Mixing and transport processes associated with slow viscous flows are studied in the context of a blinking stokeslet above a plane rigid boundary. Whilst the motivation for this study comes from feeding currents due to cilia or flagella in sessile micro- organisms, other applications in physiological fluid mechanics where eddying motions occur include the enhanced mixing which may arise in ‘bolus’ flow between red blood cells, peristaltic motion and airflow in alveoli. There will also be further applications to micro-engineering flows at micron lengthscales. This study is therefore of generic interest because it analyses the opportunities for enhanced transport and mixing in a Stokes flow environment in which one or more eddies are a central feature.

The central premise in this study is that the flow induced by the beating of microscopic flagella or cilia can be modelled by point forces. The resulting system is mimicked by using an implicit map, the introduction of which greatly aids the study of the system's dynamics. In an earlier study, Blake & Otto (1996), it was noticed that the blinking stokeslet system can have a chaotic structure. Poincaré sections and local Lyapunov exponents are used here to explore the structure of the system and to give quantitative descriptions of mixing; calculations of the barriers to diffusion are also presented. Comparisons are made between the results of these approaches. We consider the trajectories of tracer particles whose density may differ from the ambient fluid; this implies that the motion of the particles is influenced by inertia. The smoothing effect of molecular diffusion can be incorporated via the direct solution of an advection–diffusion equation or equivalently the inclusion of white noise in the map. The enhancement to mixing, and the consequent ramifications for filter feeding due to chaotic advection are demonstrated.

So-called ‘moonpools’ are vertical openings through the deck and hull of ships or barges, used for marine and offshore operations, such as pipe laying or recovery of divers. In the present study rectangular moonpools of large horizontal dimensions are considered. The natural modes of oscillation of the inner free surfaces are determined, under the assumption of infinite water depth and infinite length and beam of the barges that contain the moonpools. The problem is treated in two and three dimensions, via linearized potential flow theory. Results are given for the natural frequencies and the associated shapes of the free surface, for wide ranges of the geometric parameters. Simple quasi-analytical approximations are derived that yield the natural frequencies. The most striking result is that the natural frequencies of the longitudinal sloshing modes increase without bounds when both the draught and the width decrease to zero, the length of the moonpool being kept constant. As a corollary the problem of waves travelling in a channel through a rigid ice sheet is addressed and their dispersion equation is derived. The same behaviour is obtained: the waves travel increasingly faster as both the draught and the width of the channel are reduced.

Ensemble-averaged equations are derived for small-amplitude acoustic wave propagation through non-dilute suspensions. The equations are closed by introducing effective properties of the suspension such as the compressibility, density, viscoelasticity, heat capacity, and conductivity. These effective properties are estimated as a function of frequency, particle volume fraction, and physical properties of the individual phases using a self-consistent, effective-medium approximation. The theory is shown to be in excellent agreement with various rigorous analytical results accounting for multiparticle interactions. The theory is also shown to agree well with the experimental data on concentrated suspensions of small polystyrene particles in water obtained by Allegra & Hawley and for glass particles in water obtained in the present study.

Kolmogorov's equation, which relates second- and third-order moments of the velocity increment, provides a simple method for estimating the mean energy dissipation rate 〈ε〉 for homogeneous and isotropic turbulence. However, this equation is usually not verified in small to moderate Reynolds number flows. This is due partly to the lack of isotropy in either sheared or non-sheared flows, and, more importantly, to the influence, which is flow specific, of the inhomogeneous and anisotropic large scales. These shortcomings are examined in the context of the central region of a turbulent channel flow. In this case, we propose a generalized form of Kolmogorov's equation, which includes some additional terms reflecting the large-scale turbulent diffusion acting from the walls through to the centreline of the channel. For moderate Reynolds numbers, the mean turbulent energy transferred at a scale r also contains a large-scale contribution, reflecting the non-homogeneity of these scales. There is reasonable agreement between the new equation and hot-wire measurements in the central region of a fully developed channel flow.

