The paper describes the phenomenon of drop suspension above an air-blown porous surface, which is similar to the well-known levitation of a drop evaporating above a hot plate (‘spheroidal state’ or the Leidenfrost phenomenon). It has been shown that the basis of this similarity is the close analogy between the hydrodynamic mechanisms of drop suspension. Together with the viscous mechanism, the effect of gas- or heat-flow choking under the drop plays an important role here. The latter conditions the threshold character of the above phenomena. A mathematical model of cool- and hot-drop suspension is offered which does not contain any a priori assumptions about the drop form and can be applied to the critical range of parameters. An approximation has also been considered in which the bottom of the drop is assumed to be flat, which allows us to carry out an analysis within a wide range of parameters. The simplest version of this approximation is a disk model, where the problems considered are found to be similar. This version allows analytical solution. The model developed has been verified in a ‘cool’ experiment. The theoretical and experimental data have been shown to be in qualitative (and in some respects in quantitative) agreement.

Flows in water-filled rectangular cavities due to an applied horizontal temperature gradient are examined for Rayleigh numbers, based on height, R [ges ] 106 and aspect ratios (length/height) L [ges ] 2. Laser Doppler velocimetry measurements of the horizontal velocity distribution throughout the core are complemented with local temperature measurements and interferometry observations. The results show that the core stream function is aspect-ratio dependent when R > O(105) and the Prandtl number is fixed. When R [les ] 106 it appears that the layers on the horizontal surfaces fill the cavity depth if L [ges ] 3. For values of R [ges ] 107 and L = 2 the motion in the core is extremely small and the mass flux occurs in layers adjacent to the horizontal walls. Computations of the heat transfer across the cavity are in good agreement with numerical solutions of the vertical boundary-layer equations.

An experimental investigation is carried out into the characteristics of the velocity and thermal fields in free and wall jets in which the buoyancy force opposes the flow. The flow configuration considered is that of a negatively buoyant two-dimensional jet discharged adjacent to a vertical surface, as well as that discharged away from the boundaries of the region. Such convective flows are frequently encountered in heat-rejection processes and in enclosure fires, where, at various locations, the buoyancy force is upward while the flow is downward, resulting in negative buoyancy. An experimental system is developed to study the downward penetration of such jets in which the buoyancy force opposes the externally induced flow. The penetration distance is measured and related to the inflow conditions, particularly the temperature and velocity at the discharge location. A steady state is simulated by allowing the fluid to flow out of the enclosure at the open top. The velocity and temperature distributions are also measured, in order to understand the basic nature of such flows. Several other effects, such as the entrainment into the flow, are also considered in this study.

The emergence of a shockwave from the open end of a shock tube is studied, with special emphasis on test fluids of high molar heat capacity, i.e. retrograde fluids. A variety of wavelike vapour-liquid phase changes are observed in such fluids, including the liquefaction shock, mixture-evaporation shock, condensation waves associated with shock splitting and liquid-evaporation waves (these phenomena have analogues in the polymorphic phase changes of solids; only the first two are treated in this paper). The open end of the shock-tube test section discharges into an observation chamber where photographs of the emerging flow are taken. Calculations were performed with the Benedict-Webb-Rubin, van der Waals and other equations of state. Numerical (finite-difference) predictions of the flow were made for single-phase and two-phase flows: solutions were tested against the experimental shock diffraction and vortex data of Skews. The phase-change properties of the test fluid can be quantified by the ‘retrogradicity’ r(T), measuring the difference in slope between the P, T isentrope and the vapour-pressure curve, and the ‘kink’ k(T), measuring the difference between the single-phase and mixture sound speeds. Mixture-evaporation (i.e. rarefaction) shocks appear to have a sonic-sonic or double Chapman-Jouguet structure and show agreement with amplitude predictions based on k(T). Liquefaction shocks are found to show a reproducible transition from regular, smooth shock fronts to irregular, chaotic shock fronts with increasing shock Mach number. This transition can be correlated with published stability limits.

