It is shown theoretically that the crest of a steep, irrotational gravity wave, considered in isolation, is unstable. There exists just one basic mode of instability, whose exponential rate of growth β equals 0.123(g / R)½, where g denotes gravity and R is the radius of curvature at the undisturbed crest. A volume of water near the crest is shifted towards the forward face of the wave; the ‘toe’ of the instability is at a horizontal distance 0.45R ahead of the crest. The instability may represent the initial stage of a spilling breaker. On small scales, the ‘toe’ may be a source of parasitic capillary waves.

The interval of Rayleigh numbers in Bénard convection corresponding to cellular motion is determined in the case of free-free boundaries, rigid-free boundaries and rigid-rigid boundaries, taking into account the variation of the kinematic viscosity with temperature. Neglecting the effect of surface tension, it is shown that this interval is largest for the rigid-rigid case. The most important feature from the obtained formula (6.1) is, however, that the interval is extremely dependent on the depth of the fluid layer. To obtain a cellular pattern it is therefore necessary to have very small fluid depths. For example, with Silicone oil and a fluid depth of 7 mm, cellular motion will, according to the theory, be observed for Rayleigh numbers larger than the critical value and less than 1·07 times the critical value. For a fluid depth of 5 mm, however, the formula (6.1) gives that cellular motion will be observed for Rayleigh numbers up to 1·54 times the critical value.

An investigation into the magnitude and direction of the convective velocity in a plane air jet was performed. Convective velocities were obtained from cross-correlation measurements. They are defined as the ratio of the spacing between two hot-wire probes and the time delay between their signals to reach maximum correlation. These velocities were larger in magnitude than the local mean velocities for lateral distances greater than the half-width of the jet. Frequency analysis of the convective velocity indicates that the large-scale eddies move slower than the mean flow while the small scales move faster. Based on the convective velocity vector, broadband ‘convection lines’ were defined and found to point outward with respect to the streamlines for all values of y/b [Gt ] 0·5. Likewise, frequency investigation indicates that ‘convection lines’ point outward for all y/b [Lt ] 1·3 and then inward for larger values of y/b.

We report on a novel method to generate fast transient microjets and study their characteristics. The simple device consists of two electrodes on a substrate with a hole in between. The side of the substrate with the electrodes is submerged in a liquid. Two separate microjets exit through the tapered hole after an electrical discharge is induced between the electrodes. They are formed during the expansion and collapse of a single cavitation bubble. The cavitation bubble dynamics as well as the jets were studied with high-speed photography at up to 500 000 f.p.s. With increasing jet velocity they become unstable and spray formation is observed. The jet created during expansion (first jet) is in most cases slower than the jet created during bubble collapse, which can reach up to $400~\text{m}~\text{s}^{-1}$. The spray exiting the orifice is at least in part due to the presence of cavitation in the microchannel as observed by high-speed recording. The effect of viscosity was tested using silicone oil of 10, 50 and 100 cSt. Interestingly, for all liquids the transition from a stable to an unstable jet occurs at $We\sim 4600$. We demonstrate that these microjets can penetrate into soft material; thus they can be potentially used as a needleless drug delivery device.

The stability with respect to two- and three-dimensional perturbations of natural-convection flow of air in a square enclosure with differentially heated vertical walls and periodic boundary conditions in the lateral direction has been investigated. The horizontal walls are either conducting or adiabatic. The solution is numerically approximated by Chebyshev–Fourier expansions. In contrast to the assumption made in earlier studies, three-dimensional perturbations turn out to be less stable than two-dimensional perturbations, giving a lower critical Rayleigh number in the three-dimensional case for the onset of transition to turbulence. Both the line-symmetric and line-skew-symmetric three-dimensional perturbations are found to be unstable. The most unstable wavelengths in the lateral direction typically are of the same size as the enclosure. In the nonlinear solution new symmetry breaking occurs, giving either a steady or an oscillating final state. The three-dimensional structures in the nonlinear saturated solution consist of counter-rotating longitudinal convection rolls along the horizontal walls. The energy balance shows that the three-dimensional instabilities have a combined thermal and hydrodynamic nature. Besides the stability calculations, two- and three-dimensional direct numerical simulations of the weakly turbulent flow were performed for the square conducting enclosure at the Rayleigh number 108. In the two-dimensional case, the time-dependent temperature shows different dominant frequencies in the horizontal boundary layers, vertical boundary layers and core region, respectively. In the three-dimensional case almost the same frequencies are found, except for the horizontal boundary layers. The strong three-dimensional mixing leaves no, or only very weak, three-dimensional structures in the time-averaged nonlinear solution. Three-dimensional effects increase the maximum of the time- and depth-averaged wall-heat transfer by 15%.

