This paper presents a study of the steady, axisymmetric, creeping translation of a fluid sphere in a tube for the case in which surfactant is adsorbed onto the fluid sphere interface. Marangoni stresses caused by the convective redistribution of surfactant are computed perturbatively in the limit of sorption-controlled uniform retardation, and fully converged numerical solutions of the creeping-flow equations including the Marangoni stress are obtained by a collocation technique.

The results indicate that when the fluid sphere moves in a liquid which is at rest at infinity, the Marangoni stress retards the particle velocity. This retardation generally increases with the sphere to tube diameter ratio up to a value of approximately 0.6, whereupon the retardation begins to level off or even become reduced. When the sphere is suspended in a Poiseuille flow, stagnation rings develop on the sphere surface, and the Marangoni stresses that derive from this surface convection pattern can accelerate the fluid particle when the particle velocity is small with respect to the Poiseuille centreline velocity, but in the same direction as that velocity.

The maximum angle of attack for unseparated flow over an airfoil of chord c with finite nose radius of curvature r is shown to be 0.818(r/c)½

This paper describes analytical and numerical solutions for the flow of a Bingham plastic in an eccentric annulus. Analytical solutions are obtained by expanding in powers of δ, the ratio of the difference in radii of the bounding cylinders to their mean. The solution over most of the annulus is similar to that in a slot of uniform width, containing a central plug-like region over which the velocity is independent of the radial variable. However, unlike the uniform-slot solution, the velocity in the plug varies around the annulus and the stress exceeds the yield stress. This simple structure is supplemented by true plugs (over which the velocity is constant and the stress is below the yield stress) at the widest and, in some cases, the narrowest parts of the annulus. A simple criterion is given for conditions under which the fluid ceases to flow on the narrow side and bounds are obtained for the extent of the motionless region and for the true plugs.

The predictions of the theory have been compared to numerical results over a wide range of eccentricities, radius ratios, fluid properties and flow parameters. Good quantitative agreement has been reached for radius ratios in excess of about 0.7. In particular the extent and location of pseudo-plugs and true plugs are confirmed.

In this paper we study the quasi-stationary turbulent state developed by an incompressible flow submitted to a constant periodic force. The turbulent state is described by its Lyapunov exponents and Lyapunov dimension. Our aim is to investigate in particular if the dimension is, as several lines of argument seem to indicate, given in one way or another by the number of ‘turbulent’ modes present in the flow.

The exponents may be viewed as the asymptotic limits of local exponents, which are the divergence rates between the actual state of the fluid and nearby states. These local exponents fluctuate as the fluid suffers chaotic changes: they are systematically larger during bursts of excitation at large scale. In one case, we find that the flow oscillates repeatedly, on a very long timescale, between two distinct turbulent states, which have distinct local Lyapunov spectra and also distinct energy spectra. At the moderate resolutions studied here (642, 1282 and 163), we find that the dimension of the attractor in flows with standard dissipation is bounded above by the number of degrees of freedom contained in the large scales, i.e. at wavenumbers smaller than that of the injection wavenumber kf. We also consider two-dimensional flows with artificial terms (hyperviscosity), which concentrate small-scale dissipation in a narrow band of wavenumbers, and allow the formation of an inertial range. The dimension in these flows is no longer bounded by the number of large scales; it grows with, but remains significantly smaller than, the number of modes contained in the inertial range. It is conjectured that this is due to the presence of coherent structures in the flow, and that at higher resolutions also, the presence of coherent structures in turbulent flows will lead to an effective attractor dimension significantly lower than the estimations based on self-similar Kolmogorov theories.

The paper describes studies of the turbulence of the liquid in a bubbly, grid-generated turbulent flow field. Laser-Doppler and hot-film anemometry are used for the experimental investigation. It is found that the turbulent kinetic energy increases strongly with the void fraction α. Roughly speaking, there exist two distinct regimes: the first one corresponds to low value of α, where hydrodynamic interactions between bubbles are negligible, and the second one to higher values, for which, owing to their mutual interactions, the bubbles transfer a greater amount of kinetic energy to the liquid. The Reynolds stress tensor shows that the quasi-isotropy is not altered. At low enough values of α, the difference between the turbulent kinetic energy in the liquid phase and the energy associated with the grid-generated turbulence proves to be approximately equal to the intensity of the pseudo-turbulence, defined as the fluctuating energy that would be induced by the motion of the bubbles under non-turbulent conditions. The one-dimensional spectra exhibit a large range of high frequencies associated with the wakes of the bubbles and the classical $-\frac{5}{3}$ power law is progressively replaced by a $-\frac{8}{3}$ dependence.

