Drifting convection rolls in a rapidly rotating cylindrical annulus with conical endwalls exhibit different transitional modes to chaotic flows at different Prandtl numbers. Three transition sequences for Prandtl numbers 0.3, 1.0 and 7.0 are studied for a moderately large Coriolis parameter and a wavenumber near the critical value using an initial-value code. As the Rayleigh number increases, each transition sequence first leads to a vacillating flow, and then to an aperiodic flow, the route of which is Prandtl-number dependent. From the low Prandtl number to the high Prandtl number, the transitions take different routes of torus folding, period doubling, and mode-locking intermittency.

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

Experiments are described in which a rectangular tube filled with a stratified fluid and tilted at an angle α (about 12°) is rocked at the critical frequency of waves on a slope, σ = N sin α, where N is the uniform buoyancy frequency of the fluid in the central section of the tube. Localized overturns with axes transverse to the flow are observed with a scale comparable with the tube height, producing convective motions and mixing. The overturns have a periodic structure along the tube and, although occurring on each forcing cycle, they alternate in position, so that they reoccur at a given position only every two cycles, that is at the frequency of the first subharmonic of the forcing frequency. The wavelength and vertical structure of the disturbance are consistent with the presence of an internal wave mode with a frequency half that of the forcing, and this is indicative of a parametric instability. The parameters of the regions where static instability occurs show that, as observed, the fluid is more likely to be unstable to convective motions than in earlier experiments (Thorpe 1994b) on standing waves.

We consider the thermocapillary motion of a well-mixed suspension of non-conducting spherical bubbles of negligible viscosity in a viscous conducting liquid under conditions of vanishingly small Reynolds and Marangoni numbers. Recently, Acrivos, Jeffrey & Saville (1990) showed that when all the bubbles are of identical size, the ensemble-averaged migration velocity $\bar{U}_{1}$ of a test bubble of radius a1 within the suspension equals $U^{(0)}_1[1-\frac{3}{2}c_1+O(c^2_1)]$, where c1 is the volume fraction of the bubbles and U1(0) is the thermocapillary velocity of single bubble given by Young, Goldstein & Block (1959). Here we extend this result to a bi-disperse suspension containing bubbles of radii a1 and a2 ≡ λa1 in which case $\bar{U}_1 = U^{(0)}_1[1-\frac{3}{2}c_1 - S(\lambda)c_2+\ldots]$, where c1 and c2 are the corresponding volume fractions of the two sets of bubbles. Values for S(Λ) are presented for some typical size ratios Λ, and asymptotic expressions for S(Λ) are derived for Λ → 0 and for Λ → ∞.

We extend the results of a previous paper to fluids of finite depth. We consider the Hamiltonian theory of waves on the free surface of an incompressible fluid, and derive the canonical transformation that eliminates the leading order of nonlinearity for finite depth. As in the previous paper we propose using the Lie transformation method since it seems to include a nearly correct implementation of short waves interacting with long waves. We show how to use the Eikonal method for slowly varying currents and/or depths in combination with the nonlinear transformation. We note that nonlinear effects are more important in water of finite depth. We note that a nonlinear action conservation law can be derived.

The Lagally theorem is used to obtain an expression for the Bjerknes force acting on a bubble in terms of the singularities of the fluid velocity potential, defined within the bubble by analytic continuation. This expression is applied to transient cavity collapse in the neighbourhood of boundaries, allowing analytical estimates to be made of the Kelvin impulse of the cavity. The known result for collapse near a horizontal rigid boundary is recovered, and the Kelvin impulse of a cavity collapsing in the neighbourhood of a submerged and partially submerged sphere is estimated. A numerical method is developed to deal with more general body shapes and in particular, bodies of revolution. Noting that the direction of the impulse at the end of the collapse phase generally indicates the direction of the liquid jet that may form, the behaviour of transient cavities in these geometries is predicted. In these examples the concept of a zone of attraction is introduced. This is a region around the body, within which the Kelvin impulse at the time of collapse, and consequent jet formation, is expected to be directed towards the body. Outside this zone the converse is true.