The general dynamical problem for bubbles moving in an infinite expanse of perfect liquid is discussed from the standpoint of Hamiltonian theory, which is appreciated as a basis for linking symmetries with conservation laws and for identifying variational principles that describe steady motions. Allowance is made for surface tension and for an arbitrary gas law relating the pressure and volume of the bubble contents, but particular attention is paid to models where the volume is constant.

In §, the most detailed part of the paper, a comprehensive theory is developed which represents the free surface parametrically and so applies globally in time. Conservation laws for energy and for linear and angular components of impulse are shown to follow simply from respective symmetries; consequences of Galilean in variance and of a scaling symmetry are also explored. Finally in §, variational characterizations of steady translational, spinning and spiralling motions are explained. In §3 a formally simpler Hamiltonian theory is shown to derive from the mildly restrictive assumption that the free surface can be represented in an orthogonal coordinate system; and some special details attending the use of cylindrical coordinates are noted. For bubbles steadily translating along an axis of symmetry, approximate calculations supported by Rayleigh's principle are presented in §4.1. Steadily spiralling motions are treated in §5; estimates based on spheroidal approximations to shape are presented in §5.1; and some speculations about stability are discussed in §5.2. A brief account of generalizations dealing with multiply connected bubbles is given in §6.

The small-amplitude free vibrations of a spherical drop immersed in an outer immiscible fluid and in partial contact with a solid support are considered when both fluids are assumed to be viscous and incompressible, while gravity effects are neglected. Using the normal-mode decomposition and the Green-function method, the solution of the linearized Navier-Stokes equations is reduced to the solution of an eigenvalue problem. The model includes as particular cases the viscous model for a free drop proposed by Prosperetti (1980) and the inviscid model for a supported drop previously proposed by the authors.

The influence of the viscosity and of the support size are analysed both for the bubble and for the drop. At large values of the viscosity, the free drop shows significant differences with respect to the unsupported drop and a singular behaviour of the eigenvalue problem as the support size tends to zero.

The comparison with the available experimental data shows a quite satisfactory agreement for both the vibration frequency and the damping constant, provided that the support angle is not too large.

This paper investigates the stability and large-displacement post-buckling behaviour of liquid-lined elastic rings. The fluid flow and the wall deformation are described by the free-surface Navier–Stokes equations and by geometrically nonlinear shell theory, respectively. The fluid–structure interaction problem is solved numerically by a finite element method. The compressive load on the ring is a combination of the external pressure and the effect of surface tension. Once this combined load exceeds a critical value, the subsequent non-axisymmetric collapse of the ring is controlled by the dynamics of the surface-tension-driven redistribution of fluid in the liquid lining. It is shown that, for sufficiently large surface tension, the ring can undergo a catastrophic collapse which leads to a complete occlusion of its lumen. A novel lubrication theory model, which ensures exact volume conservation for flows on strongly curved substrates, is developed and found to be capable of accurately describing the motion of the air–liquid interface and the fluid–structure interaction in the large-displacement regime, even in cases where the film thickness is large.

The findings have important implications for the occurrence of airway closure in lung diseases (such as oedema) which cause an increase in the thickness of the airways' liquid lining. It is shown that under such conditions, airways can become occluded even if the volume of fluid in their liquid lining is much smaller than that required to occlude them in their axisymmetric state.