The detailed experimental study conducted by Félix Savart in 1833 has revealed the existence of water bells when a cylindrical jet of diameter D0 impacts with the velocity U0 normally on to a disc of diameter Di. We continue this study with a Newtonian fluid characterized by its density, ρ, kinematic viscosity, v, and surface tension, σ. We first show that for a given Reynolds number, Re ≡ U0D0/v, and Weber number, We ≡ ρU20D0/σ, the domain where the bells exist in terms of the diameter ratio, X ≡ Di/D0, extends from the minimum value, X−:

formula here

up to the maximum value, X+:

formula here

In the domain, X ∈]X−, X+[, the liquid film which results from the impact of the jet detaches at the edge of the disc, forming an angle ψ0 with the direction of the jet. In the non-viscous limit, we show that this angle is determined by the nonlinear equation

formula here

where ψmax0 corresponds to the limit of ψ0 for We >> 1. In that limit, we find that cos (ψmax0) ≈ 1 − 0.352X2, for X < 1, and cos (ψmax0) ≈ 0.1 for X > 1.

The shape of the resulting bell is shown to be a catenary, first analytically described by Joseph Boussinesq in 1869. This shape results from the equilibrium between surface tension and centrifugal acceleration and is characterized by the length L ≡ D0We/16. This solution holds in the low-gravity limit, gL/U20 >> 1, and when the pressure difference, p, across the liquid sheet is small, pL/(2σ) >> 1. Considering the dynamics of formation of that catenary, we show that it is characterized by a quasi-constant velocity along the jet axis.

Finally, we show that these bells are not always stationary and may even undergo self-sustained oscillations. Studying their stability, we derive a general stability criterion and show the sensitivity of the bells to both the pressure difference across the liquid sheet and to the ejection angle. In this latter case, we find a critical angle of ejection above which the bell is periodically destroyed and created. The period of the cycle is shown to scale linearly with the formation time of the bell.

This paper aims at a description of boundary-layer flow which is subjected to free-stream turbulence in the range from 1–6% and is based on both flow visualization results and extensive hot-wire measurements. Such flows develop streamwise elongated regions of high and low streamwise velocity which seem to lead to secondary instability and breakdown to turbulence. The initial growth of the streaky structures is found to be closely related to algebraic or transient growth theory. The data have been used to determine streamwise and spanwise scales of the streaky structures. Both the flow visualization and the hot-wire measurements show that close to the leading edge the spanwise scale is large as compared to the boundary-layer thickness, but further downstream the spanwise scale approaches the boundary-layer thickness. Wavenumber spectra in both the streamwise and the spanwise directions were calculated. A scaling for the streamwise structure of the disturbance was found, which allows us to collapse the spectra from different downstream positions. The scaling combines the facts that the streaky structures increase their streamwise length in the downstream direction which becomes proportional to the boundary-layer thickness and that the energy growth is algebraic, close to proportional to the downstream distance.

The unsteady travelling ‘spots’ or spot-like disturbances are produced, in an otherwise planar boundary layer, by an initial impulse/blip, from wall forcing or from nearby external forcing. Theory and computations are described for the evolving spot-like structure, yielding initial-value problems for inviscid spot-like disturbances, commencing near the onset of an adverse pressure gradient. A transient stage incorporates the initial conditions, following which adverse pressure gradient effects become significant. Leading and trailing critical layers then form, which confine and define the spot-like disturbance, and these depart from the wall downstream accompanied by disturbance amplification and mean flow distortion. The interplay of adverse pressure gradient effects with three-dimensionality, nonlinearity and non-parallelism is considered in turn.

Three-dimensional effects provoke a universal closed planform of spot-like disturbance, which has a different side behaviour from the zero-gradient case. Nonlinear interactions eventually change the internal structure, particularly at the spot-like disturbance leading edge, while pointing to the mean-flow alteration underhanging the spot-like disturbance and to a pressure-feedback alteration for the region behind the spot-like disturbance. These two alterations offer complementary mechanisms for describing the calmed region trailing a spot-like disturbance, in which an attached thinned wall layer is identified. Non-parallel effects lead to enhanced spot-like disturbance growth and larger-scale/shorter-scale interactive behaviour downstream. The approach to separation is also considered, yielding maximal growth for small spot-like disturbances at 5/6 of the way from the minimum pressure position to the separation position. Links with recent experiments on adverse-gradient spot-like disturbances and with findings on calmed region properties are investigated, as well as the unsteady forcing effects from an incident relatively thick vortical wake outside the boundary layer.