Model experiments are performed in a long, thin-walled, fluid-filled, latex tube in which the fluid is locally suddenly stopped, starting from a steady flow, thus simulating the wave phenomenon generated by the final closure of the aortic valve. The resulting waveform is determined as it propagates upstream. The effect of a local step-wise change in compliance close to the valve, representing the aortic sinus section, is investigated. The observed phenomena are analysed by means of a quasi-one-dimensional model, solved by the method of characteristics, taking into account the influence of nonlinearities, wall shear stress, viscoelastic wall properties and wave reflections. The theoretical computations are well confirmed by the experimental results. The pressure jump, induced by the valve closing, appeared to be slightly affected by nonlinearities. The decrease of the pressure jump while propagating upstream and the gradual pressure increase that follows the pressure jump are caused by the effect of wall shear stress. The local change in compliance generates the expected wave reflections and has a strong influence on the rise-time of the wave front. The experiments confirmed the prediction that wall viscoelasticity is the dominant factor in the gradual decay of the slope of the wave.

The flow induced by injection of a given amount of buoyancy or hot fluid from a localized source in a viscous fluid is investigated for conditions under which the Reynolds number Re is small compared with one, and the dimensionless buoyancy or Rayleigh number Ra is large compared with one. Laboratory experiments show that the buoyant fluid rises in the form of an extremely viscous ‘thermal’ which enlarges with time as a result of entrainment of surrounding fluid. The formation of a stable ‘chemical ring’ or torus of passive tracer similar in appearance to high Reynolds-number vortex rings is a notable feature of the creeping flow for high Rayleigh numbers. The possibility of large variations of viscosity due to temperature differences is included. A self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes’ flow. Experiments at 2.5 × 102 < Ra < 2.5 × 104 and Re < 10−2 demonstrate the nature of extremely viscous thermals, support the similarity solution and enable evaluation of a proportionality constant. Possible applications of the results to dispersion by viscous drops and particularly to thermal convection in the Earth's solid mantle are mentioned.

The motions of fluid particles within and around a mass of hot, buoyant material (a thermal) rising through an extremely viscous, unbounded environment are computed using a simple kinematic model. The model is based on a similarity solution by Griffiths (1986a) and allows for growth of thermals due to outward diffusion of heat. Particle motions are also computed for the case of a non-expanding, isothermal sphere, such as a bubble of relatively low-viscosity fluid, in Stokes flow. Motions induced in the surroundings lead to large vertical displacements: the ‘total drift’ function and hydrodynamic mass corresponding to those defined for the inviscid case by Darwin (1953) and Lighthill (1956) are infinite in this unbounded geometry. Rotation of initially horizontal fluid elements (strain) in the surroundings is discussed.

All material lying within an expanding thermal becomes confined at later times to a torus (dye ring) if the Rayleigh number for the thermal is large, to a central tapered blob if Ra < 50, or to an umbrella-shaped cap with narrow stem if Ra takes intermediate values. The ‘mushroom’ shape widely observed for tracers within laminar elements in thermal convection is predicted for intermediate-to-large Rayleigh numbers. Buoyancy and heat, on the other hand, are assumed to remain evenly distributed throughout an enlarging sphere. Laboratory experiments illustrate and confirm the predictions of the model.

The time-harmonic three-dimensional finite-depth floating-body problem is reformulated as a boundary integral equation. Using the elementary fundamental solution that satisfies the boundary condition on the sea bottom but not the linearized free surface condition, the integral equation extends over both the ship hull and the free surface. It is shown that this integral equation is free of irregular frequencies, that is, it has at most one solution.

The nonlinear equations describing convection in the form of thermal Rossby waves in a rotating annulus are solved both by an analytical perturbation theory and by a numerical method. It is shown that even in the absence of curvature of the surfaces bounding the fluid annulus in the axial direction a mean flow is generated by Reynolds stresses. The good agreement between analytical expressions and numerical results indicates that the former are valid over a larger domain of the parameter space than may be expected on the basis of the analysis of convection rolls in a non-rotating layer. This is caused in part by the reduced release of potential energy accompanying the reduced convective heat transport owing to the drift of the convection columns. The effect of curvature causes the replacement of the basic mode of convection by a different mode characterized by a double roll structure. The associated zonal mean flow is typically stronger than in the case without curvature.

We consider three-dimensional water-wave diffraction and radiation by a structure consisting of a number of separate (vertically) non-overlapping members in the context of linearized potential flow. An interaction theory is developed which solves the complete problem, predicting wave exciting forces, hydrodynamic coefficients and second-order drift forces, but is based algebraically on the diffraction characteristics of single members only. This method, which includes also the diffraction interaction of evanescent waves, is in principle exact (within the context of linearized theory) for otherwise arbitrary configurations and spacings. This is confirmed by a number of numerical examples and comparisons involving two or four axisymmetric legs, where full three-dimensional diffraction calculations for the entire structures are also performed using a hybrid element method. To demonstrate the efficacy of the interaction theory, we apply it finally to an array of 33 (3 by 11) composite cylindrical legs, where experimental data are available. The comparison with measurements shows reasonable agreement.