Sufficient conditions for the stability of steady solutions of a multi-layer model are found. The basic flow may be either parallel, axisymmetric or non-parallel. The lower boundary of the model may be either rigid, including the possibility of topography, or soft. The latter, ‘reduced gravity’, case represents an ideal situation in which the active layers are on top of an infinitely deep, motionless one.

Two conditions are sufficient to assure the stability of the basic flow. It is conjectured that unstable flows for which only the first or second condition is violated decay through Rossby-like or Poincaré-like growing perturbations, respectively.

In order to understand the meaning of both conditions, assume that a quite general O(α) ‘wave’ is superimposed on the basic flow: an O(α2) energy integral, δ2E can be calculated. This wave energy is neither conserved, because the wave might exchange energy with the O(a2) varying part of the the ‘mean flow’, nor positive definite, because the perturbation might lower the total energy by increasing the speed where it decreases the thickness, and vice versa. Now, the first condition determines that δ2E has an upper bound, and the second one implies that δ2E is positive definite; hence the stability of the basic solution. In the particular case of two-dimensional divergenceless flow, as well as for quasi-geostropic models, δ2E is a priori positive definite, and therefore the first condition suffices to guarantee the stability of the more basic solution. The conditions found here are indeed valid for more general perturbations, e.g. they prevent inertial (or 'symmetric’) instability, a phenomenon for which there is no distinction between wave and the varying part of the mean flow.

We consider inviscid, incompressible flow down a backward-facing step. Using thin-aerofoil theory, a model is proposed in which the separated region downstream of the back face of the step consists of a constant-pressure zone immediately behind the step, followed by a Prandtl–Batchelor constant-vorticity region. The motivation for this model is a series of experimental studies which showed the pressure just downstream of the step to be almost constant in some upstream portion of the separated region. Previous models have ignored this constant-pressure region and agreement with experiment has not been good. Agreement with experiment is clearly superior using the constant-pressure/constant-vorticity model, though it is possible that the comparison could be improved still further by consideration of the behaviour of the shear layer after reattachment. Some discussion of such models is given.

This paper presents results obtained from a numerical solution to a stream function–vorticity formulation of the Navier–Stokes equations for the flow around a circular cylinder in planar oscillating flow at small Keulegan–Carpenter numbers (KC) in the subcritical Reynolds number (Re) range. The equations are solved by finite-difference methods. For very small KC ([les ] 1), the numerical results coincide with analytical solutions. As KC is increased, the incipient separation and instability leading to an asymmetrical flow with vortex shedding are predicted. Computed flow fields at small KC values are compared to flow visualizations, and good agreement is found for moderate β-values (≈ 250). The well-documented flow regimes with the transverse vortex street, single-, double- and three-pair shedding, are predicted by the model. Although the flow is not fully resolved for the highest Re values, comparisons of calculated drag and inertia coefficients with experimental data for three different values of the frequency parameter β in the range 196–1035 for 0 < KC < 26 show good agreement.

The linear, free-surface oscillations of an inviscid fluid in a cylindrical basin subject to the contact-line condition en∇ζ = ζt (ζ is the free-surface displacement and c is a complex constant) are determined through a boundary-layer approximation for l/a [Lt ] 1, where a is a characteristic length of the cross-section and l is the capillary length. The primary result is ω2 = ω2n [1 + (l/a) [Fscr ] (ζn; c/ωnl)] where ω is the frequency of a free oscillation, ωn is the natural frequency for a particular normal mode, ζ = ζn in the limit l/a → 0, and l/a→0, and [Fscr ](ζn;c/ωnl)] is a corresponding form factor. The imaginary part of [Fscr ] is positive (for the complex time dependence exp (iωt) if Re (c) > 0, which implies positive dissipation through contact-line motion. Explicit results are derived for circular and rectangular cylinders and compared with Graham-Eagle's (1983) results for the circular cylinder for c = 0 and Hocking's (1987) results for the two-dimensional problem. The exact eigenvalue equation for the circular cylinder and a variational approximation for an arbitrary cross-section are derived on the assumption that the static meniscus is negligible.

