A generalized theoretical analysis and finite-difference solutions of the Navier-Stokes equations of the initial-value problem are applied to obtain the linear internal wave fields generated by a density perturbation and two rotational velocity perturbations in an inviscid linearly stratified fluid. The velocity perturbations are those due to an axisymmetric swirl and a vortex pair. Solutions obtained correspond to the strong stratification limit.

The theoretical results of the rotational perturbation cases show an oscillating non-propagating disturbance, which is absent in the density-perturbation case. The swirl-flow solution shows an oscillatory behaviour in both the angular momentum deposited in the fluid and in the torque exerted by the external gravitational force field. The vortex-flow solution shows a vertical ray pattern.

The equi-partitioning of energy is reached at about 0.4 of a Brunt-Väisälä (B.V.) period. The potential energy-kinetic energy conversion, or vice versa, takes place between 0.15 and 0.3 B.V. periods.

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Görtler vortices are possible causes of transition to turbulence. In this paper, the effect of fully nonlinear Taylor-Görtler vortices on the growth of small-amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external-boundary-layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien-Schlichting waves considered have the asymptotic structure of lower-branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the former modes are simpler to analyse and more relevant to the boundary-layer problem. The effect of fully nonlinear Taylor-Görtler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

In this paper we set out to calculate the self-diffusivity of a Brownian particle in a concentrated suspension. The problem is treated by regarding the neighbours of a test particle as forming a ‘cage’. For short time t < tc, say, the particle is partially constrained by the cage and an equation is proposed to describe the coupled dynamics of particle and cage. The equation is shown to be asymptotically exact in some cases and acceptably accurate for other simple systems by comparing with Monte Carlo simulations. For times t > tc, the particle diffuses sufficiently far to escape its original cage (and finds itself in a new one). A quantitative estimate for tc is proposed and verified for a system of rod-like particles by numerical simulation. By combining these two ingredients an estimate of the long-time (t [Gt ] tc) self-diffusivity of a particle is made. For rod-like particles tc is the reptation time, and the result here is compared with the theory of Doi & Edwards (1978a, b), and with experiment. For a system of spheres comparison is made with the tracer light-scattering experiments of Kops-Werkhoven & Fijnaut (1982). In both cases good agreement is found when the particle concentration is sufficiently high.

The effect of currents flowing across a bar field on resonant reflection of surface waves by the bars is investigated. Using a multiple-scale expansion, evolution equations for the amplitudes of linear waves are derived and used to investigate the reflection of periodic wave trains with steady amplitude for both normal and oblique incidence. The presence of a current is found to shift resonant frequencies by possibly significant amounts and is also found to enhance reflection of waves by bar fields due to the additional effect of the perturbed current field.

We present a theoretical study of the localization phenomenon of gravity waves by a rough bottom in a one-dimensional channel. After recalling localization theory and applying it to the shallow-water case, we give the first study of the localization problem in the framework of the full potential theory; in particular we develop a renormalized-transfer-matrix approach to this problem. Our results also yield numerical estimates of the localization length, which we compare with the viscous dissipation length. This allows the prediction of which cases localization should be observable in and in which cases it could be hidden by dissipative mechanisms.

We present experimental evidence of the localization of linear gravity waves on a rough (i.e. random) bottom in a one-dimensional channel. The localization phenomenon is observed through very precise measurements in a wave tank. Viscous dissipation and rough-bed finite-size effects are examined. The experimental estimation of the localization lengths are compared with the theoretical predictions of Devillard, Dunlop & Souillard (1988). Finally, the resonant modes due to the disorder are directly observed for the first time.

We examine various perturbations of Jeffery-Hamel flows, the exact solutions of the Navier-Stokes equations governing the steady two-dimensional motions of an incompressible viscous fluid from a line source at the intersection of two rigid plane walls. First a pitchfork bifurcation of the Jeffery-Hamel flows themselves is described by perturbation theory. This description is then used as a basis to investigate the spatial development of arbitrary small steady two-dimensional perturbations of a Jeffery-Hamel flow; both linear and weakly nonlinear perturbations are treated for plane and nearly plane walls. It is found that there is strong interaction of the disturbances up- and downstream if the angle between the planes exceeds a critical value 2α2, which depends on the value of the Reynolds number. Finally, the problem of linear temporal stability of Jeffery-Hamel flows is broached and again the importance of specifying conditions up- and downstream is revealed. All these results are used to interpret the development of flow along a channel with walls of small curvature. Fraenkel's (1962) approximation of channel flow locally by Jeffery-Hamel flows is supported with the added proviso that the angle between the two walls at each station is less than 2α2.

We find certain exact solutions of Jeffery-Hamel type for the boundary-layer equations for film flow over certain beds. If β is the angle of the bed with the horizontal and S is the arclength these beds have equation sin β = (const.)S−3, and allow a description of flows on concave and convex beds. The velocity profiles are markedly different from the semi-Poiseuille flow on a plane bed.

We also find a class of beds in which the Jeffery-Hamel flows appear as a first approximation throughout the flow field, which is infinite in streamwise extent. Since the parameter γ specifying the Jeffery-Hamel flow varies in the streamwise direction this allows a description of flows over curved beds which are slowly varying, as described in the theory, in such a way that the local approximation is that Jeffery-Hamel flow with the local value of γ. This allows the description of flows with separation and reattachment of the main stream in some cases.