Two Lagrangian-mean measures crucial to the accurate estimation of mean particle velocities in wavy or turbulent shear flows are considered. The measures are the pseudomomentum and generalized Stokes drift and of particular interest is their expression in terms of quantities directly measurable by fixed instruments. To proceed, the measures are first calculated for broad spectra of progressive symmetric rotational wave pairs of small amplitude. Both discrete and continuous spectra are considered and the waves may grow or decay. The expressions are then cast into a form composed of quantities that are measurable in a fixed reference frame, such as the surface slope spectrum of surface gravity waves or space–time velocity correlations in the interior of wavy shear flows. Finally, an example is given in which the measures are calculated for a plane channel flow subject to a broad spectrum of discrete progressive waves, specifically a numerical simulation of turbulent channel flow. It is seen that while the streamwise component of pseudomomentum is everywhere negative in the flow, the generalized Stokes drift changes sign, giving rise to an enhanced mass transport close to the boundary and a reduction in transport some distance from it. The sign change occurs 12.5 viscous units from the wall, near the centre of a 15 viscous units thick highly sheared layer of Stokes drift.

Prandtl–Meyer flows with embedded oblique shock waves due to excessive heat release from condensation (supercritical flows) are considered by extending the subcritical asymptotic solution of Delale & Crighton (1998). The embedded shock origin is located by the construction of the envelope of the family of characteristics emanating either from the corner or from the deflected wall in the parabolic approximation. A shock fitting technique for embedded oblique shock waves is introduced in the small deflection angle approximation and the law of deflection of a streamline through an embedded oblique shock wave is established within the same approximation. The network of characteristics downstream of the embedded shock front is constructed and the solution for the flow field therein is evaluated by utilizing the asymptotic solution of the rate equation along streamlines downstream of the shock front together with the equations of motion in characteristic form. Results obtained by employing the classical nucleation equation and the Hertz–Knudsen droplet growth law, compared with the supercritical experiments of Smith (1971) for moist air expansions, show that supercritical Prandtl–Meyer flows can only be realized locally when the embedded shock lies sufficiently far downstream of the throat, where the corner is located.

Numerical simulations are used to study the formation of vortex rings that are generated by applying a non-conservative force of long duration, simulating experimental vortex ring generation with large stroke ratio. For sufficiently long-duration forces, we investigate the extent to which properties of the leading vortex ring are invariant to the force distribution. The results confirm the existence of a universal ‘formation number’ defined by Gharib, Rambod & Shariff (1998), beyond which the leading vortex ring is separated from a trailing jet. We find that the formation process is governed by two non-dimensional parameters that are formed with three integrals of the motion (energy, circulation, and impulse) and the translation velocity of the leading vortex ring. Limiting values of the normalized energy and circulation of the leading vortex ring are found to be around 0.3 and 2.0, respectively, in agreement with the predictions of Mohseni & Gharib (1998). It is shown that under this normalization smaller variations in the circulation of the leading vortex ring are obtained than by scaling the circulation with parameters associated with the forcing. We show that by varying the spatial extent of the forcing or the forcing amplitude during the formation process, thicker rings with larger normalized circulation can be generated. Finally, the normalized energy and circulation of the leading vortex rings compare well with the same properties for vortices in the Norbury family with the same mean core radius.

