The translational and rotational drag coefficients for a cylinder undergoing uniform translational and rotational motion in a model lipid bilayer membrane is calculated from the appropriate linearized Navier–Stokes equations. The calculation serves as a model for the lateral and rotational diffusion of membrane-bound particles and can be used to infer the ‘microviscosity’ of the membrane from the measured diffusion coefficients. The drag coefficients are obtained exactly using dual integral equation techniques. The region of validity of an earlier asymptotic solution obtained by Saffman (1976) is elucidated.

A dead fish is propelled upstream when its flexible body resonates with oncoming vortices formed in the wake of a bluff cylinder, despite being well outside the suction region of the cylinder. Within this passive propulsion mode, the body of the fish extracts sufficient energy from the oncoming vortices to develop thrust to overcome its own drag. In a similar turbulent wake and at roughly the same distance behind a bluff cylinder, a passively mounted high-aspect-ratio foil is also shown to propel itself upstream employing a similar flow energy extraction mechanism. In this case, mechanical energy is extracted from the flow at the same time that thrust is produced. These results prove experimentally that, under proper conditions, a body can follow at a distance or even catch up to another upstream body without expending any energy of its own. This observation is also significant in the development of low-drag energy harvesting devices, and in the energetics of fish dwelling in flowing water and swimming behind wake-forming obstacles.

Approximate constitutive equations are derived for a dilute suspension of rigid spheroidal particles with Brownian rotations, and the behaviour of the approximations is explored in various flows. Following the suggestion made in the general formulation in part 1, the approximations take the form of Hand's (1962) fluid model, in which the anisotropic microstructure is described by a single second-order tensor. Limiting forms of the exact constitutive equations are derived for weak flows and for a class of strong flows. In both limits the microstructure is shown to be entirely described by a second-order tensor. The proposed approximations are simple interpolations between the limiting forms of the exact equations. Predictions from the exact and approximate constitutive equations are compared for a variety of flows, including some which are not in the class of strong flows analysed.

In an experimental study of the vortex-induced oscillations of a long flexible circular cylinder, the observed stationary amplitudes describe an hysteresis loop partially different from earlier studies. Each branch of the loop is associated with a vortex shedding mode and, as a jump from one branch to the other occurs, the phase difference between the cylinder displacement and the vortex shedding undergoes an abrupt change. The critical flow velocities at which the jump occurs concur with the flow visualization observations of Williamson & Roshko (1988) on the vortex shedding modes near the fundamental synchronization region. Impulsive regimes, obtained at a given flow velocity with the cylinder initially at rest or pre-excited, and progressive regimes resulting from a variation of the flow velocity, are examined. The occurrence of bifurcations is detected for a flow velocity range in the case of the impulsive regimes. The coordinates of the bifurcations define a boundary between two vortex shedding modes, a boundary that verifies the critical curve obtained by Williamson & Roshko (1988). The experimental set-up of this study simulates half the wavelength of a vibrating cable, eliminates the end effects present in oscillating rigid cylinder set-up and has one of the lowest damping ratios reported for the study of this phenomenon.

The development of a small three-dimensional disturbance in plane Poiseuille flow is considered. Its kinetic energy is expressed in terms of the velocity and vorticity components normal to the wall. The normal vorticity develops according to the mechanism of vortex stretching and is described by an inhomogeneous equation, where the spanwise variation of the normal velocity acts as forcing. To study specifically the effect of the forcing, the initial normal vorticity is set to zero and the energy density in the wavenumber plane, induced by the normal velocity, is determined. In particular, the response from individual (and damped) Orr–Sommerfeld modes is calculated, on the basis of a formal solution to the initial-value problem. The relevant timescale for the development of the perturbation is identified as a viscous one. Even so, the induced energy density can greatly exceed that associated with the initial normal velocity, before decay sets in. Initial conditions corresponding to the least-damped Orr–Sommerfeld mode induce the largest energy density and a maximum is obtained for structures infinitely elongated in the streamwise direction. In this limit, the asymptotic solution is derived and it shows that the spanwise wavenumbers at which the largest amplification occurs are 2.60 and 1.98, for symmetric and antisymmetric normal vorticity, respectively. The asymptotic analysis also shows that the propagation speed for induced symmetric vorticity is confined to a narrower range than that for antisymmetric vorticity. From a consideration of the neglected nonlinear terms it is found that the normal velocity component cannot be nonlinearly affected by the normal vorticity growth for structures with no streamwise dependence.