A simple water-wave instability induced by a shear flow is re-examined, using a cubic equation first derived by Stern & Adam (1973) for a piecewise constant vorticity model. The instability criteria and the growth rate are computed. It is found that this mechanism is effective only if the surface drift velocity exceeds the minimum wave speed for capillary–gravity waves, and only if the drift-layer thickness lies within a band which depends on the wavelength and the drift velocity.

This paper summarizes and extends some mathematical results for a model for a class of water-entry problems characterized by the geometrical property that the impacting body is nearly parallel to the undisturbed water surface and that the impact is so rapid that gravity can be neglected. Explicit solutions for the pressure distributions are given in the case of two-dimensional flow and a variational formulation is described which provides a simple numerical algorithm for three-dimensional flows. We also pose some open questions concerning the well-posedness and physical relevance of the model for exit problems or when there is an air gap between the impacting body and the water.

A parametric study is made of chaotic Rayleigh–Bénard convection over moderate Rayleigh numbers. As a basis for comparison over the Rayleigh number (Ra) range we consider mean quantities, r.m.s. fluctuations, Reynolds number, probability distributions and power spectra. As a further means of investigating the flow we use the Karhunen–Loéve procedure (empirical eigenfunctions, proper orthogonal decomposition). Thus, we also examine the variation in eigenfunctions with.Ra. This in turn provides an analytical basis for describing the manner in which the chaos is enriched both temporarily and spatially as Ra increases. As Ra decreases, the significant mode count decreases but, in addition, the eigenfunctions tend more nearly to the eigenfunctions of linearized theory. As part of this parametric study a variety of scaling properties are investigated. For example it is found that the empirical eigenfunctions themselves show a simple scaling in Ra.

A study is made of the number of dimensions needed to specify chaotic Rayleigh–Bénard convection, over a range of Rayleigh numbers (γ = Ra/Rac < 102). This is based on the calculation of Lyapunov dimension over the range, as well as the notion of Karhunen–Loéve dimension. An argument suggesting a universal relation between these estimates and supporting numerical evidence is presented. Numerical evidence is also presented that the reciprocal of the largest Lyapunov exponent and the correlation time are of the same order of magnitude. Several other universal features are suggested. In particular it is suggested that the intrinsic attractor dimension is $O(Ra^{\frac{2}{3}})$, which is sharper than previous results.

The influence of finite-amplitude perturbations on the unsteady vortex shedding past an impulsively started circular cylinder is investigated by means of a numerical model. The computational scheme is a mixed spectral–finite analytic technique, in which the fast-Fourier-transform algorithm is used for the evaluation of the nonlinear terms in the two-dimensional time-dependent Navier–Stokes equations in their stream function–vorticity transport form (the Helmholtz formulation) at Re = 1000. The vortex shedding is promoted by imposing at t = 0 a small rotational field to the initially irrotational flow. Attention is focused on the strength of the perturbation vortex, which affects the way in which the vortex shedding develops in time. The results of the simulations are presented by means of computer-generated drawings of absolute streamlines, relative streamlines and vorticity fields; it appears that, when the strength of the initial perturbation assumes the minimum value that has been tested, the vortex shedding phenomenon develops in a way different from that resulting from other numerical experiments of the same kind.

Experimental results are presented that reveal key features of the large-scale organized structures in a supersonic, turbulent boundary layer. Measurements were obtained in a Mach 3 zero-pressure-gradient boundary layer using a crossed-wire probe and arrays of normal hot wires with vertical, spanwise, and streamwise separations ranging from 0.1 to 0.6δ. Space–time correlation results indicate the existence of large-scale structures of a size comparable to δ, with a spanwise extent only slightly less than the vertical scale. The convection velocity of the large-scale motions is nearly constant across 80% of the boundary layer and is equal to approximately 0.9U∞.

It is shown that positive events detected with the VITA conditional sampling technique correspond to steep gradients in the streamwise mass flux which extend across most of the boundary layer. These sharp gradients appear to be the upstream interfaces of large-scale turbulent ‘bulges’, similar to those seen in incompressible boundary layers. In a reference frame moving with the convection speed of the sharp gradients, low-momentum fluid is observed rotating on a large-scale, while high-momentum fluid forms a saddle point on the upstream edge of the large-scale motion. These motions are associated with elevated levels of the shear product, emphasizing their role in the dynamics of the boundary layer.