The present method is valid for a large class of arrays of arbitrary individual geometries, number and configuration of bodies with non-intersecting vertical projections. Its application should make it unnecessary to perform full diffraction computations for many multiple-member structures and arrays.

A turbulent spot is induced by a spark triggered in one of the laminar boundary layers in the entrance region of a two-dimensional duct flow. The development of the spot is studied using ensemble-averaged velocity and wall shear stress in the plane of symmetry of the spot. Following an initial growth of the spot, the potential-flow field associated with this spot triggers a second spot on the opposite wall of the duct. This new spot propagates at the same convection velocity as the original spot and grows until the turbulent regions occupied by the two spots completely fill the width of the duct. This transition mechanism differs significantly from that observed for a plane Poiseuille flow, where the spot fills the duct almost immediately after it is generated.

Mixed-layer deepening in stratified fluid has been studied experimentally in mean-shear-free turbulence generated by an oscillating grid. Conditions were varied over a wide range and both two-layered and constant-gradient fluid systems were considered. It is shown that the mixed-layer deepening rate is represented well by power laws, and when local scaling is used all the data can be collapsed on an entrainment relation E = K Ri−n with n = 1.50±0.05 when Ri [gsim ] 7. This power law suggests that the turbulent kinetic energy is made available for mixing on a buoyancy timescale characteristic of eddy recoil or internal-wave breaking rather than a turbulent-eddy overturning timescale. In the constant-gradient situation internal waves are generated which radiate energy away from the interface. An evaluation of the radiated energy indicates, however, that generally energy radiation does not affect the entrainment rate. The coefficient K therefore has the same value (K ≈ 3.8) in linearly stratified fluid as in the two-layer situation. The interface thickness is found to be a function of stability, but reaches an asymptotic value of h/D = 0.055 when Ri is very large. There is some indication that the interface thickness is also a weak function of Reynolds number.

The thermocapillary flow field in a uniformly rotating liquid cylinder heated from above is calculated using linear boundary-layer theory appropriate for small values of the Ekman number. The results show that the thermocapillary flow is confined to a thin layer at the liquid-gas interface if the temperature difference across the cylinder is sufficiently small. The interior flow is a uniform rotation with the endplates.

The flow due to centrifugal buoyancy is also analysed using the same theory. The magnitude of this flow compared with the thermocapillary motion is small in typical circumstances. However, it does influence the temperature field in the interior of the cylinder, whereas the thermocapillary motion does not. Full details of these flows and the first-order corrections to the interface shape are presented.

We study the deformation and breakup of a low-viscosity slender drop in a linear flow, $\overline{\boldmath v}^{\infty} \overline{\boldmath L}\cdot\overline{\boldmath x} $, assuming that the drop remains axisymmetric. We find that the drop stretches as if it were immersed in an axisymmetric extensional flow with a strength $\overline{\boldmath D}:\overline{\boldmath m}\overline{\boldmath m} $, where $\overline{\boldmath D} = \frac{1}{2}(\overline{\boldmath L}+\overline{\boldmath L}^T)$, and $\overline{\boldmath m} $ is the orientation of the drop, and rotates as if it were a material element in a hypothetical flow $\overline{\boldmath M}=G\overline{\boldmath D}+\overline{\Omega} $, where $\overline{\Omega} = \frac{1}{2}(\overline{\boldmath L}^T - \overline{\boldmath L})$, and G is a known function of the drop length. The approximations involved in the model are quite good when $\overline{\boldmath M}$ has only one eigenvalue with a positive real part, and somewhat less precise when $\overline{\boldmath M}$ has two eigenvalues with positive real parts. In the suitable limits the model reduces to Buckmaster's (1973) model for axisymmetric extensional flow and to the linear-axis version of the more general model proposed by Hinch & Acrivos (1980) for simple shear flow. In establishing a criterion for breakup for all linear flows, we find that the relevant quantity that specifies the flow is the largest positive real part of the eigenvalues of $\overline{\boldmath M}$, which depends on the drop length and the imposed flow. Our predictions are in reasonable agreement with the recent experimental data of Bentley (1985) for general two-dimensional linear flows and those of Grace (1971) for simple shear and hyperbolic extensional flow. We also present calculations for a class of three-dimensional flows as an illustration of the behaviour of three-dimensional flows in general.