We study the impact and subsequent retraction of liquid droplets upon high-speed impact on hydrophobic surfaces. Extensive experiments show that the drop retraction rate is a material constant and does not depend on the impact velocity. We show that on increasing the Ohnesorge number, $\Oh\,{=}\,\eta/\sqrt{\rho R_{\rm I} \gamma}$, the retraction, i.e. dewetting, dynamics crosses from a capillary-inertial regime to a capillary-viscous regime. We rationalize the experimental observations by a simple but robust semi-quantitative model for the solid-liquid contact line dynamics inspired by the standard theories for thin-film dewetting.

The purpose of this article is to show that the Navier–Stokes equations can be rewritten as a set of linearized inhomogeneous Euler equations (in convective form) with source terms that are exactly the same as those that would result from externally imposed shear-stress and energy-flux perturbations. These results are used to develop a mathematical basis for some existing and potential new jet noise models by appropriately choosing the ‘base flow’ about which the linearization is carried out.

The flow around a circular cylinder has been examined over the Reynolds number range 105 to 7·5 × 105, Reynolds number being based on cylinder diameter. Narrow-band vortex shedding has been observed up to a Reynolds number of 5·5 × 105, i.e. well into the critical régime. At this Reynolds number the Strouhal number reached the unusually high value of 0·46. Spectra of the velocity fluctuations measured in the wake are presented for several values of Reynolds number.

A turbulent mixing layer in a water channel was observed at Reynolds numbers up to 3 × 106. Flow visualization with dyes revealed (once more) large coherent structures and showed their role in the entrainment process; observation of the reaction of a base and an acid indicator injected on the two sides of the layer, respectively, gave some indication of where molecular mixing occurs. Autocorrelations of streamwise velocity fluctuations, using a laser-Doppler velocimeter (LDV) revealed a fundamental periodicity associated with the large structures. The surprisingly long correlation times suggest time scales much longer than had been supposed; it is argued that the mixing-layer dynamics at any point are coupled to the large structure further downstream, and some possible consequences regarding the effects of initial conditions and of the influence of apparatus geometry are discussed.

Direct numerical simulations have been carried out for a fully developed turbulent channel flow with a smooth upper wall and a lower wall consisting of square bars separated by a rectangular cavity. A wide range of $w/k$, the cavity width to roughness height ratio, was considered. For $w/k\,{\ge}\,7$, recirculation zones occur immediately upstream and downstream of each element while mean streamlines and spatial distributions of the skin frictional drag indicate that each element is virtually isolated. The maximum form drag occurs at $w/k\,{=}\,7$ and coincides with the minimum skin frictional drag. The dependence on $w/k$ of the Clauser roughness function reflects that of the form drag.

‘Bioconvection’ is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. ‘Gyrotaxis’ describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are non-spherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamydomonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5–6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth.

Experiments were conducted in a two-stream planar mixing layer at convective Mach numbers, Mc, of 0.28, 0.42, 0.50, 0.62 and 0.79. Planar laser Mie scattering (PLMS) from a condensed alcohol fog and planar laser-induced fluorescence (PLIF) of nitric oxide were used for flow visualization in the side, plan and end views. The PLIF signals were also used to characterize the turbulent mixture fraction fluctuations.

