Nonlinear instability of a zonal flow of slightly viscous Boussinesq fluid in a rapidly rotating frame is studied mathematically by the method of normal mode cascade, the flow being along a rectangular channel with horizontal and vertical rigid walls. Viscosity is represented approximately by supposing that its only effects occur in Ekman layers near the top and bottom walls of the channel, after the linear model of Barcilon. Self-interaction of one slightly unstable mode is found to lead to equilibration with supercritical instability. Also, interactions of two slightly unstable modes plausibly lead to equilibration. These results are related to the literature of experiments on differentially heated, rotating annuli.

A theoretical and experimental study is made of the second-order resonant interaction between triads of linearly damped waves, one common member of which is continuously forced. In the case of a single triad, if the forced wave exceeds a critical amplitude defined by properties of the triad members, energy proceeds irreversibly to the other two waves. A stable limit state is reached where all power in excess of that required to sustain a critical amplitude in the forced wave is transferred to the other waves, which also reach steady terminal amplitudes.

It is shown that when two or more triads are simultaneously at resonance the only stable limit state is one wherein the forced wave has fallen to the lowest critical amplitude, and the only other two waves remaining are those of the triad possessing this critical amplitude. Regardless of their initial amplitudes, all other waves not externally forced ultimately disappear.

The theory is applied to the interaction of standing internal gravity waves in a linearly stratified liquid. The experiments described here quantitatively confirm the major predictions.

The solutions of the incompressible quasi-cylindrical momentum-integral equations describing the flow in the viscous core of a wing-tip vortex are obtained in a closed form and are shown to have two distinct branches. The discontinuities of these solutions have infinite axial gradients and therefore, following Hall, are assumed to signal the inception of the vortex breakdown. Benjamin's finite transition, with its excess flow force dissipated, is shown to give results equivalent to a sudden cross-over, upstream of the discontinuity, from one branch solution to another. The critical point of such a cross-over is downstream from the cross-over, at the discontinuity. Sarpkaya's experimental data, and the nature of the solutions ahead of the discontinuity, suggest that the physical manifestation of the discontinuity is the spiral breakdown, whereas the cross-over seems to be related to the rapidly expanding and subsequently contracting axisymmetric bubble. This therefore implies that the beginning of the spiral breakdown is the all important disturbance which triggers off not only the downstream asymmetric departure of the flow from its quasi-cylindrical form but also the formation of the upstream axisymmetric cross-over bubble. Solutions for the turbulent flow downstream from the spiral breakdown indicate that the wing-tip vortex breakdown can result in an appreciable reduction of the maximum circumferential velocity and should thus lessen the danger that trailing vortices present to following aircraft.

The deformation of initially spherical drops of radius r0 subjected to an external flow of velocity u is experimentally examined for a large range of Weber and Bond numbers. Observations of the changes in the response are compared with recent analytical predictions. The data show that beyond a critical Weber number the response ceases to be vibratory and becomes monotonic with time. Subsequently, it is found that, although the response is unstable, the deformation imposed by the external aerodynamic pressure distribution remains the dominant factor. Measurements of the drag coefficient yield a mean value ofCD = 2·5 over a large Reynolds-number range. The time at which Taylor instability occurs is shown to be inversely proportional to Bond number to the one-quarter power. There is little evidence of the instability occurring until a normalized time $t^{*} = (\epsilon^{\frac{1}{2}}\bar{t}u/r_0)$ is approximately unity; here ε is the gas/liquid density ratio (ρ/ρ′) and $\bar{t}$ is the real time.

An approximate solution is developed for the system of equations describing flow and ion transport in a diffuse electrical double layer slightly perturbed from equilibrium. The approximation is valid only when the potential difference across the diffuse layer is small, less than about 25 mV. When the approximate solution is used to study wave motion of low-tension interfaces, it is found chat ion transport in diffuse layers slows down interfacial motion in both stable and unstable situations. Although the slowing effect is relatively small (a few per cent or less) when the small potential approximation applies, the form of the solution suggests that the effect could be significant for potential differences in the 50–100 mV range, which exist in many systems of interest. There are also indications that the slowing effect can significantly influence wave motion of thin liquid films with diffuse layers, e.g. soap films, although a detailed analysis of the thin-film situation is not carried out.