In this paper continuation methods are applied to the axisymmetric Navier-Stokes equations in order to investigate how the stability of spherical Couette flow depends on the gap size σ. We find that the flow loses its stability due to symmetry-breaking bifurcations and exhibits a transition with hysteresis into a flow with one pair of Taylor vortices if the gap size is sufficiently small, i.e. if σ [les ] σB.

In wider gaps, i.e. for σB < σ [les ] σF, both flows, the spherical Couette flow and the flow with one pair of Taylor vortices, are stable. We predict that the latter exists in much wider gaps than previous experiments and calculations showed. Taylor vortices do not exist if σ > σF. The numbers σB and σF are computed by calculating the instability region of the spherical Couette flow and the region of existence of the flow with one pair of Taylor vortices.

A pseudo-spectral numerical scheme is used to study two-dimensional, single-cell, time-dependent convection in a square cross-section of fluid saturated porous material heated from below. With increasing Rayleigh number R convection evolves from steady S to chaotic NP through the sequence of bifurcations S→P(1)→QP2→P(2)→NP, where P(1) and P(2) are simply periodic regimes and QP2 is a quasi-periodic state with two basic frequencies. The transitions (from onset of convection to chaos) occur at Rayleigh numbers of 4π2, 380–400, 500–520, 560–570, and 850–1000. In the first simply periodic regime the fundamental frequency f1 varies as $R^{\frac{7}{8}} $ and the average Nusselt number $\overline{Nu}$ is proportional to $R^{\frac{2}{3}}$; in P(2), f1 varies as $R^{\frac{3}{2}}$ and $\overline{Nu}\propto R^{\frac{11}{10}}$. Convection in QP2 exhibits hysteresis, i.e. if the QP2 state is reached from P(1) (P(2)) by increasing (decreasing) R then the frequency with the largest spectral power is the one consistent with the extrapolation of f1 according to $R^{\frac{7}{8}}(R^{\frac{3}{2}})$. The chaotic states are characterized by spectral peaks with at least 3 fundamental frequencies superimposed on a broadband background noise. The time dependence of these states arises from the random generation of tongue-like disturbances within the horizontal thermal boundary layers. Transition to the chaotic regime is accompanied by the growth of spectral components that destroy the centre-symmetry of convection in the other states. Over-truncation can lead to spurious transitions and bifurcation sequences; in general it produces overly complex flows.

The method of matched asymptotic expansions is used to extend the short-wave asymptotics of the transmission coefficient T by the addition of the terms of order 1/N5, (logN)2/N6 and log N/N6 as N → ∞ (where N = wavenumber times cylinder radius). The result is the formula \begin{eqnarray*} T &=& \frac{2{\rm i}}{\pi N^4}\exp (-2{\rm i}N)\left[1+\frac{4\log N}{\pi N}-\frac{4}{\pi N} \bigg(2-\gamma-\log 2+\frac{{\rm i}\pi}{8}\bigg)+\frac{8(\log N)^2}{\pi^2N^2}\right.\\ && \left.-\frac{8\log N}{\pi^2N^2}\bigg(5-2\gamma - \log 4+\frac{{\rm i}\pi}{4}\bigg)\right] + O\bigg(\frac{1}{N^6}\bigg)\quad {\rm as}\;N\rightarrow \infty \end{eqnarray*} (where γ = Euler's constant). The first term above is that derived rigorously by Ursell (1961) using an integral-equation method; the second term is that added by Leppington (1973) using matched asymptotic expansions; and the next three terms are those derived in this paper. Significant agreement between numerical values of T obtained from the completed fifth-order asymptotics and those obtained using Ursell's multipole expansions is demonstrated for 8 [les ] N [les ] 20 (table 2). The extensions of the perturbation expansions for the potential in the various fluid sub-domains (used in the method of matched expansions) provide some interesting cross-checks, between the solutions for potentials occurring later in the series and determined at advanced matching stages, with those for potentials occurring earlier on and determined independently at an earlier stage in the matching process. Some examples are given.