Visualizations using PLMS indicate a transition in the turbulent structure from quasi-two-dimensionality at low convective Mach number, to more random three-dimensionality for $M_c\geqslant 0.62$. A transition is also observed in the core and braid regions of the spanwise rollers as the convective Mach number increases from 0.28 to 0.62. A change in the entrainment mechanism with increasing compressibility is also indicated by signal intensity profiles and perspective views of the PLMS and PLIF images. These show that at Mc = 0.28 the instantaneous mixture fraction field typically exhibits a gradient in the streamwise direction, but is more uniform in the cross-stream direction. At Mc = 0.62 and 0.79, however, the mixture fraction field is more streamwise uniform and with a gradient in the cross-stream direction. This change in the composition of the structures is indicative of different entrainment motions at the different compressibility conditions. The statistical results are consistent with the qualitative observations and suggest that compressibility acts to reduce the magnitude of the mixture fraction fluctuations, particularly on the high-speed edge of the layer.

Experimental observations of the collapse of initially vertical columns of small grains are presented. The experiments were performed mainly with dry grains of salt or sand, with some additional experiments using couscous, sugar or rice. Some of the experimental flows were analysed using high-speed video. There are three different flow regimes, dependent on the value of the aspect ratio $a\,{=}\,h_i/r_i$, where $h_i$ and $r_i$ are the initial height and radius of the granular column respectively. The differing forms of flow behaviour are described for each regime. In all cases a central, conically sided region of angle approximately $ 59^\circ$, corresponding to an aspect ratio of 1.7, remains undisturbed throughout the motion. The main experimental results for the final extent of the deposit and the time for emplacement are systematically collapsed in a quantitative way independent of any friction coefficients. Along with the kinematic data for the rate of spread of the front of the collapsing column, this is interpreted as indicating that frictional effects between individual grains in the bulk of the moving flow only play a role in the last instant of the flow, as it comes to an abrupt halt. For $a\,{<}\,1.7$, the measured final runout radius, $r_\infty$, is related to the initial radius by $r_\infty \,{=}\, r_i(1\,{+}\,1.24a)$; while for $1.7\,{<}\,a$ the corresponding relationship is $r_\infty \,{=}\,r_i(1\,{+}\,1.6a^{1/2})$. The time, $t_\infty$, taken for the grains to reach $r_\infty$ is given by $t_\infty \,{=}\,3(h_i/g)^{1/2}\,{=}\,3(r_i/g)^{1/2}a^{1/2}$, where $g$ is the gravitational acceleration. The insights and conclusions gained from these experiments can be applied to a wide range of industrial and natural flows of concentrated particles. For example, the observation of the rapid deposition of the grains can help explain details of the emplacement of pyroclastic flows resulting from the explosive eruption of volcanoes.

The behaviour of heavy particles in isotropic, homogeneous, decaying turbulence has been experimentally studied. The settling velocity of the particles has been found to be much larger than in a quiescent fluid. It has been determined that the enhancement of the settling velocity depends on the particle loading, increasing as the volume fraction of particles in the flow increases. The spatial and temporal distribution of the particle concentration field is shown to exhibit large inhomogeneities. As the particles interact with the underlying turbulence they concentrate preferentially in certain regions of the flow. A characteristic dimension of these particle clusters is found to be related to the viscous scales of the flow. Measurements of the settling velocity conditioned on the local concentration of particles in the flow have shown that there is a monotonic increase in the settling velocity with the local concentration (the relation being quasi-linear). A simple phenomenological model is proposed to explain this behaviour.

