A finite-difference procedure for three-dimensional parabolic flows is used to predict the development of Taylor vortices in the flow between concentric rotating cylinders, resulting from the growth of small disturbances of a Couette flow. Predictions of such flows are presented in the developing and fully developed region. A precise calculation of the wavelength of the vortices has been possible by employing a periodic boundary condition on the pressure field. The predicted torque coefficient compares well with experimental data. The critical Taylor number has also been predicted with good accuracy.

We describe observations of slowly decelerating stratified flow over undular bottom topography in an estuary. The flow, which initially was supercritical with respect to the first internal wave mode, approached a resonance after it had become subcritical. A series of acoustic images showed large-amplitude first-mode trapped waves during this phase of the tide. We derive a criterion for quasi-steady response, and present an extension of Yih's class II linear finite-amplitude solutions that accounts for the waves observed.

This paper is concerned with oscillations of a body of water in basins of variable depth, employing a system of linearized equations which can be obtained from the theory of a directed fluid sheet for an incompressible, homogeneous, inviscid fluid (Green & Naghdi 1976a, 1977). For free oscillations over a level bottom, an assessment of the range of validity of the linearized theory is made by an appropriate comparison with a corresponding well-known exact solution (Lamb 1932). This assessment indicates an ‘intermediate’ range of validity for the linearized theory not covered by usual classical approximations for long waves. Encouraged by this assessment, we apply the linear theory of a directed fluid sheet to basins of variable depth; and, in particular, consider a class of basin profiles whose equilibrium depth (along its width) varies in one direction only. By a method of asymptotic integration, a general solution is obtained which is relatively simple and accounts for the effect of vertical inertia. The solution is sinusoidal in time, periodic along the breadth direction and involves Bessel functions of the first order in the width direction. For two special basin profiles, detailed comparisons are made between the predictions of the asymptotic solution (i.e. the frequencies in the lowest modes of oscillations) with corresponding results obtained by other procedures.

Consider the directional solidification of a binary alloy rejecting a heavy solute as it solidifies upward. If the solidification front is planar, the fluid melt ahead of the front is stably stratified and convection is not expected. In this paper we analyse the linear stability of planar solidification asymptotically in the limit of large solutal Rayleigh number, R. Three distinct linear modes are found which correspond to internal waves, buoyancy edge waves, or morphological modes. Of these three modes, only the morphological modes are subject to an instability. We find that for large Rayleigh number this instability first occurs at long wavelengths with wavenumbers that scale on R−1/14. The scalings derived from the linear analysis are used to construct a nonlinear theory for the morphological instability in the large Rayleigh number limit. Similarity solutions are found which describe steadily convecting, non-planar growth reminiscent of an observed phenomenon known as steepling.

The structure of the steady magnetohydrodynamic switch-on shock wave is investigated for several orderings of the four diffusivities involved in the problem. The various orderings are approximated to by allowing one or more of the appropriate diffusivities to approach zero, and approximate solutions that are uniformly valid to order unity are sought. In general, singular perturbation problems are encountered, the number occurring (from zero to a maximum of three) depending upon the ordering of the diffusivities and the magnitude of the downstream velocity normal to the shock relative to certain critical velocities downstream of the shock. Where necessary, the approximate solutions are rendered uniformly valid to first order by the insertion of boundary layers, for which the approximate equations are determined to first order. For most of the cases considered, the limiting forms of the integral curves are determined and they are sketched in appropriate three-dimensional phase spaces.

The crest of a steep, symmetric gravity wave is shown to be closely approximated by the expression \[ x+iy = \frac{\alpha +\gamma i\chi}{(\beta + i\chi)^{\frac{1}{3}}}, \] where x, y are co-ordinates in the vertical plane, χ is the complex velocity potential and α, β, γ are certain constants. This expression is asymptotically correct both for small and for large values of |χ|; and the free surface agrees with the exact profile calculated by Longuet-Higgins & Fox (1977) everywhere to within 1·5 per cent. The pressure at the surface is constant to within 5 per cent.

Aviscoelastic model (a mixing-length model with relaxation) is developed to investigate the effect of turbulent advection on the mean flow perturbation and the drag force induced by turbulent shear flow over an undulating surface. The relaxation term is proportional to the ratio of eddy turnover time to travelling time; accordingly, near the surface, the relaxation model reduces to an eddy-viscosity or mixing-length model, while far from the surface it reduces to a rapid-distortion model.

The linear governing equations are transformed into streamline coordinates and solved through matched asymptotic expansions. According to order-of-magnitude estimates in Belcher, Newley & Hunt (1993), the drag force contributed by nonlinear shear stress is of the same order as that contributed by asymmetric pressure arising from the leeward thickening of the perturbed boundary layer. The nonlinear analysis in the present model confirms this estimate in most cases. Our analytical results show a dip in shear stress at the interface between the inner and outer layers and provide evidence that this dynamical feature is related to eddy advection. Numerical calculation using a shooting method gives results that compare well with the analysis.

