An expression which predicts the inertial lift, to lowest order, on a rigid sphere translating in a linear shear flow field near a flat infinite wall has been derived. This expression may be used when the wall lies within the inner region of the sphere's disturbance flow. It is valid even when the gap is small compared to the radius of the sphere. When the sphere is far from the wall, the lift force predicted by the present analysis converges to the value predicted by earlier analyses which consider the sphere as a point force or a force doublet singularity. The effect of rotation of the sphere on the lift has also been analysed.

The flow of a two-dimensional viscous film falling from the edge of an inclined plane exhibits a distinctive set of phenomena which, in various combinations, have been referred to as the teapot effect. This paper makes plain that three basic mechanisms are at the root of these phenomena: deflection of the liquid sheet by hydrodynamic forces, contact-angle hysteresis, and multiple steady states that give rise to a purely hydrodynamic hysteresis. The evidence is drawn from Galerkin/finite-element analysis of the Navier-Stokes system, matched to a one-dimensional asymptotic approximation of the sheet flow downstream, and is corroborated experimentally by flow visualization and measurements of free-surface profiles and contact line position. The results indicate that the Gibbs inequality condition quantifies the inhibiting effect of sharp edges on spreading of static contact lines, even in the presence of flow nearby. The branchings, turning points, and isolas of families of solutions in parameter space explain abrupt flow transitions observed experimentally, and illuminate the stability of predicted flow states.

The problem of a periodic planar arrangement of a large number of co-flowing, interacting jets is investigated. It is shown that this interaction gives rise to strong nearfield oscillations of large-scale spatial coherence and to far-field inhomogeneities. In the experiments performed, the jets were produced behind a flat plate perforated by holes arranged in a square or triangular periodic pattern and placed perpendicular to a uniform flow. At moderate Reynolds numbers, the interaction results in a remarkable low-frequency oscillation of the merging distance of the jets downstream of the plate. A detailed description of the recirculating flow in the cavities between the jets emphasizes the role of the backflow in the cavities on the oscillatory behaviour. This description is supported by measurements of the local fluctuating velocity and pressure, two-point correlation measurements and quantitative flow visualizations. These experimental observations suggest a new formulation for the instability dynamics of such unstable recirculating flows. This formulation, based on the nonlinear delayed saturation of the jet's shear layer instability (NLDS model) predicts successfully the dependence of the oscillation of the merging distance on the jet Reynolds number and on the local geometrical features of the confinement of the jets. Furthermore, it is shown that the diffusion of mass coming from one jet, seeded with an inert dye, gives rise to an exponential diffusion front over a distance corresponding to a few mesh sizes indicating a strong local coupling of the jets. At the scale of the whole jet assembly, the oscillations are organized as large-scale travelling waves, propagating from the boundaries of the domain to its centre. This symmetry-breaking property is discussed and supplemented by a spatio-temporal simulation of an array of coupled oscillators.

Conclusive experimental evidence is presented for the existence of a low-dimensional temporal dynamical system in an open flow, namely the near field of an axisymmetric, subsonic free jet. An initially laminar jet (4 cm air jet in the Reynolds number range 1.1 × 104 [Lt ] ReD × 9.1 × 104) with a top-hat profile was studied using single-frequency, longitudinal, bulk excitation. Two non-dimensional control parameters – forcing frequency StD (≡fexD/Ue, where fez is the excitation frequency, D is the jet exit diameter and Ue is the exit velocity) and forcing amplitude af (≡ u’f/Ue, where u’f is the jet exit r.m.s. longitudinal velocity fluctuation at the excitation frequency) – were varied over the ranges 10-4 < af < 0.3 and 0.3 < StD < 3.0 in order to construct a phase diagram. Periodic and chaotic states were found over large domains of the parameter space. The periodic attractors correspond to stable pairing (SP) and stable double pairing (SDP) of rolled-up vortices. One chaotic attractor, near SP in the parameter space, results from nearly periodic modulations of pairing (NPMP) of vortices. At large scales (i.e. approximately the size of the attractor) in phase space, NPMP exhibits approximately quasi-periodic behaviour, including modulation sidebands around ½fex in u-spectra, large closed loops in its Poincaré sections, correlation dimension v ∼ 2 and largest Lyapunov exponent λ1 ∼ 0. But investigations at smaller scales (i.e. distances greater than, but of the order of, trajectory separation) in phase space reveal chaos, as shown by v > 2 and λ1 > 0. The other chaotic attractor, near SDP, results from nearly periodic modulations of the first vortex pairing but chaotic modulations of the second pairing and has a broadband spectrum, a dimension 2.5 [Lt ] v [Lt ] 3 and the largest Lyapunov exponent 0.2 [Lt ] λ1 [Lt ] 0.7 bits per orbit (depending on measurement locations in physical and parameter spaces).