Velocity measurements in a 51 mm diameter turbulent jet are presented. The measurement programme is conducted in two parts. The first part is devoted to the validation of laser velocimeter (LV) data. This consists of comparative measurements with the LV and a hot-wire anemometer. The second part consists of a survey of the jet flow field at Mach 0·28, 0·90, and 1·37 under ambient temperature conditions. Radial and centre-line distributions of the axial and radial, mean and fluctuating velocities are obtained. The distributions indicate a decrease in the spreading rate of the mixing layer with increasing Mach number and a corresponding lengthening of the potential core. The results further indicate that these two parameters vary with the square of the jet Mach number. Radial distributions collapse when plotted in terms of ση*, where σ = 10.7/(1 - 0.273 MJ2) and η* = (r − r0·5)/x. This is true for distributions in planes located as far downstream as two potential core lengths. The collapsed data of mean velocity can be approximated by a Görtler error function profile: \[ U/U_J = 0.5[1-{\rm erf}(\sigma\eta^{*})]. \] Centre-line distributions at various Mach numbers also collapse if plotted in terms of x/xc, xc being the potential core length. A general equation for the collapsed data of mean velocity is given by: U/UJ = 1 - exp{1.35/(1 - x/xc)}, for the range of Mach numbers 0·3-1·4, where xc = 4.2 + 1.1 MJ2.

A description of MHD turbulence at low magnetic Reynolds number and large interaction parameter is proposed, in which attention is focussed on the role of insulating walls perpendicular to a uniform applied magnetic field. The flow is divided in two regions: the thin Hartmann layers near the walls, and the bulk of the flow. In the latter region, a kind of electromagnetic diffusion along the magnetic field lines (a degenerate form of Alfvén waves) is displayed, which elongates the turbulent eddies in the field direction, but is not sufficient to generate a two-dimensional dynamics. However the normal derivative of velocity must be zero (to leading order) at the boundaries of the bulk region (as at a free surface), so that when the length scale l⊥ perpendicular to the magnetic field is large enough, the corresponding eddies are necessarily two-dimensional. Furthermore, if l⊥ is not larger than a second limit, the Hartmann braking effect is negligible and the dynamics of these eddies is described by the ordinary Navier-Stokes equations without electromagnetic forces. MHD then appears to offer a means of achieving experiments on two-dimensional turbulence, and of deducing velocity and vorticity from measurements of electric field.

In this paper, two-dimensional problems of the diffraction and stability of shock waves are investigated using an approximate theory in which disturbances to the flow are treated as a wave propagation on the shocks. These waves carry changes in the slope and the Mach number of the shock. The equations governing the wave propagation are analogous in every way to the non-linear equations for plane waves in gas dynamics, and their solutions can be deduced by the same mathematical techniques. Since the propagation speed of the waves is found to be an increasing function of Mach number, waves carrying an increase in Mach number will eventually break and form what we may call a ‘shock’, corresponding to the breaking of a compression wave into a shock in the ordinary plane wave case. Such a ‘shock’ moving on the shock is called a shock-shock. The shock-shock is a discontinuity in Mach number and shock slope, and it must be fitted in to satisfy the appropriate relations between these are interpreted as the trace of cylindrical sound waves in the flow behind the shock. In particular a shock-shock is the trace of a genuine shock in the flow behind, and thus corresponds to Mach reflection.

The general theory of the wave propagation is set out in § 2. The subsequent sections contain applications of the theory to specific problems, including the motion of a shock along a curved wall, diffraction by a wedge, stability of plane shocks and the instability of a converging cylindrical shock.

From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.

When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.

Diffuse-interface models may be used to compute moving contact lines because the Cahn–Hilliard diffusion regularizes the singularity at the contact line. This paper investigates the basic questions underlying this approach. Through scaling arguments and numerical computations, we demonstrate that the Cahn–Hilliard model approaches a sharp-interface limit when the interfacial thickness is reduced below a threshold while other parameters are fixed. In this limit, the contact line has a diffusion length that is related to the slip length in sharp-interface models. Based on the numerical results, we propose a criterion for attaining the sharp-interface limit in computing moving contact lines.

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter . When , viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When , vertical viscous shearing is important so that (lh is a characteristic horizontal length scale). The parameter is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when : the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of but they tend to be smooth for < 1, while for > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for < 1 but tends to isotropy as increases above unity. When < 1, the horizontal and vertical energy spectra are very steep while, when > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.