We consider steady, two-dimensional viscous flow of two fluids near a corner. The two fluids meet at the wedge vertex and are locally in contact with each other along a straight line emanating from the corner. The double wedge, treated in polar coordinates, admits separable solutions with bounded velocities at the corner. We seek local solutions which satisfy all local boundary conditions, as well as partial local solutions which satisfy all but the normal-stress boundary conditions. We find that local solutions exist for a wide range of total wedge angles and that a class of individual wedge angles and stress exponents is selected. Partial local solutions exist for all combinations of individual wedge angles and the stress exponents are determined as functions of these angles and the viscosity ratio. In both cases, Moffatt vortices can be found. Our aim in this work is to describe local two-fluid flow by determining for which wedge angles solutions exist, identifying singularities in the stress at the corner, and identifying conditions under which Moffatt vortices can be present in the flow. Furthermore, for the single-wedge geometry, we identify for small capillary number non-uniformities present in solutions valid near the corner.

The origin and evolution of spatially stationary streamwise vortical structures in plane mixing layers with laminar initial boundary layers were recently examined quantitatively (Bell & Mehta 1992). When both initial boundary layers were made turbulent, such spatially-stationary streamwise structures were not measured which is indicative of the high sensitivity of these structures to initial conditions. In the present study, the effects of four different types of spanwise perturbations at the origin of the mixing layer were investigated. The wavelengths of the imposed perturbations were chosen to be comparable to the initial Kelvin-Helmholtz wavelength. For the first two perturbations, the boundary layers were otherwise left undisturbed. A serration on the splitter plate trailing edge was found to have a relatively small effect on the formation and development of the streamwise structures. The introduction of cylindrical pegs in the high-speed side boundary layer not only generated a regular array of vortex pairs, but also affected the mixing-layer growth rate and turbulence properties in the far-field region. For the other two perturbations, the initial boundary layers were tripped on the splitter plate. An array of vortex generators mounted in the high-speed boundary layer and a corrugated surface attached to the splitter plate trailing edge had essentially the same effects. Both imposed a regular array of relatively strong streamwise vortices in counter-rotating pairs upon the mixing layer. This resulted in large spanwise distortions of the mixing layer mean properties and Reynolds stresses. While the vorticity injection increased the growth rate in the near-field region as expected, the far-field growth rate was reduced by a factor of about two, together with the peak Reynolds stress levels. This result is attributed to the effect of the relatively strong streamwise vorticity in making the spanwise structures more three-dimensional and hence reducing entrainment during the pairing process. The imposed streamwise vorticity did not follow the pattern of increasing spanwise spacing seen in the ‘naturally occurring’ streamwise vorticity. The mean streamwise vorticity decayed with increasing streamwise distance in all cases, albeit at different rates.

A direct numerical simulation of a fully developed, low-Reynolds-number turbulent flow in a square duct is presented. The numerical scheme employs a time-splitting method to integrate the three-dimensional, incompressible Navier-Stokes equations using spectral/high-order finite-difference discretization on a staggered mesh; the nonlinear terms are represented by fifth-order upwind-biased finite differences. The unsteady flow field was simulated at a Reynolds number of 600 based on the mean friction velocity and the duct width, using 96 × 101 × 101 grid points. Turbulence statistics from the fully developed turbulent field are compared with existing experimental and numerical square duct data, providing good qualitative agreement. Results from the present study furnish the details of the corner effects and near-wall effects in this complex turbulent flow field; also included is a detailed description of the terms in the Reynolds-averaged streamwise momentum and vorticity equations. Mechanisms responsible for the generation of the stress-driven secondary flow are studied by quadrant analysis and by analysing the instantaneous turbulence structures. It is demonstrated that the mean secondary flow pattern, the distorted isotachs and the anisotropic Reynolds stress distribution can be explained by the preferred location of an ejection structure near the corner and the interaction between bursts from the two intersecting walls. Corner effects are also manifested in the behaviour of the pressure-strain and velocity-pressure gradient correlations.

An envelope model is applied to the case of a two-dimensional channel with ciliated parallel walls. The formulation assumes identical values of the longitudinal and transverse amplitudes, frequency and wavelength of the two walls; it allows for arbitrary phase relations and arbitrary (not too small) spacing, and it includes an externally imposed pressure gradient. General results of a second-order perturbation analysis of creeping flow are presented. The time-averaged steady mean velocity may be viewed as the sum of two contributions: that of the pressure gradient (Poiseuille flow), and that of ciliary-driven motion which, owing to nonlinearities, also depends on the pressure gradient and reduces to pure streaming in the absence of a pressure gradient. For zero pressure gradient, the ratio of the streaming velocity of the channel and that of a single sheet shows the degree to which streaming is augmented or impeded by flow interaction. This ratio increases for the symplectic and peristaltic cases, but decreases for the antiplectic case, as the width of the channel decreases for fixed values of phase relation and amplitudes. The net flow arising from streaming and pressure gradient is shown as pump characteristics, and associated efficiencies are given. The results indicate that propulsion (pumping) is greatest and most effective for symplectic metachronism in ciliated channels with predominantly transverse waves, that it is nearly as good for peristaltic motion, but that it is considerably inferior for antiplectic metachronism in channels with predominantly longitudinal waves.