Ray theory is used to investigate the interaction of a short high-frequency progressive internal wave of infinitesimal amplitude with a long progressive near-inertial wave of arbitrary amplitude. Weak-interaction theory would, if applicable, predict that the largest changes in short-wave properties occur when the resonance condition c = cg is satisfied, where c is the phase velocity of the long wave and cg is the group velocity of the short wave. The present calculation confirms this prediction only when the long wave has exceedingly small amplitude (peak velocities of order 0.1 cm/s).

However, when the background velocity has a realistic amplitude (e.g. oceanic values are of order 20 cm/s) the resonance condition fails to be relevant. For example, waves which initially have c = cg become trapped in low-shear regions and consequently experience very small changes in wavenumber. Other short waves, which initially have cg [Lt ] c and hence violate the resonance condition, exhibit large and permanent changes in vertical wavenumber.

Remarkably, it is found that these permanent changes are much more likely to be decreases, rather than increases, in wavenumber. This can be explained as follows. Short waves which enter an inertial-wave packet experience both increases and decreases in wavenumber. However, at times when the wavenumber is relatively large, the group velocity is relatively small and the short wave is unlikely to escape from the inertial packet, whereas small wavenumber and large group velocity assist the escape of the short-wave group. Consequently the short waves that leave the inertial packet tend to have a smaller average wavenumber than those that enter. Thus the net effect of a near-inertial packet on a collection of short waves appears to be an increase in vertical wavelength and frequency.

The stability (i) of fully three-dimensional magnetostatic equilibria of arbitrarily complex topology, and (ii) of the analogous steady solutions of the Euler equations of incompressible inviscid flow, are investigated through construction of the second variations δ2M and δ2K of the magnetic energy and kinetic energy with respect to a virtual displacement field η(x) about the equilibrium configuration. The expressions for δ2M and δ2K differ because in case (i) the magnetic lines of force are frozen in the fluid as it undergoes displacement, whereas in case (ii) the vortex lines are frozen, so that the analogy between magnetic field and velocity field on which the existence of steady flows is based does not extend to the perturbed states. It is shown that the stability condition δ2M > 0 for all η(x) for the magnetostatic case can be converted to a form that does not involve the arbitrary displacement η(x), whereas the condition δ2K > 0 for all η for the stability of the analogous Euler flow cannot in general be so transformed. Nevertheless it is shown that, if δ2M and δ2K are evaluated for the same basic equilibrium field, then quite generally \[ \delta^2 M + \delta^2 K > 0\quad \hbox{(all non-trivial}\;\eta). \] A number of special cases are treated in detail. In particular, it is shown that the space-periodic Beltrami field \[ {\boldmath B}^{\rm E} = (B_3\cos \alpha z+B_2\sin \alpha y, B_1\cos \alpha x + B_3\sin \alpha z, B_2\cos \alpha y + B_1 \sin \alpha x) \] is stable (i.e. δ2M > 0 for all η) and that the medium responds in an elastic manner to perturbations on a scale large compared with α−1. By contrast, it is shown that δ2K is indefinite in sign for the analogous Euler flow, and it is argued that the flow is unstable to certain large-scale helical perturbations having the same sign of helicity as the unperturbed flow. It is conjectured that all topologically non-trivial Euler flows are similarly unstable.

It is known that when two different species of small particles with radii in the range 10–100 μm are dispersed uniformly in fluid and are settling under gravity, there may be a tendency for the particles of each species to gather together and develop a bulk vertical streaming motion, which results in much larger magnitudes of the mean velocity of at least one of the two types of particle. After a review of the published data on such streaming motions we describe new visual and photographic observations of the evolution of the internal structures (which are sometimes more globular than columnar) in a large number of different bidisperse systems. It appears that the observed structures result from instability of the statistically homogeneous dispersion to small concentration disturbances for certain combinations of values of the ratios of the sizes and densities of the two types of particle and the volume fractions of the two species.

The condition for growth of a sinusoidal disturbance of the homogeneous dispersion is derived from the two particle-conservation equations and is found to involve the dependence of the two mean particle velocities on the two particle concentrations in a homogeneous dispersion. Previously calculated values of these mean velocities for a dilute dispersion suggest that the condition for instability is indeed satisfied for not-too-small particle concentrations and certain combinations of the size and density ratios of the two particle species. The results of the instability theory are generally consistent with the observed features of the structures, regarded as finite-amplitude forms of the small disturbances with maximum growth rate.