant improvement in the agreement between experiment and theory; and (2) close to the source the fluctuations of concentration are of the order of the mean values.

A biplane grid with a mesh spacing of 10.8 cm was towed horizontally in a towing tank to generate turbulence in a non-stratified fluid and in stratified fluids with different constant density gradients. Turbulence velocity components and density fluctuations were measured using an array of cross-film and conductivity probes. Based on the mesh size of the grid, the nominal values of the (internal) Froude numbers were ∞, 80 and 40, and the corresponding Reynolds number was 4.3 × 104. The decay rates of the (turbulence) kinetic, potential and total energies and the dissipation rates of the kinetic and potential energies were calculated from the experimental data. For each of these quantities, the decay may be represented as a function of the downstream distance raised to a given power. The kinetic energy and its dissipation rate are lower for the stratified cases than for the non-stratified case but are almost compensated for by the corresponding potential energy and its dissipation rate. Our results are consistent with those of direct numerical simulations and agree reasonably well with those obtained in stratified wind and water tunnels. However, the results differ from laboratory results obtained using an optical method to measure the turbulent motion of tracer particles in the wake of a vertically towed grid; these latter results show an abrupt reduction in the decay rate of the turbulence kinetic energy after one Brunt–Väisälä period. A similar trend is also observed in results obtained in facilities with fairly high background turbulence or internal waves. This discrepancy is discussed and an explanation is presented. Furthermore, it is demonstrated that strongly stratified thin sheets with density gradients larger than that of the undisturbed fluid may be generated by local but incomplete mixing. The persistence of such thin sheets is proportional to the Schmidt number (≈ 500) in stratified salt water or the Prandtl number (≈ 0.71) in thermally stratified air.

This paper is concerned with the downstream evolution of a resonant triad of initially non-interacting linear instability waves in a boundary layer with a weak adverse pressure gradient. The triad consists of a two-dimensional fundamental mode and a pair of equal-amplitude oblique modes that form a subharmonic standing wave in the spanwise direction. The growth rates are small and there is a well-defined common critical layer for these waves. As in Goldstein & Lee (1992), the wave interaction takes place entirely within this critical layer and is initially of the parametric-resonance type. This enhances the spatial growth rate of the subharmonic but does not affect that of the fundamental. However, in contrast to Goldstein & Lee (1992), the initial subharmonic amplitude is assumed to be small enough so that the fundamental can become nonlinear within its own critical layer before it is affected by the subharmonic. The subharmonic evolution is then dominated by the parametric-resonance effects and occurs on a much shorter streamwise scale than that of the fundamental. The subharmonic amplitude continues to increase during this parametric-resonance stage – even as the growth rate of the fundamental approaches zero – and the subharmonic eventually becomes large enough to influence the fundamental which causes both waves to evolve on the same shorter streamwise scale.

Experiments are performed on steady and impulsively started flow in an approximately two-dimensional closed channel, with one wall locally indented. In plan the indentation is a long trapezium which halves the channel width: the inclination of the sloping walls is approximately 5.7°, and these tapered sections merge smoothly into the narrowest section via rounded corners. The Reynolds number $ Re = a_0\overline{u}_0/\nu $ (a0 = unindented channel width, $\overline{u}_0$ = steady mean velocity in the unindented channel) lies in the range 300 [les ] Re [les ] 1800. In steady flow, flow visualization reveals that separation occurs on the lee slope of the indentation, at a distance downstream of the convex corner which decreases (tending to a non-zero value) as Re increases. There is no upstream separation, and there is some evidence of three-dimensionality of the flow in the downstream separated eddy. Pressure measurements agree qualitatively but not quantitatively with theoretical predictions. Unsteady flow visualization reveals that, as in external flow, wall-shear reversal occurs over much of the lee slope (at dimensionless time $\tau = \overline{u}_0t/a_0 \approx 4$) before there is any evidence of severe boundary-layer thickening and breakaway. Then, at τ ≈ 5.5, a separated eddy develops, and its nose moves gradually upstream from the downstream end of the indentation to its eventual (τ ≈ 75) steady-state position on the lee slope. At about the same time as the wall-shear reversal, wavy vortices appear at the edge of the boundary layer on both walls of the channel, and (for Re < 750) subsequently disappear again; these are interpreted as manifestations of inflection-point instability and not as intrinsic aspects of boundary-layer separation. Pressure measurements are made to investigate the discrepancy between the actual pressure drop across the lee slope and that predicted on the assumption that energy dissipation is quasi-steady. This discrepancy has a maximum value of approximately $1.5\rho \overline{u}^2_0$ (ρ = fluid density), and decays to zero by the time τ ≈ 7.