Several aspects of the growth and departure of bubbles from a submerged needle are considered. A simple model shows the existence of two different growth regimes according to whether the gas flow rate into the bubble is smaller or greater than a critical value. These conclusions are refined by means of a boundary-integral potential-flow calculation that gives results in remarkable agreement with experiment. It is shown that bubbles growing in a liquid flowing parallel to the needle may detach with a considerably smaller radius than in a quiescent liquid. The study also demonstrates the critical role played by the gas flow resistance in the needle. A considerable control on the rate and size of bubble production can be achieved by a careful consideration of this parameter. The effect is particularly noticeable in the case of small bubbles, which are the most difficult ones to produce in practice.

During the collapse of an initially spherical cavitation bubble near a rigid wall, a reentrant jet forms from the side of the bubble farthest from the wall. This re-entrant jet impacts and penetrates the bubble surface closest to the wall during the final stage of the collapse. In the present paper, this phenomenon is modelled with potential flow theory, and a numerical approach based on conventional and hypersingular boundary integral equations is presented. The method allows for the continuous simulation of the bubble motion from growth to collapse and the impact and penetration of the reentrant jet. The numerical investigations show that during penetration the bubble surface is transformed to a ring bubble that is smoothly attached to a vortex sheet. The velocity of the tip of the re-entrant jet is always directed toward the wall during penetration with a speed less than its speed before impact. A high-pressure region is created around the penetration interface. Theoretical analysis and numerical results show that the liquid-liquid impact causes a loss in the kinetic energy of the flow field. Variations in the initial distance from the bubble centre to the wall are found to cause large changes in the details of the flow field. No existing experimental data are available to make a direct comparison with the numerical predictions. However, the results obtained in this study agree qualitatively with experimental observations.

The propagation of weakly nonlinear acoustic waves in a non-uniform medium is treated. It is assumed that the waves are one-dimensional. Non-uniformities arising from variable cross-section and stratification are included. The effect of non-uniformities on unidirectional waves on an infinite interval and resonant waves on a finite interval is discussed for a near-uniform reference state (geometrical acoustics limit) and for stronger non-uniformities in the finite-interval case. Nonlinearities are taken into account up to quadratic and, wherever necessary, cubic order in the wave amplitude.

Unidirectional waves in the geometrical acoustics limit can formally be reduced to the behaviour in a uniform system described by a kinematic wave equation with constant coefficients. For illustration acceleration waves in a weakly non-uniform medium are treated. The resonance case in the geometrical acoustics limit is closely related to resonance in a uniform system so that the methods developed for that situation require only slight modification. For larger influence of non-uniformity the geometrical acoustics limit does not apply and the resonance problem may lead to a Duffing oscillator type of behaviour.

It is known that the response of a cylindrical acoustic resonator to excitation by an oscillating piston can contain shock waves if the detuning is sufficiently small. However, the response of a spherical annular resonator is continuous, with an amplitude that depends on the detuning in the same way as does a Duffing equation. This paper discusses the response in resonators that deviate from being cylindrical and shows that, in general, the detuning range in which shocks are possible decreases as the geometrical imperfection increases.

A variational formulation for three-dimensional waves in a continuously stratified shear flow is used to derive the equations governing a resonant triad of waves. It is argued that in general, critical layers are necessary for the existence of explosive resonant triads.

The nonlinear dynamical behaviour of a conducting drop in a time-periodic electric field is studied. Taylor's (1964) theory on the equilibrium shape is extended to derive a dynamical equation in the form of an ordinary differential equation for a conducting drop in an arbitrary time-dependent, uniform electric field based on a spheroidal approximation for the drop shape and the weak viscosity effect. The dynamics is then investigated via the classical two-timing analysis and the Poincaré map analysis of the resulting dynamical equation. The analysis reveals that in the neighbourhood of a stable steady solution, an $O(\epsilon^{\frac{1}{3}})$ time-dependent change of drop shape can be obtained from an O(ε) resonant forcing. It is also shown that the probability of drop breakup via chaotic oscillation can be maximized by choosing an optimal frequency for a fixed forcing amplitude. As a preliminary analysis, the effect of weak viscosity on the oscillation frequency modification in a steady electric field is also studied by using the domain perturbation technique. Differently from other methods based on the theory of viscous dissipation, the viscous pressure correction is directly obtained from a consideration of the perturbed velocity field due to weak viscosity.