A definition that distinguishes between physically and dynamically open flows is proposed and justified by our experimental results. The most important conclusion of this study is that a physically open flow, even one that is apparently dynamically open due to convective instability, can exhibit dynamically closed behaviour as a result of feedback. A conceptual model for transitional jets is proposed based on twodimensional instabilities, subharmonic resonance and feedback from downstream vortical structures to the nozzle lip. Feedback was quantified and shown to affect the exit fundamental–subharmonic phase difference ϕ – a crucial variable in subharmonic resonance and, hence, vortex pairing. The effect of feedback, the sensitivity of pairings to ϕ, the phase diagram, and the documented periodic and chaotic attractors demonstrate the validity of the proposed conceptual model.

The stability to linearized two-dimensional disturbances of plane Poiseuille flow of a Bingham fluid is considered. Bingham fluids exhibit a yield stress in addition to a plastic viscosity and this description is typically applied to drilling muds. A non-zero yield stress results in an additional parameter, a Bingham number, and it is found that the minimum Reynolds number for linear instability increases almost linearly with increasing Bingham number.

We give a theory of permanent roll waves on a shallow layer of fluid mud which is modelled as a power-law fluid. Based on the long-wave approximation, Kármán's momentum integral method is applied to derive the averaged continuity and the momentum equations. Linearized instability analysis of a uniform flow shows that the growth rate of unstable disturbances increases monotonically with the wavenumber, and therefore is insufficient to suggest a preferred wavelength for the roll wave. Nonlinear roll waves are obtained next as periodic shocks connected by smooth profiles with depth increasing monotonically from the rear to the front. Among all wavelengths only those longer than a certain threshold correspond to positive energy loss across the shock, and are physically acceptable. This threshold also implies a minimum discharge, viewed in the moving system, for the roll wave to exist. These facts suggest that a roll wave developed spontaneously from infinitesimal disturbances should have the shortest wavelength corresponding to zero dissipation across the shock, though finite dissipation elsewhere. The discontinuity at the wave front is a mathematical shortcoming needing a local requirement. Predictions for the spontaneously developed roll waves in a Newtonian case are compared with available experimental data. Longer roll waves, with dissipation at the discontinuous fronts, cannot be maintained if the uniform flow is linearly stable, when the fluid is slightly non-Newtonian. However, when the fluid is highly non-Newtonian, very long roll waves may still exist even if the corresponding uniform flow is stable to infinitesimal disturbances. Numerical results are presented for the phase speed, wave height and wavenumber, and wave profiles for a representative value of the flow index of fluid mud.

The trajectories of electrically charged tracer particles travelling around a charged sphere subjected to a simple shear flow have been calculated. This is a limiting case of the relative trajectories of two unequal-sized spheres when the radius ratio a1/a2 approaches zero. Until now these trajectories have been calculated by assuming the additivity of hydrodynamic and electrostatic forces, while neglecting the electroviscous coupling forces. These electroviscous forces are long range and can significantly alter the relative trajectories of spheres. When a1/a2 → 0, it is found that these trajectories depend on two parameters, α and β, which depend on the surface charge density of the tracer particle and the sphere. The relative trajectories of charged particles are qualitatively different from those of neutral particles. There exist six intervals of α-values for which the trajectories of the tracer particle show different features. Several new types of trajectory appear, besides the open and closed trajectories for neutral particles, which we refer to as uni- and bidirectional infinite length trajectories, uni- and bidirectional finite length trajectories, open returning trajectories, and prolate, oblate and circular closed trajectories. This richness of possible trajectories is the result of three electrokinetic phenomena, affecting particle motion: electro-osmotic slip, electrophoretic and diffusiophoretic motion.

This paper develops a three-dimensional infinite series solution for the Stokes flow through a parallel walled channel which is obstructed by a thin planar barrier with periodically spaced rectangular orifices of arbitrary aspect ratio B’/d’ and spacing D’. Here B’ is the half-height of the channel and d’ is the half-width of the orifice. The problem is motivated by recent electron microscopic studies of the intercellular channel between vascular endothelial cells which show a thin junction strand barrier with discontinuities or breaks whose spacing and width vary with the tissue. The solution for this flow is constructed as a superposition of Hasimoto's (1958) general solution for the two-dimensional flow through a periodic slit array in an infinite plane wall and a new three-dimensional solution which corrects for the top and bottom boundaries. In contrast to the well-known solutions of Sampson (1891) and Hasimoto (1958) for the flow through zero-thickness orifices of circular or elliptic cross-section or periodic slits in an infinite plane wall, which exhibit characteristic viscous velocity profiles, the present bounded solutions undergo a fascinating change in behaviour as the aspect ratio B’/d’ of the orifice opening is increased. For B’/d’ [Lt ] 1 and (D’ –- d’)/B’ of O(1) or greater, which represents a narrow channel, the velocity has a minimum at the orifice centreline, rises sharply near the orifice edges and then experiences a boundary-layer-like correction over a thickness of O(B’) to satisfy no-slip conditions. For B’/d’ of O(1) the profiles are similar to those in a rectangular duct with a maximum on the centreline, whereas for B’/d’ [Gt ] 1, which describes widely separated channel walls, the solution approaches Hasimoto's solution for the periodic infinite-slit array. In the limit (D’ –- d’)/B’ [Lt ] 1, where the width of the intervening barriers is small compared with the channel height, the solutions exhibit the same behaviour as Lee & Fung's (1969) solution for the flow past a single cylinder. The drag on the zero-thickness barriers in this case is nearly the same as for the cylinder for all aspect ratios.