A laser-Doppler velocimeter is used to analyse volume-cycled oscillatory flow of a Newtonian viscous fluid in a straight circular tube. The axial velocity is measured at radial positions across the diameter of the tube for a wide range of amplitude A = stroke distance/tube radius (2.4 [les ] A [les ] 21.6) and Womersley parameter (9 < α < 33). Transition to turbulence is detected during the decelerating phase of fluid motion for 500 < Rδ < 854, where Rδ = αA √2 is the Reynolds number based on Stokes-layer thickness. The turbulence is confined to an annular region which is a few times the Stokes-layer thickness near the wall. Hot-film anemometer measurements indicate the core flow remains stable when the boundary layer becomes turbulent for Rδ up to 1310.

The shape oscillations of drops in another fluid with or without surfactants has been analysed by normal mode expansions. The effects of surfactants are accommodated by considering the Gibbs elasticity, associated with the redistribution of surfactants, and a Boussinesq surface fluid with two surface viscosities. A general transcendental equation for the complex frequency of the free oscillations is derived. Explicit dispersion relations are given for fluids of small bulk viscosities and an interface of small, moderate, and large interfacial properties by a perturbation method. We have found that the oscillation always damps out faster for an interface exhibiting interfacial properties other than the interfacial tension, and the Gibbs elasticity is the most important parameter that alters the free-oscillation frequency and the damping constant. Moreover, the energy dissipation for an extensible interface can be much higher than that of an inextensible interface owing to the strong vorticity generated in the boundary layers.

Three-dimensional motion of a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated. The deflections of the core centreline are not restricted to be small compared with the core radius. We first derive the equation of the vortex motion, correct to the second order in the ratio of the core radius to that of curvature, by a matching procedure, which recovers the results obtained by Moore & Saffman (1972). An asymptotic formula for the linear dispersion relation is obtained up to the second order. Under the assumption of localized induction, the equation governing the self-induced motion of the vortex is reduced to a nonlinear evolution equation generalizing the localized induction equation. This new equation is equivalent to the Hirota equation which is integrable, including both the nonlinear Schrödinger equation and the modified KdV equation in certain limits. Therefore the new equation is also integrable and the soliton surface approach gives the N-soliton solution, which is identical to that of the localized induction equation if the pertinent dispersion relation is used. Among other exact solutions are a circular helix and a plane curve of Euler's elastica. This local model predicts that, owing to the existence of the axial flow, a certain class of helicoidal vortices become neutrally stable to any small perturbations. The non-local influence of the entire perturbed filament on the linear stability of a helicoidal vortex is explored with the help of the cutoff method valid to the second order, which extends the first-order scheme developed by Widnall (1972). The axial velocity is found to discriminate between right- and left-handed helices and the long-wave instability mode is found to disappear in a certain parameter range when the successive turns of the helix are not too close together. Comparison of the cutoff model with the local model reveals that the non-local induction and the core structure are crucial in making quantitative predictions.

Resonance phenomena of a conducting drop forced by an alternating electric field are studied by perturbation analysis. Although the motions are assumed to be irrotational, weak viscous effects are included in the boundary condition of the normal stress balance. Without an external field, the first-order expansion of the domain perturbations yields the same result as that obtained by Lamb (1932) for the viscous decay of the free oscillation modes. A primary resonance occurs in the first-order forced oscillation problem. Under strong excitation, superharmonic, sub-harmonic, and coupled resonances are revealed in the second-order solutions. Hence, large-amplitude drop oscillations may occur even if the excitation frequencies are away from the characteristic drop frequencies and the spatial forms of the excitation modes do not directly match the drop shape modes. In order to obtain comparable response amplitudes, however, the magnitudes of external forcing required to excite secondary resonances are shown to be about an order greater than that for the primary resonances.

Vortex shedding from flat plates with square leading and trailing edges having chord-to-thickness ratios 3–16 at Reynolds numbers (1–3) × 103 is investigated experimentally in low-speed wind tunnels. It is shown that vortex shedding from flat plates with square leading and trailing edges is characterized by the impinging-shear-layer instability where the separated shear layer becomes unstable in the presence of a sharp trailing edge corner. The Strouhal number which is based on the plate's chord is approximately constant and equal to 0.6 for chord-to-thickness ratios 3–5. With further increase in the ratio it increases stepwise to values that are approximately equal to integral multiples of 0.6.