It is shown that the exponential growth rate of the fast kinematic dynamo instability can be related to the Lagrangian stretching properties of the underlying chaotic flow. In particular, a formula is obtained relating the growth rate to the finite time Lyapunov numbers of the flow and the cancellation exponent κ. (The latter quantity characterizes the extremely singular nature of the magnetic field with respect to fine-scale spatial oscillation in orientation.) The growth rate formula is illustrated and tested on two examples: an analytically soluble model, and a numerically solved spatially smooth temporally periodic flow.

Boiling and natural-convection processes in a horizontal, fluid-saturated porous layer are investigated. The layer is heated uniformly from below and is cooled from above. Volumetric cooling is also allowed. The thermodynamic structure consists of a liquid region overlying a two-phase region. Numerical techniques are used to solve the transient, two-dimensional equations in the liquid and two-phase regions, and at the phase-change interface. A parametric study is carried out in terms of the liquid-phase Rayleigh number (Ra) and the non-dimensional bottom heat flux (Qb). Three solution regimes are observed: conduction-dominated at low Ra, convection-dominated at intermediate Ra, and oscillatory convection at high Ra. In the convection-dominated regime, transitions to multiple cell patterns are observed as Qb is increased. Oscillatory convection appears to be triggered by asymmetric disturbances in the system. The effects of initial conditions and the stability of the solutions to perturbations are also investigated. The heat transfer correlations and qualitative flow patterns are in agreement with experiments.

In this paper we discuss the possibility that concentric vortex rings, with-associated circulation of opposite sign, can propagate steadily as a coherent pair. Inviscid flow considerations suggest that such a configuration, which we define as a vortex ring pair, may be possible. Numerical solutions of the Navier-Stokes equations for incompressible, laminar flow show that, although diffusion results in a continual redistribution of vorticity, a quasi-steadily propagating vortex ring pair could be attained in practice. Experiments are reported to test this idea by generating counterrotating vortex ring pairs by impulsive fluid motion through an annular orifice. Depending on the normalized impulse and the orifice radius ratio, the vortex rings are observed either (i) to propagate together until diffusive effects or vortex ring instability destroys the coherent motion, or (ii) the inner ring propagates to some maximum axial distance where it reverses its direction and returns to the orifice wall, leaving the outer ring free to continue its forward motion unabated. Numerical simulation shows that the stable flow of the vortex ring trajectories can be reasonably well reproduced. The boundary separating motion (i) from (ii) and the normalized inner ring penetration distance are found over the range of impulse and radius ratio covered by the experiments. Other observed features of vortex ring motion including self-similar trajectories of the spiral core centres during vortex sheet roll-up and ring instability are also presented.

The present numerical simulation explores a thermal–convective mechanism for oscillatory thermocapillary convection in a shallow rectangular cavity for a Prandtl number 6.78 fluid. The computer program developed for this simulation integrates the two-dimensional, time-dependent Navier–Stokes equations and the energy equation by a time-accurate method on a stretched, staggered mesh. Flat free surfaces are assumed. The instability is shown to depend upon temporal coupling between large-scale thermal structures within the flow field and the temperature sensitive free surface. A primary result of this study is the development of a stability diagram presenting the critical Marangoni number separating the steady from the time-dependent flow states as a function of aspect ratio for the range of values between 2.3 and 3.8. Within this range, a minimum critical aspect ratio near 2.3 and a minimum critical Marangoni number near 20 000 are predicted, below which steady convection is found.

It is known that stationary fluid with density that varies sinusoidally with small amplitude and wavenumber κ in the vertical direction is unstable to disturbances that are sinusoidal in a horizontal direction with wavenumber α. Small values of α/κ are the most unstable in the sense that a neutral disturbance exists at sufficiently small α/κ however small the Rayleigh number may be. The non-uniformity of density in the undisturbed state may be regarded as being a consequence of non-uniformity of concentration of extremely small solid particles in fluid. This paper is concerned with the corresponding instability of such a non-uniform dispersion when the particle size is not so small that the fall speed relative to the fluid is negligible. In the undisturbed state, which is an outcome of the well-known primary instability of a uniform fluidized bed with particle volume fraction ϕ0, the sinusoidal distribution of concentration propagates vertically, and in the steady state relative to this kinematic wave particles fall with speed V (= |ϕdU/dϕ|ϕ0), where U(ϕ) is the mean speed of fall of particles, relative to zero-volume-flux axes, in a uniform dispersion with volume fraction ϕ. This particle convection with speed V transports particle volume and momentum and tends to even out variations of a disturbance in the vertical direction and thereby to suppress a disturbance, especially one with small α/κ. Analysis of the behaviour of a disturbance is based on the equation of motion of the mixture of particles and fluid and an assumption that the disturbance velocities of the particles and the fluid are equal (as is suggested by the relatively small relaxation time of particles). The method of solution used in the associated pure-fluid problem is also applicable here, and values of the Rayleigh number as a function of α/κ for a neutral disturbance and a given value of the new non-dimensional parameter involving V are found. Particle convection with only modest values of V stabilizes all disturbances for which α/κ < 1 and increases significantly the Rayleigh number for a neutral disturbance when α/κ > 1. It appears that under practical conditions disturbances with α/κ above unity are unstable, although ignorance of the values of parameters characterizing a fluidized bed hinders quantitative conclusions.