Here we explore the effect of a series of low-permeability tilted baffles on the ascent of a buoyant fluid injected into a porous rock from a linear well, motivated by several industrial processes, particularly CO2 sequestration. We first consider, both theoretically and experimentally, the dynamics associated with flow past an individual inclined baffle; if the incident flux is sufficiently large then a pool of injectate grows beneath the baffle and spills over both ends, partitioning the flux, but otherwise the injectate only flows over the updip end of the baffle, leaving a stagnant zone under the downdip part of the baffle. In a multi-layered system, the flow may then be described using nonlinear recurrence relations based on the flow past an individual baffle. Using this approach, in the particular case in which there is a regular distribution of baffles, we show that, on a scale greater than individual baffles, the flow adjusts to a plume of constant width rising at an angle to the vertical, which depends on the geometry of the baffles. This constant-width plume may be described by an effective directional permeability at angles to the vertical where we find , where is the effective permeability of the regions between baffles. Within the boundaries of this plume the majority of pore space is bypassed. Indeed, the plume-scale effective porosity is largely associated with the pools of injectate which collect beneath each baffle. We show that since the constant-width plume has such a large lateral extent, the total pore space occupied by the plume, per unit height, is larger than for a homogeneous formation. Furthermore, these pools lead to a large surface area between injectate and formation water, which enhances the reaction of the injectate with the formation water. However, we also show that, in steady state, it may be hard to determine this plume-scale effective porosity since the ratio of the effective Darcy speed and the effective interstitial speed of the plume only relates to the porosity of the updip part of these pools, in which the injectate is flowing.

A two-dimensional, incompressible, irrotational, linearized flow model is employed in this analysis of two supercavitating, flat-plate hydrofoils in the presence of a free surface. The cavities are taken to have finite lengths, and gravity is neglected. The ensuing boundary-value problem is converted, by means of conformal mapping, to a mixed-boundary-value problem for the complex velocity in the upper half-plane. This altered problem is solved by use of the methods of thin-aerofoil theory and the solution involves digital-computer evaluation of a large number of incomplete elliptic integrals of the first and third kinds. Typical results are presented in graphs, and the results of the present work are compared with Yim's (1964) theory for a single supercavitating body near a free surface.

Most treatments of magnetohydrodynamics waves have confined physical interpretation to cases when the Alfvén velocity a1 is small compared with the sound velocity a0. Here we consider the ‘low-beta situation’, in which a1, the only possible waves are longitudinal ones, propagated unidirectionally along lines of magnetic force with velocity a0. These can be interpreted as sound waves, confined to effectively rigid magnetic tubes of force. Hall-current effects do not alter these conclusions (in contrast to the high-beta situation), and finite conductivity introduces only small dissipation.

An application is made to the flow pattern around a body moving through the F2 layer of the ionosphere, where, although neutral particles have a very large mean free path, charged particles interact electrostatically and, it is argued, may be regarded as forming a continuous fluid whose movement is independent of that of the neutral particles. A body moving at satellite speed or below would then excite the above-mentioned unidirectional sound waves, but no waves at much faster Alfvén velocity. These considerations suggest that its movement would be accompanied by a [xvee ]-shaped pattern of electron density (figure 2), which might be in part responsible for some anomalous radar echoes that have been reported.

The response of an initially neutral rough-wall turbulent boundary layer to a change in wall temperature is investigated experimentally. The change causes a localized peak in stable stratification that diffuses and moves away from the wall with downstream distance. The streamwise and wall-normal components of turbulent velocity fluctuations are damped at similar rates, even though the stratification only directly impacts the wall-normal component. The Reynolds shear profiles reveal the growth of an internal layer that scales approximately with the bulk Brunt–Väisälä frequency.

A one-dimensional model with no magnetic field is considered. It is supposed that the plasma starts in thermal equilibrium and then a current is forced to grow. Instability leads to the growth of waves, which are shown to stir the distribution in phase space, but only over a limited range of velocity. It is concluded that in order to restore stability the energy in the wave must become comparable to the energy of drift.

The Hartmann oscillator problem is studied using the hydraulic analogy between compressible gasdynamics and incompressible flows with a free surface. Extensive photographs, taken of the oscillator in order to visualize the periodic flow, show that qualitative features of the flow agree well with the observations made with a cavity in an air jet.

The construction and operation of the water channel built and used at the University of California at Los Angeles for this study is also described. Continuous ‘shooting’ water flow was found to be analogous to supersonic isentropic gas flow; a static depth of approximately ¼ in. of water appeared to be satisfactory. It seems that the most valuable aspect of the hydraulic analogy is its ability to easily present excellent visualization of flow phenomena, both steady and unsteady.

We study the respective effects of shear rate and of external field intensity and direction on the contribution to the bulk stress of Brownian dipolar axisymmetric particles suspended in a steady macroscopically homogeneous shear flow of an incompressible Newtonian fluid. Towards this end we obtain the steady orientational distribution and make use of existing general dynamic theories of dilute suspensions. The calculation focuses on the limit of weak rotary diffusion. Thus, unlike previous analyses, the present contribution is not restricted to weak shear effects.