Thermal convection experiments in a liquid gallium layer subject to a uniform rotation and a uniform vertical magnetic field are carried out as a function of rotation rate and magnetic field strength. Our purpose is to measure heat transfer in a low-Prandtl-number (Pr = 0.023), electrically conducting fluid as a function of the applied temperature difference, rotation rate, applied magnetic field strength and fluid-layer aspect ratio. For Rayleigh–Bénard (non-rotating, non-magnetic) convection we obtain a Nusselt number–Rayleigh number law Nu = 0.129Ra0.272±0.006 over the range 3.0 × 103 < Ra < 1.6 × 104. For non-rotating magnetoconvection, we find that the critical Rayleigh number RaC increases linearly with magnetic energy density, and a heat transfer law of the form Nu ∼ Ra1/2. Coherent thermal oscillations are detected in magnetoconvection at ∼ 1.4RaC. For rotating magnetoconvection, we find that the convective heat transfer is inhibited by rotation, in general agreement with theoretical predictions. At low rotation rates, the critical Rayleigh number increases linearly with magnetic field intensity. At moderate rotation rates, coherent thermal oscillations are detected near the onset of convection. The oscillation frequencies are close to the frequency of rotation, indicating inertially driven, oscillatory convection. In nearly all of our experiments, no well-defined, steady convective regime is found. Instead, we detect unsteady or turbulent convection just after onset.

We consider the steady two-dimensional thin-film flow of a viscoplastic material, modelled as a biviscosity fluid with a yield stress, round the outside of a large horizontal stationary or rotating cylinder. In both cases we determine the leading- order solution both when the ratio of the viscosities in the ‘yielded’ and ‘unyielded’ regions is of order unity and when this ratio approaches zero in the appropriate distinguished limit. When the viscosity ratio is of order unity the flow consists, in general, of a region of yielded fluid adjacent to the cylinder and a region of unyielded fluid adjacent to the free surface, separated by the yield surface. In the distinguished limit the flow consists, in general, of a region of yielded fluid adjacent to the cylinder whose stress is significantly above the yield stress and a pseudo-plug region adjacent to the free surface, in which the leading-order azimuthal component of velocity varies azimuthally but not radially, separated by the pseudo-yield surface; the pseudo-plug is itself, in general, divided by the yield surface into a region of yielded fluid whose stress is only just above the yield stress and a region of unyielded fluid adjacent to the free surface whose stress is significantly below the yield stress. The solution for a stationary cylinder represents a curtain of fluid with prescribed volume flux falling onto the top of and off at the bottom of the cylinder. If the flux is sufficiently small then the flow is unyielded everywhere, but when it exceeds a critical value there is a yielded region. In the distinguished limit the yielded region always extends all the way round the cylinder, but the unyielded region does so only when the flux is sufficiently small. For a rotating cylinder a film with finite thickness everywhere is possible only when the flux is sufficiently small. Depending on the value of the flux and the speed of rotation the flow may be unyielded everywhere, have a yielded region on the right of the cylinder only, or have yielded regions on both the right and left of the cylinder. At the critical maximum flux the maximum supportable weight of fluid on the cylinder is attained and the pseudo-yield, yield and free surfaces all have a corner. In the distinguished limit there are rigid plugs (absent in the stationary case) near the top and bottom of the cylinder.

The convective flow in a vertical slot with differentially heated walls and vertical temperature gradient is considered for very large Rayleigh numbers. Gravity is taken to be vertical and to consist of both a mean and a harmonic modulation (‘jitter’) at a given frequency and amplitude. The time-dependent Boussinesq equations governing the two-dimensional convection are solved numerically. To this end an economic operator-splitting scheme is devised combined with internal iterations within a given time step. The approximation of the nonlinear terms is conservative and no scheme viscosity is present in the approximation. The flow is investigated for a range of Prandtl numbers from Pr = 1000 when fluid inertia is insignificant and only thermal inertia plays a role to Pr = 0.73 when both are significant and of the same order. The flow is governed by several parameters. In the absence of jitter, these are the Prandtl number, Pr, the Rayleigh number, Ra, and the dimensionless critical stratification, τB. Simulations are reported for Pr = 103 and a range of τB and Ra, with emphasis on mode selection and finite-amplitude states. The presence of jitter adds two more parameters, i.e. the dimensionless jitter amplitude ε and frequency ω, rendering the flow susceptible to new modes of parametric instability at a critical amplitude εc. Stability maps of εc vs. ω are given for a range of ω. Finally we investigate the response of the system to jitter near the neutral curves of the various instability modes.