The near-wall flow structure of a zero-pressure-gradient flat-plate turbulent boundary layer with a single-layer viscoelastic compliant surface was visualized using the hydrogen-bubble technique. The compliant materials were made by mixing silicone elastomer with silicone oil. The flow-visualization experiments indicated low-speed wall streaks with increased spanwise spacing and elongated spatial coherence compared to those obtained on a rigid surface. More interestingly, an intermittent relaminarization-like phenomenon was observed at low Reynolds numbers for the particular compliant surface investigated. Apparently, the observed changes in the near-wall flow structure over the compliant surface are caused by the stable interaction between the compliant surface and the turbulent flow-field. Optical holographic interferometry and laser Doppler velocimetry were also employed to obtain the basic parameters of the turbulent boundary layers and the flow-induced compliant-surface displacements to better understand the physics of the interaction between a turbulent boundary layer and a passive compliant surface.

We consider non-axisymmetric motion of a rigid particle in a cylindrical fluid-filled tube, with negligible inertial effects. The particle is assumed to fit closely in the tube, and lubrication theory is used to describe the fluid flow in the narrow gap between the particle and the tube wall. The solution to the Reynolds lubrication equation and the components of the resistance matrix are expressed in terms of a Green's function. For the case in which the gap is almost uniform, the Green's function is expanded as a power series in a small parameter δ, characteristic of the variations in gap width, and the first two terms are obtained.

The velocity of a freely suspended axisymmetric particle driven by a pressure difference along the tube is deduced from the resistance matrix. According to the results at first order in δ, in general the particle moves transversely with a constant velocity. In the absence of higher-order effects, it would eventually collide with the wall. Motion along the tube axis is a neutrally stable solution to the equations of motion at first order. However, if effects at second order in δ are included, motion of an axisymmetric particle along the tube axis is stable or unstable depending on its shape. Generally, if the particle is narrower near the front than near the rear, and the width near the middle is at least as large as the mean of the widths near the front and rear, then its motion is stable. Numerical calculations (not restricted to small δ) confirm these results for axisymmetric particles, and show that a non-axisymmetric shape similar to a red blood cell has a stable equilibrium position in the tube.

An extension of an earlier theory of the two-dimensional incompressible flow past an isolated body is described. For a crossflow cascade of bodies, each of unit size in the crossflow direction and distance 2H apart, the region of validity of the extended theory covers H [Gt ] 1. A comparison with recent numerical calculations is favourable and a tentative asymptotic structure for the case of H = O(1) is described.

The paper is devoted to the creation of an original model of propagation of weak but finite-amplitude waves initiating an exothermic process connected with chemical reaction or relaxation of a non-equilibrium medium. The medium is single phase (a fluid), or a homogeneous two-phase medium (liquid with gas bubbles). A nonlinear differential model describing wave-kinetic interaction and wave evolution is derived. The linear dispersion and dissipative features of these systems are investigated both analytically and numerically. Attention is paid to explanation of the physical mechanisms resulting in the formation of a self-sustained solitary wave, which in terms of synergetics could be called a ‘dissipative structure’.

Experiments are performed on axisymmetric spreading of viscous drops on glass plates. Two liquids are investigated: silicone oil (M-100), which spreads to ‘infinity’, and paraffin oil, which spreads to a finite-radius steady state. The experiments with silicone oil partly recover the behaviour of previous workers’ data; those experiments with paraffin oil provide new data. It is found that gravitational forces dominate at long enough times while at shorter times capillary forces dominate. When the plate is heated or cooled with respect to the ambient gas, thermocapillary forces generate flows that alter the spreading dynamics. Heating (cooling) the plate is found to retard (augment) the spreading. Moreover, in case of partial wetting, the drop radius finally approached is smaller (larger) for a heated (cooled) plate. These data are all new. All these observations are in good quantitative agreement with the related model predictions of Ehrhard & Davis (1991). A breakdown of the axisymmetric character of the flow is observed only for very long times and/or very thin liquid layers.