The perturbation of pre-existing surface gravity waves caused by the presence of an internal wave was studied both experimentally and analytically. An extensive series of experiments was performed, and quantitative results were obtained for the one-dimensional monochromatic interaction of internal waves and surface gravity waves. Internal wave-induced surface slope, amplitude and wavenumber modulations were measured for a wide range of interaction conditions. A complementary theoretical analysis, based on the conservation approach of Whitham (1962) and Longuet-Higgins & Stewart (1960,1961), was performed and a closed form solution obtained for the one-dimensional wave interaction. Both the theory and the experiment demonstrate that the effect increases with interaction distance. The maximum interaction effect is found to occur when the phase speed of the internal wave and the group velocity of the surface wave are matched. The phase of the internal wave at which maximum surface-wave modulation occurs is found to be a sensitive and continuous function of the relative wave speeds. The experimental data are in good agreement with the present theoretical analysis.

The problem of fluid motions in the form of inertial waves or inertial oscillations in an incompressible viscous fluid contained in a rotating spheroidal cavity was first formulated and studied by Poincaré (1885) and Bryan (1889). Upon realizing the limitation of Bryan's implicit solution using complicated modified spheroidal coordinates, Kudlick (1966) proposed a procedure that may be used to compute an explicit solution in spheroidal coordinates. However, the procedure requires an analytical expression for the $N$ real and distinct roots of a polynomial of degree $N$, where $N$ is a key parameter in the problem. When $0\,{\le}\,N\,{\le}\,2$, an explicit solution can be derived by using Kudlick's procedure. When $3\,{\le}\,N\,{\le}\,4$, the procedure cannot be practically used because the analytical expression for the $N$ distinct roots becomes too complicated. When $N\,{>}\,4$, Kudlick's procedure cannot be used because of the non-existence of an analytical expression for the $N$ distinct roots. For the inertial wave problem, Kudlick thus restricted his analysis to several modes for $1\,{\le}\,N\,{\le}\,2$ with the azimuthal wavenumbers $1\,{\le}\,m\,{\le}\,2$. We have found the first explicit general analytical solution of this classical problem valid for $0\,{\le}\,N\,{<}\,\infty$ and $0\,{\le}\,m\,{<}\,\infty$. The explicit general solution in spheroidal polar coordinates represents a possibly complete set of the inertial modes in an oblate spheroid of arbitrary eccentricity. The problem is solved by a perturbation analysis. In the first approximation, the effect of viscosity on inertial waves or oscillations is neglected and the corresponding inviscid solution, the pressure and the three velocity components in explicit spheroidal coordinates, is obtained. In the next approximation, the effect of viscous dissipation on the inviscid solution is examined. We have derived the first explicit general solution for the viscous spheroidal boundary layer valid for all inertial modes. The boundary-layer flux provides the solvability condition that is required to solve the higher-order interior problem, leading to an explicit general expression for the viscous correction of all inertial modes in a rapidly rotating, general spheroidal cavity. On the basis of the general explicit solution, some unusual and intriguing properties of the spheroidal inertial waves or oscillation are discovered. In particular, we are able to show that $\int_V ( {\bm u} \,{\bm \cdot}\, \nabla^2 {\bm u} )\,{\rm d}V \equiv 0$, where ${\bm u}$ is the velocity of any three-dimensional inviscid inertial waves or oscillations in an oblate spheroid of arbitrary eccentricity and $\int_V$ denotes three-dimensional integration over the volume of the spheroidal cavity.

Attention is drawn to a class of inviscid irrotational flows which satisfy the conditions at a time-dependent free surface exactly. The flows are related to the ellipsoids of Dirichlet (1860).