The velocity field of the Burgers one-dimensional model of turbulence at extremely large Reynolds numbers is expressed as a train of random triangular shock waves. For describing this field statistically the distributions of the intensity and the interval of the shock fronts are defined. The equations governing the distributions are derived taking into account the laws of motion of the shock fronts, and the self-preserving solutions are obtained. The number of shock fronts is found to decrease with time t as t−α, where α (0 [les ] α < 1) is the rate of collision, and consequently the mean interval increases as tα. The distribution of the intensity is shown to be the exponential distribution. The distribution of the interval varies with α, but it is proved that the maximum entropy is attained by the exponential distribution which corresponds to α = ½. For α = ½, the turbulent energy is shown to decay with time as t−1, in good agreement with the numerical result of Crow & Canavan (1970).

Considerable advances have been made in the past few years in treating a variety of problems in slender-body Stokes flow (Taylor 1969; Batchelor 1970; Cox 1970, 1971; Tillett 1970). However, the problem of treating the creeping motion past bluff objects, whose boundaries do not conform to a constant co-ordinate surface of one of the special orthogonal co-ordinate systems for which the Stokes slow-flow equation is simply separable, is still largely unsolved. In the slender-body Stokes flow studies mentioned above, the viscous-flow boundary-value problem is formulated approximately as an integral equation for an unknown distribution of Stokeslets over a line enclosed by the body. The theory is valid for only very extended shapes, since the error in drag decays inversely as the logarithm of the aspect ratio of the object. By contrast, the present authors show that the boundary-value problem for the axisymmetric flow past an arbitrary convex body of revolution can be formulated exactly as an integral equation for an unknown distribution of ring-like singularities over the surface of the body. The kernel in this integral equation is closely related to the fundamental separable solutions of the Stokes slow-flow equation when written in an oblate spheroidal co-ordinate system of vanishing aspect ratio. The two lowest-order appropriate spheroidal singularities are found to provide a complete description for all surface elements, except those perpendicular to the axis. Higher-order singularities of all orders are required to describe axially perpendicular surfaces, such as the ends of a cylinder or the blunt base of an object. The newly derived integral equation is solved numerically to provide the first theoretical solutions for low aspect ratio cylinders and cones. The theoretically predicted drag results are in excellent agreement with experimentally measured values.

By regarding the amplitudes of a set of orthogonal modes as the co-ordinates in an infinite-dimensional phase space, the probability distribution for an ensemble of randomly forced two-dimensional viscous flows is determined as the solution of the continuity equation for the phase flow. For a special, but infinite, class of types of random forcing, the exact equilibrium probability distribution can be found analytically from the Navier-Stokes equations. In these cases, the probability distribution is the product of exponential functions of the integral invariants of unforced inviscid flow.

Linearized equations for the mean flow and for the turbulent stresses over sinusoidal, travelling surface waves are derived using assumptions similar to those used by Bradshaw, Ferriss & Atwell (1967) to compute boundary-layer development. With the assumptions, the effects on the local turbulent stresses of advectal, vertical transport, generation and dissipation of turbulent energy can be assessed, and solutions of the equations are expected to resemble closely real flows with the same conditions. The calculated distributions of surface pressure indicate rates of wave growth (expressed as fractional energy gain during a radian advance of phase) of about 15(ρa/ρw) (τo/c2), where τo is the surface stress, co the phase velocity and ρa and ρw the densities of air and water, unless the wind velocity at height λ/2π is less than the phase velocity. The rates are considerably less than those measured by Snyder & Cox (1966), by Barnett & Wilkerson (1967) and by Dobson (1971), and arguments are presented to show that the linear approximation fails for wave slopes of order 0.1.

It is shown that the method developed by Laumbach & Probstein (1969) for an exponential atmosphere can be extended to the case of an arbitrary atmospheric density distribution. Results are given for various burst-point altitudes in a model atmosphere and compared with those corresponding to an exponential atmosphere.

A set of constitutive equations, valid for arbitrary linear bulk flows, is derived for a dilute suspension of nearly spherical, rigid particles which are subject to rotary Brownian couples. These constitutive equations are subsequently applied to find the resulting stress patterns for a variety of time-dependent bulk flow fields. The rheological responses are found to exhibit many of the same qualitative features as have been observed in recent experimental investigations of polymeric solutions and other complex materials.