Explicit results are presented for the bulk stress when the external field acts in the plane of the simple shear flow. In cases when the deterministic rotary motion possesses a single sufficiently stable node a simple unified description of the respective effects of both the intensity and azimuthal direction of the external field is provided by the boundary-layer approximation. This approximation enables a qualitative explanation of existing numerical results as well as furnishing quantitatively accurate analytical results at relatively moderate values of the rotary Peclét number and the Langevin parameter (∼10). Furthermore, at still larger values of these parameters use of the present asymptotic approximation is clearly preferable since the numerical schemes rapidly deteriorate when steep orientational gradients appear.

Singularities of the bulk stress are rationalized in terms of the corresponding deterministic rotary motion. This is particularly interesting because some of these singular phenomena (e.g. those associated with an ‘intermediate regime’ of the field intensity and direction, for which more than one stable attractor exists in the deterministic problem) have no counterparts in suspensions of dipolar spheres or torque-free axisymmetric particles.

Finally, the present results obtained for the orientational distribution are also applicable to the study of other aspects of the macroscale description of suspensions of dipolar axisymmetric particles. In this context we mention the extension of continuum modelling of suspensions of swimming micro-organisms so as to enable the analysis of fully developed bioconvection.

We consider steady two-dimensional flow in a two-layer fluid under the effects of gravity and surface tension. The upper fluid is bounded above by a free surface and the lower fluid is bounded below by a rigid bottom. We assume the fluids to be inviscid and the flow to be irrotational in each layer. Solitary wave solutions are found to the fully nonlinear problem using a boundary integral method based on the Cauchy integral formula. The behaviour of the solitary waves on the interface and free surface is determined by the density ratio of the two fluids, the fluid depth ratio, the Froude number and the Bond numbers. The dispersion relation obtained for the linearized equations demonstrates the presence of two modes: a ‘slow’ mode and a ‘fast’ mode. When a sufficiently strong surface tension is present only on the free surface, there is a region, or ‘gap’, between the two modes where no linear periodic waves are found. In-phase and out-of-phase solitary waves are computed in this spectral gap. Damped oscillations appear in the tails of the solitary waves when the value of the free-surface Bond number is either sufficiently small or large. The out-of-phase waves broaden as the Froude number tends towards a critical value. When surface tension is present on both surfaces, out-of-phase solitary waves are computed. Damped oscillations occur in the tails of the waves when the interfacial Bond number is sufficiently small. Oppositely oriented solitary waves are shown to coexist for identical parameter values.

We consider the fluid motion induced when an elliptic cylinder performs small-amplitude torsional oscillations about an axis parallel to a generator which passes through either the centre or a point on the major or minor axis of the ellipse. In common with other fluid flows dominated by oscillatory motion, a time-independent, or steady streaming flow develops. This steady streaming exhibits several unusual and unexpected features, which are confirmed by experiment.

The impact of a deep-water plunging breaker on a finite height two-dimensional structure with a vertical front face is studied experimentally. The structure is located at a fixed horizontal position relative to a wave maker and the structure’s bottom surface is located at a range of vertical positions close to the undisturbed water surface. Measurements of the water surface profile history and the pressure distribution on the front surface of the structure are performed. As the vertical position, $z_{b}$ (the $z$ axis is positive up and $z=0$ is the mean water level), of the structure’s bottom surface is varied from one experimental run to another, the water surface evolution during impact can be categorized into three classes of behaviour. In class I, with $z_{b}$ in a range of values near $-0.1\unicode[STIX]{x1D706}_{0}$, where $\unicode[STIX]{x1D706}_{0}$ is the nominal wavelength of the breaker, the behaviour of the water surface is similar to the flip-through phenomena first described in studies with shallow water and a structure mounted on the sea bed. In the present work, it is found that the water surface between the front face of the structure and the wave crest is well fitted by arcs of circles with a decreasing radius and downward moving centre as the impact proceeds. A spatially and temporally localized high-pressure region was found on the impact surface of the structure and existing theory is used to explore the physics of this phenomenon. In class II, with $z_{b}$ in a range of values near the mean water level, the bottom of the structure exits and re-enters the water phase at least once during the impact process. These air–water transitions generate large-amplitude ripple packets that propagate to the wave crest and modify its behaviour significantly. At $z_{b}=0$, all sensors submerged during the impact record a nearly in-phase high-frequency pressure oscillation indicating possible air entrainment. In class III, with $z_{b}$ in a range of values near $0.03\unicode[STIX]{x1D706}_{0}$, the bottom of the structure remains in air before the main crest hits the bottom corner of the structure. The subsequent free surface behaviour is strongly influenced by the instantaneous momentum of the local flow just before impact and the highest wall pressures of all experimental conditions are found.