Depending on a parameter P, the cross-section may take the form of a variable ellipse (P < 0), a hyperbola (P > 0) or a pair of parallel lines (P = 0). The elliptical case was investigated both theoretically and experimentally by Taylor (1960). The hyperbolic case (P > 0) is remarkable in that the flow develops a singularity when the angle between the asymptotes approaches a right-angle. It is suggested that this solution represents a possible instability near the crest of a standing gravity wave of large amplitude.

In the intermediate case (P = 0) the solution describes an open-channel flow in which the fluid filaments are stretched uniformly in a horizontal direction. The latter flow is demonstrated experimentally.

We present a theory for the three-dimensional flow of a Bingham-plastic fluid in a shallow and wide channel. Focusing attention on slow flows appropriate for gentle slopes, low discharge rates or the final stage of deposition, we ignore inertia and apply the long-wave approximation. For steady flows, the velocity distribution, total discharge, and section-averaged flux are obtained analytically in terms of the fluid property and the geometry of the channel cross-section. Nonlinear stationary waves, which connect a uniform depth upstream to another uniform depth downstream, are then investigated, for both wet and dry beds. A numerical scheme is applied to calculate the transient flow evolution. The final development of the stationary wave due to steady discharge upstream is obtained numerically and the relation between the tongue-like shape of the wave front and the fluid property is discussed. The phase speed of the stationary wave is also derived analytically. Finally, the transient spreading of a finite fluid mass released from a reservoir after a dam break is simulated numerically. The transient development of the front and the final extent of deposition are examined.

A linear stability analysis for dune and ripple formation is presented that implements a rotational two-dimensional flow model valid in the smooth as well as in the transitional and rough flow regimes. Sediment is assumed to be transported as bedload, disregarding the role of suspension. Therefore, the main mechanism driving instability, for both ripples and dunes, is the phase lag between bed shear stress and bed elevation. Ripples are shown to be confined to relatively low values of the Shields parameter and of the particle Reynolds number. For higher values of the Shields parameter and of the particle Reynolds number (and thus of the Froude number and of the roughness Reynolds number), ripples are replaced by dunes. The present analysis ultimately allows for a successful unification of the theories of dune and ripple formation and for a clarification of the debated role of ripples on the formation of dunes. A good agreement between predicted and observed wavelengths for both ripples and dunes is found.

Swimming micro-organisms often have to propel themselves in complex non-Newtonian fluids. We carry out experiments with self-propelling helical swimmers driven by an externally rotating magnetic field in shear-thinning inelastic fluids. Similarly to swimming in a Newtonian fluid, we obtain for each fluid a locomotion speed that scales linearly with the rotation frequency of the swimmer, but with a prefactor that depends on the power index of the fluid. The fluid is seen to always increase the swimming speed of the helix, up to 50 % faster, and thus the strongest of such type reported to date. The maximum relative increase is for a fluid power index of approximately 0.6. Using simple scalings, we argue that the speed increase is not due directly to the local decrease of the flow viscosity around the helical filament, but hypothesise instead that it originates from confinement-like effect due to viscosity stratification around the swimmer.

Steady Couette and pressure-driven turbulent channel flows have large regions in which the gradients of the viscous and Reynolds stresses are approximately in balance (stress gradient balance regions). In the case of Couette flow, this region occupies the entire channel. Moreover, the relevant features of pressure-driven channel flow throughout the channel can be obtained from those of Couette flow by a simple transformation. It is shown that stress gradient balance regions are characterized by an intrinsic hierarchy of ‘scaling layers’ (analogous to the inner and outer domains), filling out the stress gradient balance region except for locations near the wall. The spatial extent of each scaling layer is found asymptotically to be proportional to its distance from the wall.

There is a rigorous connection between the scaling hierarchy and the mean velocity profile. This connection is through a certain function $A(y^+)$ defined in terms of the hierarchy, which remains $O(1)$ for all $y^+$. The mean velocity satisfies an exact logarithmic growth law in an interval of the hierarchy if and only if $A$ is constant. Although $A$ is generally not constant in any such interval, it is arguably almost constant under certain circumstances in some regions. These results are obtained completely independently of classical inner/outer/overlap scaling arguments, which require more restrictive assumptions.