This work is part of an investigation dealing with the effect of external flow in a melt on dendritic growth. In this paper, we will consider steady growth with zero surface tension. Assuming that the Prandtl number Pr is large, we are able to obtain a uniformly valid asymptotic solution for the steady state in the whole growth region.

The flow field induced by a sphere oscillating horizontally in a linearly stratified fluid is studied using a series of laboratory experiments. The resulting flows are shown to depend on the Stokes number β, the Keulegan–Carpenter number KC and the internal Froude number Fr. For Fr [clubs ] 0.2, it is shown that the nature of the resulting flow field is approximately independent of Fr and, based on this observation, a flow regime diagram is developed in the (β, KC)-plane. The flow regimes include: (i) fully-attached flow; (ii) attached vortices; (iii) local vortex shedding; and (iv) standing eddy pair. An internal-wave flow regime is also identified but, for such flows, the motion field is a function of Fr as well. Some quantitative measures are given to allow for future comparisons of the present results with analytical and/or numerical models. Wherever possible, the results are compared with the experiments of Tatsuno & Bearman (1990) on right circular cylinders oscillating in homogeneous fluids.

We report on experiments to measure the shock-induced growth of sinusoidal perturbations on thick interfaces separating two gases of different densities. The results show that the growth rates are reduced as the interface thickness is increased. A model that accounts for the growth rate reduction caused by the presence of a finite density gradient on the interface is proposed and good agreement is obtained with the experimental results.

A numerical study of the three-dimensional internal waves excited by topography in the flow of a stratified fluid is described. In the resonant flow of a nearly two-layer fluid, it is found that the time-development of the nonlinearly excited waves agrees qualitatively with the solution of the forced KP equation or the forced extended KP equation. In this case, the upstream-advancing solitary waves become asymptotically straight crested because of abnormal reflection at the sidewall similar to Mach reflection. The same phenomenon also occurs in the subcritical flow of a nearly two-layer fluid. However, in the subcritical flow of a linearly stratified Boussinesq fluid, the two-dimensionalization of the upstream waves can be interpreted as the separation of the lateral modes due to the differences in the group velocity of the linear wave, although this does not mean in general that the generation of upstream waves is describable by the linearized equation.

The present paper deals with the plane unsteady problem of the penetration of a blunt solid contour into an ideal compressible liquid. At the initial instant of time, the solid body touches the liquid free boundary at a single point. At the initial stage, the duration of which depends on the body geometry, the displacements of liquid particles are small and the disturbed fluid motion is described within the framework of the acoustic approximation. The main feature of the problem is the existence of a contact line between the free surface of the liquid and the solid-body surface. The position of this line is not known in advance and is to be determined together with the solution of the problem. A brief description of the method that provides a solution of complicated nonlinear problems such as this is given. The pressure distribution and the velocity field in the liquid are shown to be given in quadratures and in the case of a parabolic entering contour in an explicit form. For a parabolic entering contour the pressure at the top of the contour calculated using the model of an incompressible liquid is observed to deviate from a precise value by not more than 10% of the latter after the first expansion waves have passed the contact point. The solution analysis enabled us to distinguish the regions in which the acoustic approximation fails and the liquid flow becomes essentially nonlinear.

This paper analyses instabilities on the cryolite/aluminium interface in an aluminium reduction cell. The simplified cell model is a finite rectangular tank containing the two fluid layers, and carrying a uniform normal current. The magnetic field is assumed to be a linear function of position. Several previous studies have considered waves consisting of a single Fourier component but here we consider perturbations which are a general combination of the normal gravity-wave modes. We derive a system of coupled ordinary differential equations for the time-development of the mode amplitudes, and show that instability can occur via mode interactions, the electromagnetic perturbation force due to one mode feeding energy into the other. Growth rates are determined by computing the eigenvalues of an interaction matrix, and an approximate method using only the three leading diagonals is developed. If two modes have similar frequencies they may resonate and become unstable at a very low threshold current. We consider the influence of various cell parameters and draw some general conclusions about cell design.

Numerical simulations of geostrophic turbulence above topography are used to compare (a) nonlinear generation of system entropy, S, (b) selective damping of enstrophy and (c) development of vorticity–topography correlation. In the damped cases, S initially increases, approaching a quasi-equilibrium (maximum S subject to the instantaneous, though decaying, energy and enstrophy). When strongly scale-selective damping is applied, onset of the vorticity–topography correlation follows the timescales for enstrophy decay. During the period of decay, it is shown that nonlinear interaction continues to generate S, offsetting in part the loss of S to explicit damping.