The difference in turbulent diffusion between active scalar (heat) and passive scalar (mass) in a stable thermally stratified flow is investigated both experimentally and numerically. The experiments are conducted in an unsheared thermally stratified water flow downstream of a turbulence-generating grid. Passive mass is released into the stable thermally stratified flow from a point source located 60 mm downstream from the grid. Instantaneous streamwise and vertical velocities, the temperature of the active scalar and the concentration of the passive scalar are simultaneously measured using a combined technique with a two-component laser-Doppler velocimeter (LDV), a resistance thermometer and a laser-induced fluorescence (LIF) method. From the measurements, turbulent heat and mass fluxes and eddy diffusivities for both active heat and passive mass are estimated. To investigate the Prandtl or Schmidt number effects on the difference in turbulent diffusion between active heat and passive mass, a three-dimensional direct numerical simulation (DNS) based on a finite difference method is applied to stable thermally stratified flows of both water and air behind the turbulence grid. The Schmidt number of passive mass in the DNS is set to the same value as the Prandtl number of active heat.

The results show that stable stratification causes a large difference in eddy diffusivities between active heat and passive mass. The numerical predictions by the DNS are in qualitative agreement with the measurements despite the assumption of the same molecular diffusivity for active heat and passive mass. The difference suggests that the assumption of identical eddy diffusivity for active heat and passive mass, used in conventional turbulence models, gives significant errors in estimating heat and mass transfer in a plume under stably stratified conditions.

A nonlinear evolution equation for pulsating Chapman–Jouguet detonations with chain-branching kinetics is derived. We consider a model reaction system having two components: a thermally neutral chain-branching induction zone governed by an Arrhenius reaction, terminating at a location where conversion of fuel into chain radical occurs; and a longer exothermic main reaction layer or chain-recombination zone having a temperature-independent reaction rate. The evolution equation is derived under the assumptions of a large activation energy in the induction zone and a slow evolution time based on the particle transit time through the induction zone, and is autonomous and second-order in time in the shock velocity perturbation. It describes both stable and unstable solutions, the latter leading to stable periodic limit cycles, as the ratio of the length of the chain-recombination zone to chain-induction zone, the exothermicity of reaction, and the specific heats ratio are varied. These dynamics correspond remarkably well with numerical solutions conducted earlier for a model three-step chain-branching reaction.

An error in a recent paper on bubble and drop oscillations by Temkin (J. Fluid Mech. vol. 380 (1999), pp. 1–38) is pointed out and corrected. In this way, his results are shown essentially to reduce to earlier ones in the literature. A concise derivation of these earlier results is presented for the case of a gas bubble.

Journal of Fluid Mechanics, vol. 380 (1999), pp. 1–38

I am indebted to Professor Prosperetti and to Dr Ren for finding a significant error in my detailed analysis of the temperature and pressure fluctuations in a pulsating particle. As they point out, the error invalidates the results of that analysis for the temperature fluctuations in gas bubbles. Those results are withdrawn. The error also affects, although in a less significant manner, the results for the temperature fluctuations in pulsating liquid drops with specific heat ratio significantly different from 1. This includes the droplets chosen to illustrate the results, namely toluene droplets. However, correct results for both bubbles and droplets may be easily obtained and this is done here.

The error found by Prosperetti and Ren (see Prosperetti & Ren 2001) exists in both the internal and external solutions, but because the external fluid is in both cases a liquid having a very small value of (γf − 1), the error has no effect there. Thus, fortuitously, the external solution is valid, provided the correct values of the interfacial temperature and pressure fluctuations are used. In addition, the simple solution presented in Appendix B, as well as the general expressions for derived quantities, such as those for the energy dissipation rates and the attenuation coefficients, are not affected by the error, provided, again, that the values of the temperature and pressure ratios appearing in them are properly evaluated. This is done below for both drops and bubbles by complementing the analysis of Appendix B with the polytropic approximation. The required steps are straightforward.