This paper reports the result of direct simulations of fluid–particle motions in two dimensions. We solve the initial value problem for the sedimentation of circular and elliptical particles in a vertical channel. The fluid motion is computed from the Navier–Stokes equations for moderate Reynolds numbers in the hundreds. The particles are moved according to the equations of motion of a rigid body under the action of gravity and hydrodynamic forces arising from the motion of the fluid. The solutions are as exact as our finite-element calculations will allow. As the Reynolds number is increased to 600, a circular particle can be said to experience five different regimes of motion: steady motion with and without overshoot and weak, strong and irregular oscillations. An elliptic particle always turn its long axis perpendicular to the fall, and drifts to the centreline of the channel during sedimentation. Steady drift, damped oscillation and periodic oscillation of the particle are observed for different ranges of the Reynolds number. For two particles which interact while settling, a steady staggered structure, a periodic wake-action regime and an active drafting–kissing–tumbling scenario are realized at increasing Reynolds numbers. The non-linear effects of particle–fluid, particle–wall and interparticle interactions are analysed, and the mechanisms controlling the simulated flows are shown to be lubrication, turning couples on long bodies, steady and unsteady wakes and wake interactions. The results are compared to experimental and theoretical results previously published.

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.

The hydrodynamic instability of a viscous incompressible flow with a free surface is studied both numerically and experimentally. While the free-surface flow is basically two-dimensional at low Reynolds numbers, a three-dimensional secondary flow pattern similar to the Taylor vorticies between two concentric cylinders appears at higher rotational speeds. The secondary flow has periodic velocity components in the axial direction and is characterized by a distinct spatially periodic variation in surface height similar to a standing wave. A numerical method, using boundary-fitted coordinates and multigrid methods to solve the Navier–Stokes equations in primitive variables, is developed to treat two-dimensional free-surface flows. A similar numerical technique is applied to the linearized three-dimensional perturbation equations to treat the onset of secondary flows. Experimental measurements have been obtained using light sheet techniques to visualize the secondary flow near the free surface. Photographs of streak lines were taken and compared to the numerical calculations. It has been shown that the solution of the linearized equations contains most of the important features of the nonlinear secondary flows at Reynolds number higher than the critical value. The experimental results also show that the numerical method predicts well the onset of instability in terms of the critical wavenumber and Reynolds number.

The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.

In this paper we consider the dynamics of a growing capillary fountain. We assume that at some time t = 0 an inviscid incompressible fluid is ejected vertically upwards through a circular nozzle. The subsequent dynamics of the resulting fountain is studied numerically using the boundary-element technique. The rate at which fluid is discharged from the nozzle and the Bond number (a measure of gravitational and surface tension forces) are the parameters that govern the dynamics of the fountain. For small discharge rates the fountain assumes the form of a slowly growing sessile drop, with dynamic effects not modifying the shape of the drop significantly. For large discharge rates we find that close to the symmetry axis a region develops with a high curvature. This strongly curved region results in a physical instability which can take on one of two forms: either a liquid drop is ejected from the free surface or the capillary surface entrains a bubble. For intermediate values of the discharge rate we find that fluid lobes develop which fall over the side of the nozzle. A number of experimental results are also presented showing the evolution of water fountains for different Bond numbers and discharge rates. Some of our numerical predictions are confirmed by the experimental results.

The linear instability of Kirchhoff's elliptic vortex in a vertically stratified rotating fluid is investigated using the quasi-geostrophic, f-plane approximation. Any elliptic vortex is shown to be unstable to baroclinic disturbances of azimuthal wavenumber m = 1 (bending mode) and m = 2 (elliptical deformation). The axial wavenumber of the unstable bending mode approaches Λc = 1.7046 in the limit of small ellipticity, indicating that it is a short-wave baroclinic instability. The instability occurs when the bending wave rotates around the vortex axis with angular velocity identical to the rotation rate of the undisturbed elliptic vortex. On the other hand, the wavenumber of the elliptical deformation mode approaches zero in the same limit, showing that it is a long-wave sideband instability.