The possible physical implications of these theoretical results are discussed.

Results from direct numerical simulations of an unsteady turbulent shear layer with a free surface are presented. The emphasis is on the interaction dynamics of the free surface with the coherent vortices in the underlying turbulent shear flow as well as the resulting free-surface signatures. Instantaneous vortex lines and isosurfaces of enstrophy indicate that coherent horseshoe vortical structures emerge from the random initial vorticity field. These horseshoe vortices impinge, break and reconnect onto the free surface, and then appear as two vortex connections with opposite signs on the surface. The two identified vortical structures correspond to ‘splatting’ and ‘swirling’ events, which have been observed in other experiments and simulations of free-surface/turbulence flows. Though free-surface depressions form near the vertical-vorticity centres in the connection processes, only a low correlation (≈50% to 60%) is found between the free-surface roughness (vertical deformation) and the connected normal vorticity. On the other hand, the free-surface curvatures and the tangential free-surface vorticities are better correlated (≈80% to 90%). The balance of enstrophy and the vorticity transport show that stretching and viscous dissipation along the direction of the vorticity vector dominate the vortex dynamics near the free surface. These two transport mechanisms are found to be responsible for the cancellation of the spanwise vorticity of the horseshoe-vortex heads and the annihilation of the surface-connected normal vorticities.

This paper presents analyses of particle image velocimetry measurements from a boundary layer on a flat plate subject to grid-generated free-stream turbulence. The pre-transition region and early stages of breakdown to turbulent spots are explored by means of quadrant analysis and quadrant hole analysis. By isolating the contributors to the Reynolds shear stresses, it is possible to identify coherent structures within the flow that are responsible for the production of TKE. It is found that so called ‘ejection’ events are the most significant form of disturbance, exhibiting the largest amplitude behaviour with increased negative spanwise vorticity. Sweep events become increasingly large close to the wall with increased Reynolds number and intermittency.

It is well known that most gas fluidized beds of particles bubble, while most liquid fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson (1995), through direct numerical integration of the volume-averaged equations of motion for the fluid and particles, that this distinction is indeed accounted for by these equations, coupled with simple, physically credible closure relations for the stresses and interphase drag. The aim of the present study is to investigate how the model equations afford this distinction and deduce an approximate criterion for separating bubbling and non-bubbling systems. To this end, we have computed, making use of numerical continuation techniques as well as bifurcation theory, the one- and two-dimensional travelling wave solutions of the volume-averaged equations for a wide range of parameter values, and examined the evolution of these travelling wave solutions through direct numerical integration. It is demonstrated that whether bubbles form or not is dictated by the value of Ω = (ρsv3t/Ag) 1/2, where ρs is the density of particles, vt is the terminal settling velocity of an isolated particle, g is acceleration due to gravity and A is a measure of the particle phase viscosity. When Ω is large (> ∼ 30), bubbles develop easily. It is then suggested that a natural scale for A is ρsvtdp so that Ω2 is simply a Froude number.

We examine the hydromagnetic stability of a radially stratified fluid rotating between two coaxial cylinders, with particular emphasis on the case when the angular velocity greatly exceeds both buoyant and Alfvén frequencies. If the magnetic field is predominantly azimuthal instabilities then have an essentially non-axisymmetric and wavelike character. Various bounds on their phase speeds and growth rates are derived, including a ‘quadrant’ theorem analogous to Howard's semicircle theorem for Kelvin–Helmholtz instability. Their strong tendency to propagate against the basic rotation (i.e. ‘westward’), previously noted by the author in the study of a more simplified (homogeneous) model, seems relatively insensitive to the generation mechanism (e.g. unstable gradient of magnetic field, angular velocity or density), but a number of counterexamples show that this constraint need not apply if the magnetic field displays significant spatial variations of direction as well as magnitude and that eastward-propagating amplifying modes are then possible.