A new method of generating turbulence in a stratified fluid is presented. The flow is forced by a symmetric array of sources and sinks placed around the perimeter of a tank containing stratified fluid. The sources and sinks are located in a horizontal plane and the flow from the sources is directed horizontally, so that fluid is withdrawn from and re-injected at its neutral density level with some horizontal momentum. The sources and sinks are arranged so that no net impulse or angular momentum is imparted to the flow. Measurements of the mixing produced by the turbulence are made using a conductivity probe to record the vertical density profile. The flow field is measured by tracking small neutrally buoyant particles which are placed within the fluid. The tracking of the particles and analysis of the flow fields are done automatically using DigImage, a recently developed suite of particle tracking software. The characteristics of the flow are found to depend on the forcing parameter F = V/Nd, where V is the mean velocity of the flow through the source orifices, d is the diameter of the sources and sinks and N is the buoyancy frequency of the stratification. At large F three-dimensional turbulence is produced within a mixed layer centred on the level of the sources and sinks. A comparison of mixing rates measured in this and more conventional experiments is made, and it is concluded that in terms of the local turbulence parameters the entrainment rates are similar. At low F, no significant mixing occurs and the flow is approximately two-dimensional with very small vertical velocities. Under these circumstances a qualitative change in the characteristics of the flow occurs after the experiment has been running for some hours. It is observed that the scale of the motion increases until there is an accumulation of the energy at the largest scale that can be accommodated within the tank. The structure of this large-scale circulation is analysed and it is found that a form of vorticity expulsion from the interior of the circulation has occurred. These results are compared with numerical simulations of two-dimensional turbulence, and some measurements of turbulent decay are discussed.

The interaction of a turbulent round jet with the free surface was investigated experimentally. Flow visualization, free-surface curvature measurements and hot-film velocity measurements were used to study this flow. It is shown that surface waves are generated by the large-scale vortical structures in the jet flow as they interact with the free surface. These waves propagate at an angle with respect to the flow direction which increases as the Froude number is increased. Propagation of the waves in the flow direction is suppressed by the surface current produced by the jet. Farther downstream the surface motions are caused by the large-scale vortical structures. Characteristic dark circular features are observed in shadowgraph images associated with concentrated vorticity normal to the free surface. The normal vorticity is believed to be the result of vortex line reconnection processes in the turbulent flow. Measurements of the mean velocity and turbulence intensity are reported. Owing to the confinement by the free surface, the decay rate of the maximum mean velocity is reduced by a factor of √2 compared to an unconfined jet.

The Green's function method is applied to the generation of internal gravity waves by a moving point mass source. Arbitrary motion of a source of arbitrary time dependence is treated using the impulsive Green's function, while ‘classical’ approaches of uniform motion of a steady or oscillatory source are recovered using the monochromatic Green's function. Waves have locally the structure of impulsive waves, emitted at the retarded time tr, and having propagated with the group velocity; at each position and time an implicit equation defines tr, in terms of which the waves are written. A source both oscillating and moving generates two systems of waves, with respectively positive and negative frequencies, and when oscillations vanish these systems merge into one.

Three particular cases are considered: the uniform horizontal and vertical motions of a steady source, and the uniform horizontal motion of an oscillatory source. Waves spread downstream of the steady source. For the oscillatory source they can extend both upstream and downstream, depending on the ratio of the source frequency to the buoyancy frequency, and are contained inside conical wavefronts, parts of which are caustics. For horizontal motion, moreover, the steady analysis (based on the monochromatic Green's function) reveals the presence of two insignificant contributions overlooked by the unsteady analysis (based on the impulsive Green's function), but which for an extended source may become of the same order as the main contribution. Among those is an upstream columnar disturbance associated with blocking.